Monash University short course, part I
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Transcript of Monash University short course, part I
![Page 1: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/1.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
MCMC and likelihood-free methodsPart/day I: Markov chain methods
Christian P. Robert
Universite Paris-Dauphine, IUF, & CREST
Monash University, EBS, July 18, 2012
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Motivations and leading example
Computational issues in Bayesianstatistics
The Metropolis-Hastings Algorithm
The Gibbs Sampler
Population Monte Carlo
![Page 3: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/3.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
What is Bayesian statistics?
Statistical model defined by a likelihood function
f(x1, . . . , xn|θ) = L(θ|x1, . . . , xn)
[inversion of what varies]
Bayesian approach turns the likelihoodinto a conditional density:
π(θ|x1, . . . , xn) ∝ π(θ)L(θ|x1, . . . , xn)
using a reference measure (or a prior)π(θ)
[Thomas Bayes, 1701–1761]
![Page 4: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/4.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
What is Bayesian statistics?
Statistical model defined by a likelihood function
f(x1, . . . , xn|θ) = L(θ|x1, . . . , xn)
[inversion of what varies]Bayesian approach turns the likelihoodinto a conditional density:
π(θ|x1, . . . , xn) ∝ π(θ)L(θ|x1, . . . , xn)
using a reference measure (or a prior)π(θ)
[Thomas Bayes, 1701–1761]
![Page 5: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/5.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
What is Bayesian statistics?
Statistical model defined by a likelihood function
f(x1, . . . , xn|θ) = L(θ|x1, . . . , xn)
[inversion of what varies]Bayesian approach turns the likelihoodinto a conditional density:
π(θ|x1, . . . , xn) ∝ π(θ)L(θ|x1, . . . , xn)
using a reference measure (or a prior)π(θ)
[Thomas Bayes, 1701–1761]
![Page 6: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/6.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
New perspective
I Uncertainty on the parameters θ of a model modeled througha probability distribution π on Θ, called prior distribution
I Inference processed through distribution of θ conditional on x,π(θ|x), called posterior distribution
π(θ|x) =f(x|θ)π(θ)∫f(x|θ)π(θ) dθ
.
![Page 7: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/7.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
New perspective
I Uncertainty on the parameters θ of a model modeled througha probability distribution π on Θ, called prior distribution
I Inference processed through distribution of θ conditional on x,π(θ|x), called posterior distribution
π(θ|x) =f(x|θ)π(θ)∫f(x|θ)π(θ) dθ
.
![Page 8: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/8.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
Justifications
I Semantic drift from unknown to random
I Actualization of the information on θ by extracting theinformation on θ contained in the observation x
I Allows incorporation of imperfect information in the decisionprocess
I Unique mathematical way to condition upon the observations(conditional perspective)
I Penalization factor
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
Posterior distribution
π(θ|x) central to Bayesian inference
I Operates conditional upon the observation s
I Incorporates the requirement of the Likelihood Principle
I Avoids averaging over the unobserved values of x
I Coherent updating of the information available on θ,independent of the order in which i.i.d. observations arecollected
I Provides a complete inferential scope
![Page 10: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/10.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
Posterior distribution
π(θ|x) central to Bayesian inference
I Operates conditional upon the observation s
I Incorporates the requirement of the Likelihood Principle
I Avoids averaging over the unobserved values of x
I Coherent updating of the information available on θ,independent of the order in which i.i.d. observations arecollected
I Provides a complete inferential scope
![Page 11: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/11.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
Posterior distribution
π(θ|x) central to Bayesian inference
I Operates conditional upon the observation s
I Incorporates the requirement of the Likelihood Principle
I Avoids averaging over the unobserved values of x
I Coherent updating of the information available on θ,independent of the order in which i.i.d. observations arecollected
I Provides a complete inferential scope
![Page 12: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/12.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
Posterior distribution
π(θ|x) central to Bayesian inference
I Operates conditional upon the observation s
I Incorporates the requirement of the Likelihood Principle
I Avoids averaging over the unobserved values of x
I Coherent updating of the information available on θ,independent of the order in which i.i.d. observations arecollected
I Provides a complete inferential scope
![Page 13: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/13.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
abc of Bayesian perspective
Posterior distribution
π(θ|x) central to Bayesian inference
I Operates conditional upon the observation s
I Incorporates the requirement of the Likelihood Principle
I Avoids averaging over the unobserved values of x
I Coherent updating of the information available on θ,independent of the order in which i.i.d. observations arecollected
I Provides a complete inferential scope
![Page 14: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/14.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Latent structures make life harder!
Even simple models may lead to computational complications, asin latent variable models
f(x|θ) =
∫f?(x, x?|θ) dx?
I If (x, x?) observed, fine!
I If only x observed, trouble!
![Page 15: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/15.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Latent structures make life harder!
Even simple models may lead to computational complications, asin latent variable models
f(x|θ) =
∫f?(x, x?|θ) dx?
I If (x, x?) observed, fine!
I If only x observed, trouble!
![Page 16: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/16.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Latent structures make life harder!
Even simple models may lead to computational complications, asin latent variable models
f(x|θ) =
∫f?(x, x?|θ) dx?
I If (x, x?) observed, fine!
I If only x observed, trouble!
![Page 17: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/17.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
example: mixture models
Models of mixtures of distributions:
X ∼ fj with probability pj ,
for j = 1, 2, . . . , k, with overall density
X ∼ p1f1(x) + · · ·+ pkfk(x) .
n∏i=1
{p1f1(xi) + · · ·+ pkfk(xi)} .
![Page 18: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/18.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
example: mixture models
Models of mixtures of distributions:
X ∼ fj with probability pj ,
for j = 1, 2, . . . , k, with overall density
X ∼ p1f1(x) + · · ·+ pkfk(x) .
For a sample of independent random variables (X1, · · · , Xn),sample density
n∏i=1
{p1f1(xi) + · · ·+ pkfk(xi)} .
![Page 19: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/19.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
example: mixture models
Models of mixtures of distributions:
X ∼ fj with probability pj ,
for j = 1, 2, . . . , k, with overall density
X ∼ p1f1(x) + · · ·+ pkfk(x) .
n∏i=1
{p1f1(xi) + · · ·+ pkfk(xi)} .
Expanding this product of sums into a sum of products involves kn
elementary terms: too prohibitive to compute in large samples.
![Page 20: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/20.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Simple mixture (1)
−1 0 1 2 3
−1
01
23
µ1
µ 2
Case of the 0.3N (µ1, 1) + 0.7N (µ2, 1) likelihood
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Simple mixture (2)
For mixture of two normal distributions,
0.3N (µ1, 1) + 0.7N (µ2, 1) ,
likelihood proportional to
n∏i=1
[0.3ϕ (xi − µ1) + 0.7 ϕ (xi − µ2)]
containing 2n terms.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Complex maximisation
Standard maximization techniques often fail to find the globalmaximum because of multimodality or undesirable behavior(usually at the frontier of the domain) of the likelihood function.
Example
In the special case
f(x|µ, σ) = (1− ε) exp{(−1/2)x2}+ε
σexp{(−1/2σ2)(x− µ)2}
with ε > 0 known,
whatever n, the likelihood is unbounded:
limσ→0
L(x1, . . . , xn|µ = x1, σ) =∞
![Page 23: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/23.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Complex maximisation
Standard maximization techniques often fail to find the globalmaximum because of multimodality or undesirable behavior(usually at the frontier of the domain) of the likelihood function.
Example
In the special case
f(x|µ, σ) = (1− ε) exp{(−1/2)x2}+ε
σexp{(−1/2σ2)(x− µ)2}
with ε > 0 known, whatever n, the likelihood is unbounded:
limσ→0
L(x1, . . . , xn|µ = x1, σ) =∞
![Page 24: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/24.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Unbounded likelihood
−2 0 2 4 6
12
34
µ
n= 3
−2 0 2 4 6
12
34
µ
σ
n= 6
−2 0 2 4 6
12
34
µ
n= 12
−2 0 2 4 6
12
34
µ
σ
n= 24
−2 0 2 4 6
12
34
µ
n= 48
−2 0 2 4 6
12
34
µ
σ
n= 96
Case of the 0.3N (0, 1) + 0.7N (µ, σ) likelihood
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Mixture once again
press for MA Observations from
x1, . . . , xn ∼ f(x|θ) = pϕ(x;µ1, σ1) + (1− p)ϕ(x;µ2, σ2)
Prior
µi|σi ∼ N (ξi, σ2i /ni), σ2
i ∼ I G (νi/2, s2i /2), p ∼ Be(α, β)
Posterior
π(θ|x1, . . . , xn) ∝n∏
j=1
{pϕ(xj ;µ1, σ1) + (1− p)ϕ(xj ;µ2, σ2)}π(θ)
=
n∑`=0
∑(kt)
ω(kt)π(θ|(kt))
[O(2n)]
![Page 26: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/26.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Mixture once again
press for MA Observations from
x1, . . . , xn ∼ f(x|θ) = pϕ(x;µ1, σ1) + (1− p)ϕ(x;µ2, σ2)
Prior
µi|σi ∼ N (ξi, σ2i /ni), σ2
i ∼ I G (νi/2, s2i /2), p ∼ Be(α, β)
Posterior
π(θ|x1, . . . , xn) ∝n∏
j=1
{pϕ(xj ;µ1, σ1) + (1− p)ϕ(xj ;µ2, σ2)}π(θ)
=
n∑`=0
∑(kt)
ω(kt)π(θ|(kt))
[O(2n)]
![Page 27: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/27.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Mixture once again
press for MA Observations from
x1, . . . , xn ∼ f(x|θ) = pϕ(x;µ1, σ1) + (1− p)ϕ(x;µ2, σ2)
Prior
µi|σi ∼ N (ξi, σ2i /ni), σ2
i ∼ I G (νi/2, s2i /2), p ∼ Be(α, β)
Posterior
π(θ|x1, . . . , xn) ∝n∏
j=1
{pϕ(xj ;µ1, σ1) + (1− p)ϕ(xj ;µ2, σ2)}π(θ)
=n∑
`=0
∑(kt)
ω(kt)π(θ|(kt))
[O(2n)]
![Page 28: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/28.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Mixture once again (cont’d)
For a given permutation (kt), conditional posterior distribution
π(θ|(kt)) = N
(ξ1(kt),
σ21
n1 + `
)×I G ((ν1 + `)/2, s1(kt)/2)
×N
(ξ2(kt),
σ22
n2 + n− `
)×I G ((ν2 + n− `)/2, s2(kt)/2)
×Be(α+ `, β + n− `)
![Page 29: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/29.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Mixture once again (cont’d)
where
x1(kt) = 1`
∑`t=1 xkt
, s1(kt) =∑`
t=1(xkt− x1(kt))
2,x2(kt) = 1
n−`∑n
t=`+1 xkt, s2(kt) =
∑nt=`+1(xkt
− x2(kt))2
and
ξ1(kt) =n1ξ1 + `x1(kt)
n1 + `, ξ2(kt) =
n2ξ2 + (n− `)x2(kt)
n2 + n− `,
s1(kt) = s21 + s21(kt) +n1`
n1 + `(ξ1 − x1(kt))
2,
s2(kt) = s22 + s22(kt) +n2(n− `)n2 + n− `
(ξ2 − x2(kt))2,
posterior updates of the hyperparameters
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Latent variables
Mixture once again
Bayes estimator of θ:
δπ(x1, . . . , xn) =
n∑`=0
∑(kt)
ω(kt)Eπ[θ|x, (kt)]
Too costly: 2n terms
![Page 31: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/31.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The AR(p) model
AR(p) model
Auto-regressive representation of a time series,
xt|xt−1, . . . ∼ N
(µ+
p∑i=1
%i(xt−i − µ), σ2
)
I Generalisation of AR(1)
I Among the most commonly used models in dynamic settings
I More challenging than the static models (stationarityconstraints)
I Different models depending on the processing of the startingvalue x0
![Page 32: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/32.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The AR(p) model
AR(p) model
Auto-regressive representation of a time series,
xt|xt−1, . . . ∼ N
(µ+
p∑i=1
%i(xt−i − µ), σ2
)
I Generalisation of AR(1)
I Among the most commonly used models in dynamic settings
I More challenging than the static models (stationarityconstraints)
I Different models depending on the processing of the startingvalue x0
![Page 33: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/33.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The AR(p) model
Unwieldy stationarity constraints
Practical difficulty: for complex models, stationarity constraints getquite involved to the point of being unknown in some cases
Example (AR(1))
Case of linear Markovian dependence on the last value
xt = µ+ %(xt−1 − µ) + εt , εti.i.d.∼ N (0, σ2)
If |%| < 1, (xt)t∈Z can be written as
xt = µ+
∞∑j=0
%jεt−j
and this is a stationary representation.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The AR(p) model
Stationary but...
If |%| > 1, alternative stationary representation
xt = µ−∞∑j=1
%−jεt+j .
This stationary solution is criticized as artificial because xt iscorrelated with future white noises (εt)s>t, unlike the case when|%| < 1.Non-causal representation...
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The AR(p) model
Stationary but...
If |%| > 1, alternative stationary representation
xt = µ−∞∑j=1
%−jεt+j .
This stationary solution is criticized as artificial because xt iscorrelated with future white noises (εt)s>t, unlike the case when|%| < 1.Non-causal representation...
![Page 36: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/36.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The AR(p) model
Stationarity+causality
Stationarity constraints in the prior as a restriction on the values ofθ.
Theorem
AR(p) model second-order stationary and causal iff the roots of thepolynomial
P(x) = 1−p∑i=1
%ixi
are all outside the unit circle
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The AR(p) model
Stationarity constraints
Under stationarity constraints, complex parameter space: eachvalue of % needs to be checked for roots of correspondingpolynomial with modulus less than 1
E.g., for an AR(2) process withautoregressive polynomialP(u) = 1− %1u− %2u
2, constraint is
%1 + %2 < 1, %1 − %2 < 1
and |%2| < 1
●
−2 −1 0 1 2
−1.
0−
0.5
0.0
0.5
1.0
θ1
θ 2
![Page 38: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/38.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The AR(p) model
Stationarity constraints
Under stationarity constraints, complex parameter space: eachvalue of % needs to be checked for roots of correspondingpolynomial with modulus less than 1
E.g., for an AR(2) process withautoregressive polynomialP(u) = 1− %1u− %2u
2, constraint is
%1 + %2 < 1, %1 − %2 < 1
and |%2| < 1
●
−2 −1 0 1 2
−1.
0−
0.5
0.0
0.5
1.0
θ1
θ 2
![Page 39: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/39.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The MA(q) model
The MA(q) model
Alternative type of time series
xt = µ+ εt −q∑j=1
ϑjεt−j , εt ∼ N (0, σ2)
Stationary but, for identifiability considerations, the polynomial
Q(x) = 1−q∑j=1
ϑjxj
must have all its roots outside the unit circle
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The MA(q) model
Identifiability
Example
For the MA(1) model, xt = µ+ εt − ϑ1εt−1,
var(xt) = (1 + ϑ21)σ2
can also be written
xt = µ+ εt−1 −1
ϑ1εt, ε ∼ N (0, ϑ2
1σ2) ,
Both pairs (ϑ1, σ) & (1/ϑ1, ϑ1σ) lead to alternativerepresentations of the same model.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The MA(q) model
Properties of MA models
I Non-Markovian model (but special case of hidden Markov)
I Autocovariance γx(s) is null for |s| > q
![Page 42: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/42.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The MA(q) model
Representations
x1:T is a normal random variable with constant mean µ andcovariance matrix
Σ =
σ2 γ1 γ2 . . . γq 0 . . . 0 0γ1 σ2 γ1 . . . γq−1 γq . . . 0 0
. . .
0 0 0 . . . 0 0 . . . γ1 σ2
,
with (|s| ≤ q)
γs = σ2
q−|s|∑i=0
ϑiϑi+|s|
Not manageable in practice [large T’s]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The MA(q) model
Representations (contd.)
Conditional on past (ε0, . . . , ε−q+1),
L(µ, ϑ1, . . . , ϑq, σ|x1:T , ε0, . . . , ε−q+1) ∝
σ−TT∏
t=1
exp
−xt − µ+
q∑j=1
ϑj εt−j
2 /2σ2
,
where (t > 0)
εt = xt − µ+
q∑j=1
ϑj εt−j , ε0 = ε0, . . . , ε1−q = ε1−q
Recursive definition of the likelihood, still costly O(T × q)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The MA(q) model
Representations (contd.)
Encompassing approach for general time series modelsState-space representation
xt = Gyt + εt , (1)
yt+1 = Fyt + ξt , (2)
(1) is the observation equation and (2) is the state equation
Note
This is a special case of hidden Markov model
![Page 45: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/45.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The MA(q) model
Representations (contd.)
Encompassing approach for general time series modelsState-space representation
xt = Gyt + εt , (1)
yt+1 = Fyt + ξt , (2)
(1) is the observation equation and (2) is the state equation
Note
This is a special case of hidden Markov model
![Page 46: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/46.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The MA(q) model
MA(q) state-space representation
For the MA(q) model, take
yt = (εt−q, . . . , εt−1, εt)′
and then
yt+1 =
0 1 0 . . . 00 0 1 . . . 0
. . .0 0 0 . . . 10 0 0 . . . 0
yt + εt+1
00...01
xt = µ−
(ϑq ϑq−1 . . . ϑ1 −1
)yt .
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
The MA(q) model
MA(q) state-space representation (cont’d)
Example
For the MA(1) model, observation equation
xt = (1 0)yt
withyt = (y1t y2t)
′
directed by the state equation
yt+1 =
(0 10 0
)yt + εt+1
(1ϑ1
).
![Page 48: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/48.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Typology of problems
c© A typology of Bayes computational problems
(i). latent variable models in general
(ii). use of a complex parameter space, as for instance inconstrained parameter sets like those resulting from imposingstationarity constraints in dynamic models;
(iii). use of a complex sampling model with an intractablelikelihood, as for instance in some graphical models;
(iv). use of a huge dataset;
(v). use of a complex prior distribution (which may be theposterior distribution associated with an earlier sample);
(vi). use of a particular inferential procedure as for instance, Bayesfactors
Bπ01(x) =
P (θ ∈ Θ0 | x)
P (θ ∈ Θ1 | x)
/π(θ ∈ Θ0)
π(θ ∈ Θ1).
![Page 49: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/49.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Typology of problems
c© A typology of Bayes computational problems
(i). latent variable models in general
(ii). use of a complex parameter space, as for instance inconstrained parameter sets like those resulting from imposingstationarity constraints in dynamic models;
(iii). use of a complex sampling model with an intractablelikelihood, as for instance in some graphical models;
(iv). use of a huge dataset;
(v). use of a complex prior distribution (which may be theposterior distribution associated with an earlier sample);
(vi). use of a particular inferential procedure as for instance, Bayesfactors
Bπ01(x) =
P (θ ∈ Θ0 | x)
P (θ ∈ Θ1 | x)
/π(θ ∈ Θ0)
π(θ ∈ Θ1).
![Page 50: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/50.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Typology of problems
c© A typology of Bayes computational problems
(i). latent variable models in general
(ii). use of a complex parameter space, as for instance inconstrained parameter sets like those resulting from imposingstationarity constraints in dynamic models;
(iii). use of a complex sampling model with an intractablelikelihood, as for instance in some graphical models;
(iv). use of a huge dataset;
(v). use of a complex prior distribution (which may be theposterior distribution associated with an earlier sample);
(vi). use of a particular inferential procedure as for instance, Bayesfactors
Bπ01(x) =
P (θ ∈ Θ0 | x)
P (θ ∈ Θ1 | x)
/π(θ ∈ Θ0)
π(θ ∈ Θ1).
![Page 51: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/51.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Typology of problems
c© A typology of Bayes computational problems
(i). latent variable models in general
(ii). use of a complex parameter space, as for instance inconstrained parameter sets like those resulting from imposingstationarity constraints in dynamic models;
(iii). use of a complex sampling model with an intractablelikelihood, as for instance in some graphical models;
(iv). use of a huge dataset;
(v). use of a complex prior distribution (which may be theposterior distribution associated with an earlier sample);
(vi). use of a particular inferential procedure as for instance, Bayesfactors
Bπ01(x) =
P (θ ∈ Θ0 | x)
P (θ ∈ Θ1 | x)
/π(θ ∈ Θ0)
π(θ ∈ Θ1).
![Page 52: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/52.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Typology of problems
c© A typology of Bayes computational problems
(i). latent variable models in general
(ii). use of a complex parameter space, as for instance inconstrained parameter sets like those resulting from imposingstationarity constraints in dynamic models;
(iii). use of a complex sampling model with an intractablelikelihood, as for instance in some graphical models;
(iv). use of a huge dataset;
(v). use of a complex prior distribution (which may be theposterior distribution associated with an earlier sample);
(vi). use of a particular inferential procedure as for instance, Bayesfactors
Bπ01(x) =
P (θ ∈ Θ0 | x)
P (θ ∈ Θ1 | x)
/π(θ ∈ Θ0)
π(θ ∈ Θ1).
![Page 53: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/53.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
Computational issues in Bayesian statistics
Typology of problems
c© A typology of Bayes computational problems
(i). latent variable models in general
(ii). use of a complex parameter space, as for instance inconstrained parameter sets like those resulting from imposingstationarity constraints in dynamic models;
(iii). use of a complex sampling model with an intractablelikelihood, as for instance in some graphical models;
(iv). use of a huge dataset;
(v). use of a complex prior distribution (which may be theposterior distribution associated with an earlier sample);
(vi). use of a particular inferential procedure as for instance, Bayesfactors
Bπ01(x) =
P (θ ∈ Θ0 | x)
P (θ ∈ Θ1 | x)
/π(θ ∈ Θ0)
π(θ ∈ Θ1).
![Page 54: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/54.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis-Hastings Algorithm
Computational issues in Bayesianstatistics
The Metropolis-Hastings Algorithm
The Gibbs Sampler
Population Monte Carlo
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
General purpose
A major computational issue in Bayesian statistics:
Given a density π known up to a normalizing constant, and anintegrable function h, compute
Π(h) =
∫h(x)π(x)µ(dx) =
∫h(x)π(x)µ(dx)∫π(x)µ(dx)
when∫h(x)π(x)µ(dx) is intractable.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
Monte Carlo 101
Generate an iid sample x1, . . . , xN from π and estimate Π(h) by
ΠMCN (h) = N−1
N∑i=1
h(xi).
LLN: ΠMCN (h)
as−→ Π(h)If Π(h2) =
∫h2(x)π(x)µ(dx) <∞,
CLT:√N(
ΠMCN (h)−Π(h)
)L N
(0,Π
{[h−Π(h)]2
}).
Caveat conducting to MCMC
Often impossible or inefficient to simulate directly from Π
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
Monte Carlo 101
Generate an iid sample x1, . . . , xN from π and estimate Π(h) by
ΠMCN (h) = N−1
N∑i=1
h(xi).
LLN: ΠMCN (h)
as−→ Π(h)If Π(h2) =
∫h2(x)π(x)µ(dx) <∞,
CLT:√N(
ΠMCN (h)−Π(h)
)L N
(0,Π
{[h−Π(h)]2
}).
Caveat conducting to MCMC
Often impossible or inefficient to simulate directly from Π
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Importance Sampling
Importance Sampling
For Q proposal distribution such that Q(dx) = q(x)µ(dx),alternative representation
Π(h) =
∫h(x){π/q}(x)q(x)µ(dx).
Principle of importance (!)
Generate an iid sample x1, . . . , xN ∼ Q and estimate Π(h) by
ΠISQ,N (h) = N−1
N∑i=1
h(xi){π/q}(xi).
return to pMC
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Importance Sampling
Importance Sampling
For Q proposal distribution such that Q(dx) = q(x)µ(dx),alternative representation
Π(h) =
∫h(x){π/q}(x)q(x)µ(dx).
Principle of importance (!)
Generate an iid sample x1, . . . , xN ∼ Q and estimate Π(h) by
ΠISQ,N (h) = N−1
N∑i=1
h(xi){π/q}(xi).
return to pMC
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Importance Sampling
Properties of importance
ThenLLN: ΠIS
Q,N (h)as−→ Π(h) and if Q((hπ/q)2) <∞,
CLT:√N(ΠIS
Q,N (h)−Π(h))L N
(0, Q{(hπ/q −Π(h))2}
).
Caveat
If normalizing constant of π unknown, impossible to use ΠISQ,N
Generic problem in Bayesian Statistics: π(θ|x) ∝ f(x|θ)π(θ).
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Importance Sampling
Properties of importance
ThenLLN: ΠIS
Q,N (h)as−→ Π(h) and if Q((hπ/q)2) <∞,
CLT:√N(ΠIS
Q,N (h)−Π(h))L N
(0, Q{(hπ/q −Π(h))2}
).
Caveat
If normalizing constant of π unknown, impossible to use ΠISQ,N
Generic problem in Bayesian Statistics: π(θ|x) ∝ f(x|θ)π(θ).
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Importance Sampling
Self-Normalised Importance Sampling
Self normalized version
ΠSNISQ,N (h) =
(N∑i=1
{π/q}(xi)
)−1 N∑i=1
h(xi){π/q}(xi).
LLN : ΠSNISQ,N (h)
as−→ Π(h)
and if Π((1 + h2)(π/q)) <∞,
CLT :√N(ΠSNIS
Q,N (h)−Π(h))L N
(0, π {(π/q)(h−Π(h)}2)
).
c© The quality of the SNIS approximation depends on thechoice of Q
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Importance Sampling
Self-Normalised Importance Sampling
Self normalized version
ΠSNISQ,N (h) =
(N∑i=1
{π/q}(xi)
)−1 N∑i=1
h(xi){π/q}(xi).
LLN : ΠSNISQ,N (h)
as−→ Π(h)
and if Π((1 + h2)(π/q)) <∞,
CLT :√N(ΠSNIS
Q,N (h)−Π(h))L N
(0, π {(π/q)(h−Π(h)}2)
).
c© The quality of the SNIS approximation depends on thechoice of Q
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Importance Sampling
Self-Normalised Importance Sampling
Self normalized version
ΠSNISQ,N (h) =
(N∑i=1
{π/q}(xi)
)−1 N∑i=1
h(xi){π/q}(xi).
LLN : ΠSNISQ,N (h)
as−→ Π(h)
and if Π((1 + h2)(π/q)) <∞,
CLT :√N(ΠSNIS
Q,N (h)−Π(h))L N
(0, π {(π/q)(h−Π(h)}2)
).
c© The quality of the SNIS approximation depends on thechoice of Q
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
Running Monte Carlo via Markov Chains (MCMC)
It is not necessary to use a sample from the distribution f toapproximate the integral
I =
∫h(x)f(x)dx ,
![Page 66: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/66.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
Running Monte Carlo via Markov Chains (MCMC)
It is not necessary to use a sample from the distribution f toapproximate the integral
I =
∫h(x)f(x)dx ,
[notation warnin: π turned to f !]
![Page 67: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/67.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
Running Monte Carlo via Markov Chains (MCMC)
It is not necessary to use a sample from the distribution f toapproximate the integral
I =
∫h(x)f(x)dx ,
We can obtain X1, . . . , Xn ∼ f(approx) without directly simulatingfrom f , using an ergodic Markovchain with stationary distribution f
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
Running Monte Carlo via Markov Chains (MCMC)
It is not necessary to use a sample from the distribution f toapproximate the integral
I =
∫h(x)f(x)dx ,
We can obtain X1, . . . , Xn ∼ f(approx) without directly simulatingfrom f , using an ergodic Markovchain with stationary distribution f
Andreı Markov
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
Running Monte Carlo via Markov Chains (2)
Idea
For an arbitrary starting value x(0), an ergodic chain (X(t)) isgenerated using a transition kernel with stationary distribution f
![Page 70: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/70.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
Running Monte Carlo via Markov Chains (2)
Idea
For an arbitrary starting value x(0), an ergodic chain (X(t)) isgenerated using a transition kernel with stationary distribution f
I irreducible Markov chain with stationary distribution f isergodic with limiting distribution f under weak conditions
I hence convergence in distribution of (X(t)) to a randomvariable from f .
I for T0 “large enough” T0, X(T0) distributed from f
I Markov sequence is dependent sample X(T0), X(T0+1), . . .generated from f
I Birkoff’s ergodic theorem extends LLN, sufficient for mostapproximation purposes
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
Running Monte Carlo via Markov Chains (2)
Idea
For an arbitrary starting value x(0), an ergodic chain (X(t)) isgenerated using a transition kernel with stationary distribution f
Problem: How can one build a Markov chain with a givenstationary distribution?
![Page 72: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/72.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
The Metropolis–Hastings algorithm
Arguments: The algorithm uses theobjective (target) density
f
and a conditional density
q(y|x)
called the instrumental (or proposal)distribution
Nicholas Metropolis
![Page 73: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/73.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
The MH algorithm
Algorithm (Metropolis–Hastings)
Given x(t),
1. Generate Yt ∼ q(y|x(t)).
2. Take
X(t+1) =
{Yt with prob. ρ(x(t), Yt),
x(t) with prob. 1− ρ(x(t), Yt),
where
ρ(x, y) = min
{f(y)
f(x)
q(x|y)
q(y|x), 1
}.
![Page 74: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/74.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Features
I Independent of normalizing constants for both f and q(·|x)(ie, those constants independent of x)
I Never move to values with f(y) = 0
I The chain (x(t))t may take the same value several times in arow, even though f is a density wrt Lebesgue measure
I The sequence (yt)t is usually not a Markov chain
![Page 75: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/75.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties
1. The M-H Markov chain is reversible, withinvariant/stationary density f since it satisfies the detailedbalance condition
f(y)K(y, x) = f(x)K(x, y)
2. As f is a probability measure, the chain is positive recurrent
3. If
Pr
[f(Yt) q(X
(t)|Yt)f(X(t)) q(Yt|X(t))
≥ 1
]< 1. (1)
that is, the event {X(t+1) = X(t)} is possible, then the chainis aperiodic
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties
1. The M-H Markov chain is reversible, withinvariant/stationary density f since it satisfies the detailedbalance condition
f(y)K(y, x) = f(x)K(x, y)
2. As f is a probability measure, the chain is positive recurrent
3. If
Pr
[f(Yt) q(X
(t)|Yt)f(X(t)) q(Yt|X(t))
≥ 1
]< 1. (1)
that is, the event {X(t+1) = X(t)} is possible, then the chainis aperiodic
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties
1. The M-H Markov chain is reversible, withinvariant/stationary density f since it satisfies the detailedbalance condition
f(y)K(y, x) = f(x)K(x, y)
2. As f is a probability measure, the chain is positive recurrent
3. If
Pr
[f(Yt) q(X
(t)|Yt)f(X(t)) q(Yt|X(t))
≥ 1
]< 1. (1)
that is, the event {X(t+1) = X(t)} is possible, then the chainis aperiodic
![Page 78: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/78.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties (2)
4. Ifq(y|x) > 0 for every (x, y), (2)
the chain is irreducible
5. For M-H, f -irreducibility implies Harris recurrence6. Thus, for M-H satisfying (1) and (2)
(i) For h, with Ef |h(X)| <∞,
limT→∞
1
T
T∑t=1
h(X(t)) =
∫h(x)df(x) a.e. f.
(ii) and
limn→∞
∥∥∥∥∫ Kn(x, ·)µ(dx)− f∥∥∥∥TV
= 0
for every initial distribution µ, where Kn(x, ·) denotes thekernel for n transitions.
![Page 79: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/79.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties (2)
4. Ifq(y|x) > 0 for every (x, y), (2)
the chain is irreducible
5. For M-H, f -irreducibility implies Harris recurrence
6. Thus, for M-H satisfying (1) and (2)(i) For h, with Ef |h(X)| <∞,
limT→∞
1
T
T∑t=1
h(X(t)) =
∫h(x)df(x) a.e. f.
(ii) and
limn→∞
∥∥∥∥∫ Kn(x, ·)µ(dx)− f∥∥∥∥TV
= 0
for every initial distribution µ, where Kn(x, ·) denotes thekernel for n transitions.
![Page 80: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/80.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties (2)
4. Ifq(y|x) > 0 for every (x, y), (2)
the chain is irreducible
5. For M-H, f -irreducibility implies Harris recurrence6. Thus, for M-H satisfying (1) and (2)
(i) For h, with Ef |h(X)| <∞,
limT→∞
1
T
T∑t=1
h(X(t)) =
∫h(x)df(x) a.e. f.
(ii) and
limn→∞
∥∥∥∥∫ Kn(x, ·)µ(dx)− f∥∥∥∥TV
= 0
for every initial distribution µ, where Kn(x, ·) denotes thekernel for n transitions.
![Page 81: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/81.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Random walk Metropolis–Hastings
Use of a local perturbation as proposal
Yt = X(t) + εt,
where εt ∼ g, independent of X(t).The instrumental density is of the form g(y − x) and the Markovchain is a random walk if we take g to be symmetric g(x) = g(−x)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Random walk Metropolis–Hastings [code]
Algorithm (Random walk Metropolis)
Given x(t)
1. Generate Yt ∼ g(y − x(t))
2. Take
X(t+1) =
Yt with prob. min
{1,
f(Yt)
f(x(t))
},
x(t) otherwise.
![Page 83: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/83.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
The original example
Example (Random walk and normal target)
forget History! Generate N (0, 1) based on the uniform proposal [−δ, δ]The probability of acceptance is then
ρ(x(t), yt) = exp{(x(t)2 − y2t )/2} ∧ 1.
[Hastings (1970)]
![Page 84: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/84.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
The original example
Example (Random walk & normal (2))
Sample statistics
δ 0.1 0.5 1.0
mean 0.399 -0.111 0.10variance 0.698 1.11 1.06
c© As δ ↑, we get better histograms and a faster exploration of thesupport of f .
![Page 85: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/85.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
The original example
-1 0 1 2
050
100
150
200
250
(a)
-1.5
-1.0
-0.5
0.0
0.5
-2 0 2
010
020
030
040
0
(b)
-1.5
-1.0
-0.5
0.0
0.5
-3 -2 -1 0 1 2 3
010
020
030
040
0
(c)
-1.5
-1.0
-0.5
0.0
0.5
Three samples based on U[−δ, δ] with (a) δ = 0.1, (b) δ = 0.5 and (c) δ = 1.0, superimposed with the convergence of the means (15, 000 simulations)
![Page 86: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/86.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Mixtures by random walk MH
Example (Mixture models)
π(θ|x) ∝n∏j=1
(k∑`=1
p`f(xj |µ`, σ`)
)π(θ)
Metropolis-Hastings proposal:
θ(t+1) =
{θ(t) + ωε(t) if u(t) < ρ(t)
θ(t) otherwise
where
ρ(t) =π(θ(t) + ωε(t)|x)
π(θ(t)|x)∧ 1
and ω scaled for good acceptance rate
![Page 87: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/87.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Mixtures by random walk MH
Example (Mixture models)
π(θ|x) ∝n∏j=1
(k∑`=1
p`f(xj |µ`, σ`)
)π(θ)
Metropolis-Hastings proposal:
θ(t+1) =
{θ(t) + ωε(t) if u(t) < ρ(t)
θ(t) otherwise
where
ρ(t) =π(θ(t) + ωε(t)|x)
π(θ(t)|x)∧ 1
and ω scaled for good acceptance rate
![Page 88: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/88.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Mixtures by random walk MH
p
thet
a
0.0 0.2 0.4 0.6 0.8 1.0
-10
12
tau
thet
a
0.2 0.4 0.6 0.8 1.0 1.2
-10
12
p
tau
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
1.2
-1 0 1 2
0.0
1.0
2.0
theta
0.2 0.4 0.6 0.8
02
4
tau
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
p
Random walk sampling (50000 iterations)
General case of a 3 component normal mixture[Celeux & al., 2000]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Mixtures by random walk MH
−1 0 1 2 3
−1
01
23
µ1
µ 2
X
Random walk MCMC output for .7N (µ1, 1) + .3N (µ2, 1)
![Page 90: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/90.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Convergence properties
Uniform ergodicity prohibited by random walk structure
At best, geometric ergodicity:
Theorem (Sufficient ergodicity)
For a symmetric density f , log-concave in the tails, and a positiveand symmetric density g, the chain (X(t)) is geometrically ergodic.
[Mengersen & Tweedie, 1996]
no tail effect
![Page 91: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/91.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Convergence properties
Uniform ergodicity prohibited by random walk structureAt best, geometric ergodicity:
Theorem (Sufficient ergodicity)
For a symmetric density f , log-concave in the tails, and a positiveand symmetric density g, the chain (X(t)) is geometrically ergodic.
[Mengersen & Tweedie, 1996]
no tail effect
![Page 92: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/92.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
illustration of the tail effect
Example (Comparison of tails)
Random walk Metropolis Hastings
algorithms based on a N (0, 1)
instrumental for the generation of
(left) a N (0, 1) distribution and
(right) a distribution with density
ψ(x) ∝ (1 + |x|)−3 (a)
0 50 100 150 200-1
.5-1
.0-0
.50.
00.
51.
01.
5
(a)
0 50 100 150 200-1
.5-1
.0-0
.50.
00.
51.
01.
5
0 50 100 150 200
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 50 100 150 200
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
(b)
0 50 100 150 200
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 50 100 150 200
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 50 100 150 200
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
90% confidence envelopes of the means, derived from500 parallel independent chains
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Further convergence properties
Under assumptions skip detailed convergence
I (A1) f is super-exponential, i.e. it is positive with positive
continuous first derivative such that
lim|x|→∞ n(x)′∇ log f(x) = −∞ where n(x) := x/|x|.In words : exponential decay of f in every direction with ratetending to ∞
I (A2) lim sup|x|→∞ n(x)′m(x) < 0, where m(x) = ∇f(x)/|∇f(x)|In words: non degeneracy of the countour manifoldCf(y) = {y : f(y) = f(x)}
Q is geometrically ergodic, andV (x) ∝ f(x)−1/2 verifies the drift condition
[Jarner & Hansen, 2000]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Further [further] convergence properties
skip hyperdetailed convergence
If P ψ-irreducible and aperiodic, for r = (r(n))n∈N real-valued nondecreasing sequence, such that, for all n,m ∈ N,
r(n+m) ≤ r(n)r(m),
and r(0) = 1, for C a small set, τC = inf{n ≥ 1, Xn ∈ C}, andh ≥ 1, assume
supx∈C
Ex
[τC−1∑k=0
r(k)h(Xk)
]<∞,
![Page 95: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/95.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Further [further] convergence properties
then,
S(f, C, r) :=
{x ∈ X,Ex
{τC−1∑k=0
r(k)h(Xk)
}<∞
}
is full and absorbing and for x ∈ S(f, C, r),
limn→∞
r(n)‖Pn(x, .)− f‖h = 0.
[Tuominen & Tweedie, 1994]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Comments
I [CLT, Rosenthal’s inequality...] h-ergodicity implies CLTfor additive (possibly unbounded) functionals of the chain,Rosenthal’s inequality and so on...
I [Control of the moments of the return-time] Thecondition implies (because h ≥ 1) that
supx∈C
Ex[r0(τC)] ≤ supx∈C
Ex
{τC−1∑k=0
r(k)h(Xk)
}<∞,
where r0(n) =∑n
l=0 r(l) Can be used to derive bounds forthe coupling time, an essential step to determine computablebounds, using coupling inequalities
[Roberts & Tweedie, 98; Fort & Moulines, 00; Jones et al., 02]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Alternative conditions
The condition is not really easy to work with...[Possible alternative conditions]
(a) [Tuominen, Tweedie, 1994] There exists a sequence(Vn)n∈N, Vn ≥ r(n)h, such that
(i) supC V0 <∞,(ii) {V0 =∞} ⊂ {V1 =∞} and(iii) PVn+1 ≤ Vn − r(n)h+ br(n)IC .
![Page 98: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/98.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Alternative conditions
(b) [Fort 2000] ∃V ≥ f ≥ 1 and b <∞, such that supC V <∞and
PV (x) + Ex
{σC∑k=0
∆r(k)f(Xk)
}≤ V (x) + bIC(x)
where σC is the hitting time on C and
∆r(k) = r(k)− r(k − 1), k ≥ 1 and ∆r(0) = r(0).
Result (a) ⇔ (b) ⇔ supx∈C Ex{∑τC−1
k=0 r(k)f(Xk)}<∞.
![Page 99: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/99.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Langevin Algorithms
Proposal based on the Langevin diffusion Lt is defined by thestochastic differential equation
dLt = dBt +1
2∇ log f(Lt)dt,
where Bt is the standard Brownian motion
Theorem
The Langevin diffusion is the only non-explosive diffusion which isreversible with respect to f .
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Discretization
Instead, consider the sequence
x(t+1) = x(t) +σ2
2∇ log f(x(t)) + σεt, εt ∼ Np(0, Ip)
where σ2 corresponds to the discretization step
Unfortunately, the discretized chain may be transient, for instancewhen
limx→±∞
∣∣σ2∇ log f(x)|x|−1∣∣ > 1
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Discretization
Instead, consider the sequence
x(t+1) = x(t) +σ2
2∇ log f(x(t)) + σεt, εt ∼ Np(0, Ip)
where σ2 corresponds to the discretization stepUnfortunately, the discretized chain may be transient, for instancewhen
limx→±∞
∣∣σ2∇ log f(x)|x|−1∣∣ > 1
![Page 102: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/102.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
MH correction
Accept the new value Yt with probability
f(Yt)
f(x(t))·
exp
{−∥∥∥Yt − x(t) − σ2
2 ∇ log f(x(t))∥∥∥2/
2σ2
}exp
{−∥∥∥x(t) − Yt − σ2
2 ∇ log f(Yt)∥∥∥2/
2σ2
} ∧ 1 .
Choice of the scaling factor σShould lead to an acceptance rate of 0.574 to achieve optimalconvergence rates (when the components of x are uncorrelated)
[Roberts & Rosenthal, 1998; Girolami & Calderhead, 2011]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Optimizing the Acceptance Rate
Problem of choice of the transition kernel from a practical point ofviewMost common alternatives:
(a) a fully automated algorithm like ARMS;[Gilks & Wild, 1992]
(b) an instrumental density g which approximates f , such thatf/g is bounded for uniform ergodicity to apply;
(c) a random walk
In both cases (b) and (c), the choice of g is critical,
![Page 104: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/104.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Case of the random walk
Different approach to acceptance ratesA high acceptance rate does not indicate that the algorithm ismoving correctly since it indicates that the random walk is movingtoo slowly on the surface of f .
If x(t) and yt are close, i.e. f(x(t)) ' f(yt) y is accepted withprobability
min
(f(yt)
f(x(t)), 1
)' 1 .
For multimodal densities with well separated modes, the negativeeffect of limited moves on the surface of f clearly shows.
![Page 105: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/105.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Case of the random walk
Different approach to acceptance ratesA high acceptance rate does not indicate that the algorithm ismoving correctly since it indicates that the random walk is movingtoo slowly on the surface of f .If x(t) and yt are close, i.e. f(x(t)) ' f(yt) y is accepted withprobability
min
(f(yt)
f(x(t)), 1
)' 1 .
For multimodal densities with well separated modes, the negativeeffect of limited moves on the surface of f clearly shows.
![Page 106: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/106.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Case of the random walk (2)
If the average acceptance rate is low, the successive values of f(yt)tend to be small compared with f(x(t)), which means that therandom walk moves quickly on the surface of f since it oftenreaches the “borders” of the support of f
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Rule of thumb
In small dimensions, aim at an average acceptance rate of50%. In large dimensions, at an average acceptance rate of25%.
[Gelman,Gilks and Roberts, 1995]
warnin: rule to be taken with a pinch of salt!
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Rule of thumb
In small dimensions, aim at an average acceptance rate of50%. In large dimensions, at an average acceptance rate of25%.
[Gelman,Gilks and Roberts, 1995]
warnin: rule to be taken with a pinch of salt!
![Page 109: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/109.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Role of scale
Example (Noisy AR(1))
Hidden Markov chain from a regular AR(1) model,
xt+1 = ϕxt + εt+1 εt ∼ N (0, τ2)
and observablesyt|xt ∼ N (x2
t , σ2)
The distribution of xt given xt−1, xt+1 and yt is
exp−1
2τ2
{(xt − ϕxt−1)2 + (xt+1 − ϕxt)2 +
τ2
σ2(yt − x2
t )2
}.
![Page 110: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/110.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Role of scale
Example (Noisy AR(1))
Hidden Markov chain from a regular AR(1) model,
xt+1 = ϕxt + εt+1 εt ∼ N (0, τ2)
and observablesyt|xt ∼ N (x2
t , σ2)
The distribution of xt given xt−1, xt+1 and yt is
exp−1
2τ2
{(xt − ϕxt−1)2 + (xt+1 − ϕxt)2 +
τ2
σ2(yt − x2
t )2
}.
![Page 111: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/111.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Role of scale
Example (Noisy AR(1) continued)
For a Gaussian random walk with scale ω small enough, therandom walk never jumps to the other mode. But if the scale ω issufficiently large, the Markov chain explores both modes and give asatisfactory approximation of the target distribution.
![Page 112: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/112.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Role of scale
Markov chain based on a random walk with scale ω = .1.
![Page 113: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/113.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Role of scale
Markov chain based on a random walk with scale ω = .5.
![Page 114: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/114.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
MA(2)
Since the constraints on (ϑ1, ϑ2) are well-defined, use of a flatprior over the triangle as prior.Simple representation of the likelihood
library(mnormt)
ma2like=function(theta){
n=length(y)
sigma = toeplitz(c(1 +theta[1]^2+theta[2]^2,
theta[1]+theta[1]*theta[2],theta[2],rep(0,n-3)))
dmnorm(y,rep(0,n),sigma,log=TRUE)
}
![Page 115: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/115.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Basic RWHM for MA(2)
Algorithm 1 RW-HM-MA(2) sampler
set ω and ϑ(1)
for i = 2 to T dogenerate ϑj ∼ U(ϑ
(i−1)j − ω, ϑ(i−1)
j + ω)
set p = 0 and ϑ(i) = ϑ(i−1)
if ϑ within the triangle thenp = exp(ma2like(ϑ)−ma2like(ϑ(i−1)))
end ifif U < p thenϑ(i) = ϑ
end ifend for
![Page 116: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/116.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Outcome
Result with a simulated sample of 100 points and ϑ1 = 0.6,ϑ2 = 0.2 and scale ω = 0.2
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![Page 117: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/117.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Outcome
Result with a simulated sample of 100 points and ϑ1 = 0.6,ϑ2 = 0.2 and scale ω = 0.5
● ●
−2 −1 0 1 2
−1.
0−
0.5
0.0
0.5
1.0
θ1
θ 2
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![Page 118: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/118.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Metropolis-Hastings Algorithm
Extensions
Outcome
Result with a simulated sample of 100 points and ϑ1 = 0.6,ϑ2 = 0.2 and scale ω = 2.0
● ●
−2 −1 0 1 2
−1.
0−
0.5
0.0
0.5
1.0
θ1
θ 2
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
The Gibbs Sampler
skip to population Monte Carlo
Computational issues in Bayesianstatistics
The Metropolis-Hastings Algorithm
The Gibbs Sampler
Population Monte Carlo
![Page 120: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/120.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
General Principles
A very specific simulation algorithm based on the targetdistribution f :
1. Uses the conditional densities f1, . . . , fp from f
2. Start with the random variable X = (X1, . . . , Xp)
3. Simulate from the conditional densities,
Xi|x1, x2, . . . , xi−1, xi+1, . . . , xp
∼ fi(xi|x1, x2, . . . , xi−1, xi+1, . . . , xp)
for i = 1, 2, . . . , p.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
General Principles
A very specific simulation algorithm based on the targetdistribution f :
1. Uses the conditional densities f1, . . . , fp from f
2. Start with the random variable X = (X1, . . . , Xp)
3. Simulate from the conditional densities,
Xi|x1, x2, . . . , xi−1, xi+1, . . . , xp
∼ fi(xi|x1, x2, . . . , xi−1, xi+1, . . . , xp)
for i = 1, 2, . . . , p.
![Page 122: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/122.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
General Principles
A very specific simulation algorithm based on the targetdistribution f :
1. Uses the conditional densities f1, . . . , fp from f
2. Start with the random variable X = (X1, . . . , Xp)
3. Simulate from the conditional densities,
Xi|x1, x2, . . . , xi−1, xi+1, . . . , xp
∼ fi(xi|x1, x2, . . . , xi−1, xi+1, . . . , xp)
for i = 1, 2, . . . , p.
![Page 123: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/123.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Gibbs code
Algorithm (Gibbs sampler)
Given x(t) = (x(t)1 , . . . , x
(t)p ), generate
1. X(t+1)1 ∼ f1(x1|x(t)
2 , . . . , x(t)p );
2. X(t+1)2 ∼ f2(x2|x(t+1)
1 , x(t)3 , . . . , x
(t)p ),
. . .
p. X(t+1)p ∼ fp(xp|x(t+1)
1 , . . . , x(t+1)p−1 )
X(t+1) → X ∼ f
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Properties
The full conditionals densities f1, . . . , fp are the only densities usedfor simulation. Thus, even in a high dimensional problem, all ofthe simulations may be univariate
The Gibbs sampler is not reversible with respect to f . However,each of its p components is. Besides, it can be turned into areversible sampler, either using the Random Scan Gibbs sampler
see section or running instead the (double) sequence
f1 · · · fp−1fpfp−1 · · · f1
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Properties
The full conditionals densities f1, . . . , fp are the only densities usedfor simulation. Thus, even in a high dimensional problem, all ofthe simulations may be univariateThe Gibbs sampler is not reversible with respect to f . However,each of its p components is. Besides, it can be turned into areversible sampler, either using the Random Scan Gibbs sampler
see section or running instead the (double) sequence
f1 · · · fp−1fpfp−1 · · · f1
![Page 126: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/126.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
2D Gibbs sampler
Example (Bivariate Gibbs sampler)
(X,Y ) ∼ f(x, y)
Generate a sequence of observations bySet X0 = x0
For t = 1, 2, . . . , generate
Yt ∼ fY |X(·|xt−1)
Xt ∼ fX|Y (·|yt)
where fY |X and fX|Y are the conditional distributions
![Page 127: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/127.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
toy example: iid N (µ, σ2) variates
When Y1, . . . , Yniid∼ N (y|µ, σ2) with both µ and σ unknown, the
posterior in (µ, σ2) is conjugate outside a standard familly
But...
µ|Y 0:n, σ2 ∼ N
(µ∣∣∣ 1n
∑ni=1 Yi,
σ2
n )
σ2|Y 1:n, µ ∼ IG(σ2∣∣n
2 − 1, 12
∑ni=1(Yi − µ)2
)assuming constant (improper) priors on both µ and σ2
I Hence we may use the Gibbs sampler for simulating from theposterior of (µ, σ2)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
toy example: iid N (µ, σ2) variates
When Y1, . . . , Yniid∼ N (y|µ, σ2) with both µ and σ unknown, the
posterior in (µ, σ2) is conjugate outside a standard familly
But...
µ|Y 0:n, σ2 ∼ N
(µ∣∣∣ 1n
∑ni=1 Yi,
σ2
n )
σ2|Y 1:n, µ ∼ IG(σ2∣∣n
2 − 1, 12
∑ni=1(Yi − µ)2
)assuming constant (improper) priors on both µ and σ2
I Hence we may use the Gibbs sampler for simulating from theposterior of (µ, σ2)
![Page 129: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/129.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
toy example: code
R Gibbs Sampler for Gaussian posterior
n = length(Y);
S = sum(Y);
mu = S/n;
for (i in 1:500)
S2 = sum((Y-mu)^2);
sigma2 = 1/rgamma(1,n/2-1,S2/2);
mu = S/n + sqrt(sigma2/n)*rnorm(1);
![Page 130: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/130.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1
, 2, 3, 4, 5, 10, 25, 50, 100, 500
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1, 2
, 3, 4, 5, 10, 25, 50, 100, 500
![Page 132: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/132.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1, 2, 3
, 4, 5, 10, 25, 50, 100, 500
![Page 133: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/133.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1, 2, 3, 4
, 5, 10, 25, 50, 100, 500
![Page 134: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/134.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1, 2, 3, 4, 5
, 10, 25, 50, 100, 500
![Page 135: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/135.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1, 2, 3, 4, 5, 10
, 25, 50, 100, 500
![Page 136: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/136.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1, 2, 3, 4, 5, 10, 25
, 50, 100, 500
![Page 137: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/137.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50
, 100, 500
![Page 138: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/138.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50, 100
, 500
![Page 139: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/139.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Example of results with n = 10 observations from theN (0, 1) distribution
Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50, 100, 500
![Page 140: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/140.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Limitations of the Gibbs sampler
Formally, a special case of a sequence of 1-D M-H kernels, all withacceptance rate uniformly equal to 1.The Gibbs sampler
1. limits the choice of instrumental distributions
2. requires some knowledge of f
3. is, by construction, multidimensional
4. does not apply to problems where the number of parametersvaries as the resulting chain is not irreducible.
![Page 141: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/141.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Limitations of the Gibbs sampler
Formally, a special case of a sequence of 1-D M-H kernels, all withacceptance rate uniformly equal to 1.The Gibbs sampler
1. limits the choice of instrumental distributions
2. requires some knowledge of f
3. is, by construction, multidimensional
4. does not apply to problems where the number of parametersvaries as the resulting chain is not irreducible.
![Page 142: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/142.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Limitations of the Gibbs sampler
Formally, a special case of a sequence of 1-D M-H kernels, all withacceptance rate uniformly equal to 1.The Gibbs sampler
1. limits the choice of instrumental distributions
2. requires some knowledge of f
3. is, by construction, multidimensional
4. does not apply to problems where the number of parametersvaries as the resulting chain is not irreducible.
![Page 143: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/143.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
General Principles
Limitations of the Gibbs sampler
Formally, a special case of a sequence of 1-D M-H kernels, all withacceptance rate uniformly equal to 1.The Gibbs sampler
1. limits the choice of instrumental distributions
2. requires some knowledge of f
3. is, by construction, multidimensional
4. does not apply to problems where the number of parametersvaries as the resulting chain is not irreducible.
![Page 144: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/144.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Latent variables are back
The Gibbs sampler can be generalized in much wider generalityA density g is a completion of f if∫
Zg(x, z) dz = f(x)
Note
The variable z may be meaningless for the problem
![Page 145: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/145.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Latent variables are back
The Gibbs sampler can be generalized in much wider generalityA density g is a completion of f if∫
Zg(x, z) dz = f(x)
Note
The variable z may be meaningless for the problem
![Page 146: Monash University short course, part I](https://reader034.fdocuments.net/reader034/viewer/2022042614/554e7ebcb4c905f66a8b536b/html5/thumbnails/146.jpg)
MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Purpose
g should have full conditionals that are easy to simulate for aGibbs sampler to be implemented with g rather than f
For p > 1, write y = (x, z) and denote the conditional densities ofg(y) = g(y1, . . . , yp) by
Y1|y2, . . . , yp ∼ g1(y1|y2, . . . , yp),
Y2|y1, y3, . . . , yp ∼ g2(y2|y1, y3, . . . , yp),
. . . ,
Yp|y1, . . . , yp−1 ∼ gp(yp|y1, . . . , yp−1).
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Generic Gibbs sampler
The move from Y (t) to Y (t+1) is defined as follows:
Algorithm (Completion Gibbs sampler)
Given (y(t)1 , . . . , y
(t)p ), simulate
1. Y(t+1)
1 ∼ g1(y1|y(t)2 , . . . , y
(t)p ),
2. Y(t+1)
2 ∼ g2(y2|y(t+1)1 , y
(t)3 , . . . , y
(t)p ),
. . .
p. Y(t+1)p ∼ gp(yp|y(t+1)
1 , . . . , y(t+1)p−1 ).
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Mixture illustration
Example (Mixtures all over again)
Hierarchical missing data structure:If
X1, . . . , Xn ∼k∑i=1
pif(x|θi),
then
X|Z ∼ f(x|θZ), Z ∼ p1I(z = 1) + . . .+ pkI(z = k),
Z is the component indicator associated with observation x
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Mixture illustration
Example (Mixtures (2))
Conditionally on (Z1, . . . , Zn) = (z1, . . . , zn) :
π(p1, . . . , pk, θ1, . . . , θk|x1, . . . , xn, z1, . . . , zn)
∝ pα1+n1−11 . . . pαk+nk−1
k
×π(θ1|y1 + n1x1, λ1 + n1) . . . π(θk|yk + nkxk, λk + nk),
withni =
∑j
I(zj = i) and xi =∑j; zj=i
xj/ni.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Mixture illustration
Algorithm (Mixture Gibbs sampler)
1. Simulate
θi ∼ π(θi|yi + nixi, λi + ni) (i = 1, . . . , k)
(p1, . . . , pk) ∼ D(α1 + n1, . . . , αk + nk)
2. Simulate (j = 1, . . . , n)
Zj |xj , p1, . . . , pk, θ1, . . . , θk ∼k∑
i=1
pijI(zj = i)
with (i = 1, . . . , k)pij ∝ pif(xj |θi)
and update ni and xi (i = 1, . . . , k).
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
A wee problem
−1 0 1 2 3 4
−1
01
23
4
µ1
µ2
Gibbs started at random
Gibbs stuck at the wrong mode
−1 0 1 2 3
−1
01
23
µ1
µ2
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
A wee problem
−1 0 1 2 3 4
−1
01
23
4
µ1
µ2
Gibbs started at random
Gibbs stuck at the wrong mode
−1 0 1 2 3
−1
01
23
µ1
µ2
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Slice sampler as generic Gibbs
If f(θ) can be written as a product
k∏i=1
fi(θ),
it can be completed as
k∏i=1
I0≤ωi≤fi(θ),
leading to the following Gibbs algorithm:
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Slice sampler as generic Gibbs
If f(θ) can be written as a product
k∏i=1
fi(θ),
it can be completed as
k∏i=1
I0≤ωi≤fi(θ),
leading to the following Gibbs algorithm:
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Slice sampler (code)
Algorithm (Slice sampler)
Simulate
1. ω(t+1)1 ∼ U[0,f1(θ(t))];
. . .
k. ω(t+1)k ∼ U[0,fk(θ(t))];
k+1. θ(t+1) ∼ UA(t+1) , with
A(t+1) = {y; fi(y) ≥ ω(t+1)i , i = 1, . . . , k}.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Example of results with a truncated N (−3, 1) distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.00
00.
002
0.00
40.
006
0.00
80.
010
x
y
Number of Iterations 2
, 3, 4, 5, 10, 50, 100
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Example of results with a truncated N (−3, 1) distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.00
00.
002
0.00
40.
006
0.00
80.
010
x
y
Number of Iterations 2, 3
, 4, 5, 10, 50, 100
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Example of results with a truncated N (−3, 1) distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.00
00.
002
0.00
40.
006
0.00
80.
010
x
y
Number of Iterations 2, 3, 4
, 5, 10, 50, 100
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Example of results with a truncated N (−3, 1) distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.00
00.
002
0.00
40.
006
0.00
80.
010
x
y
Number of Iterations 2, 3, 4, 5
, 10, 50, 100
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Example of results with a truncated N (−3, 1) distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.00
00.
002
0.00
40.
006
0.00
80.
010
x
y
Number of Iterations 2, 3, 4, 5, 10
, 50, 100
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Example of results with a truncated N (−3, 1) distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.00
00.
002
0.00
40.
006
0.00
80.
010
x
y
Number of Iterations 2, 3, 4, 5, 10, 50
, 100
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Example of results with a truncated N (−3, 1) distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.00
00.
002
0.00
40.
006
0.00
80.
010
x
y
Number of Iterations 2, 3, 4, 5, 10, 50, 100
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Good slices
The slice sampler usually enjoys good theoretical properties (likegeometric ergodicity and even uniform ergodicity under bounded fand bounded X ).As k increases, the determination of the set A(t+1) may getincreasingly complex.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Stochastic volatility
Example (Stochastic volatility core distribution)
Difficult part of the stochastic volatility model
π(x) ∝ exp−{σ2(x− µ)2 + β2 exp(−x)y2 + x
}/2 ,
simplified in exp−{x2 + α exp(−x)
}
Slice sampling means simulation from a uniform distribution on
A ={x; exp−
{x2 + α exp(−x)
}/2 ≥ u
}=
{x;x2 + α exp(−x) ≤ ω
}if we set ω = −2 log u.Note Inversion of x2 + α exp(−x) = ω needs to be done bytrial-and-error.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Stochastic volatility
Example (Stochastic volatility core distribution)
Difficult part of the stochastic volatility model
π(x) ∝ exp−{σ2(x− µ)2 + β2 exp(−x)y2 + x
}/2 ,
simplified in exp−{x2 + α exp(−x)
}Slice sampling means simulation from a uniform distribution on
A ={x; exp−
{x2 + α exp(−x)
}/2 ≥ u
}=
{x;x2 + α exp(−x) ≤ ω
}if we set ω = −2 log u.Note Inversion of x2 + α exp(−x) = ω needs to be done bytrial-and-error.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Completion
Stochastic volatility
0 10 20 30 40 50 60 70 80 90 100−0.1
−0.05
0
0.05
0.1
Lag
Corre
lation
−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
Dens
ity
Histogram of a Markov chain produced by a slice sampler and target distribution in overlay.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Convergence
Properties of the Gibbs sampler
Theorem (Convergence)
For(Y1, Y2, · · · , Yp) ∼ g(y1, . . . , yp),
if either[Positivity condition]
(i) g(i)(yi) > 0 for every i = 1, · · · , p, implies thatg(y1, . . . , yp) > 0, where g(i) denotes the marginal distributionof Yi, or
(ii) the transition kernel is absolutely continuous with respect to g,
then the chain is irreducible and positive Harris recurrent.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Convergence
Properties of the Gibbs sampler (2)
Consequences
(i) If∫h(y)g(y)dy <∞, then
limnT→∞
1
T
T∑t=1
h1(Y (t)) =
∫h(y)g(y)dy a.e. g.
(ii) If, in addition, (Y (t)) is aperiodic, then
limn→∞
∥∥∥∥∫ Kn(y, ·)µ(dx)− f∥∥∥∥TV
= 0
for every initial distribution µ.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Convergence
Slice sampler
fast on that slice
For convergence, the properties of Xt and of f(Xt) are identical
Theorem (Uniform ergodicity)
If f is bounded and suppf is bounded, the simple slice sampler isuniformly ergodic.
[Mira & Tierney, 1997]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Convergence
A small set for a slice sampler
no slice detail
For ε? > ε?,C = {x ∈ X ; ε? < f(x) < ε?}
is a small set:Pr(x, ·) ≥ ε?
ε?µ(·)
where
µ(A) =1
ε?
∫ ε?
0
λ(A ∩ L(ε))
λ(L(ε))dε
if L(ε) = {x ∈ X ; f(x) > ε}‘[Roberts & Rosenthal, 1998]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Convergence
Slice sampler: drift
Under differentiability and monotonicity conditions, the slicesampler also verifies a drift condition with V (x) = f(x)−β, isgeometrically ergodic, and there even exist explicit bounds on thetotal variation distance
[Roberts & Rosenthal, 1998]
Example (Exponential Exp(1))For n > 23,
||Kn(x, ·)− f(·)||TV ≤ .054865 (0.985015)n (n− 15.7043)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Convergence
Slice sampler: drift
Under differentiability and monotonicity conditions, the slicesampler also verifies a drift condition with V (x) = f(x)−β, isgeometrically ergodic, and there even exist explicit bounds on thetotal variation distance
[Roberts & Rosenthal, 1998]
Example (Exponential Exp(1))For n > 23,
||Kn(x, ·)− f(·)||TV ≤ .054865 (0.985015)n (n− 15.7043)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Convergence
Slice sampler: convergence
no more slice detail
Theorem
For any density such that
ε∂
∂ελ ({x ∈ X ; f(x) > ε}) is non-increasing
then||K523(x, ·)− f(·)||TV ≤ .0095
[Roberts & Rosenthal, 1998]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
Convergence
A poor slice sampler
Example
Consider
f(x) = exp {−||x||} x ∈ Rd
Slice sampler equivalent toone-dimensional slice sampler on
π(z) = zd−1 e−z z > 0
or on
π(u) = e−u1/d
u > 0
Poor performances when d large(heavy tails)
0 200 400 600 800 1000
-2-1
01
1 dimensional run
co
rre
latio
n
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
1 dimensional acf
0 200 400 600 800 1000
10
15
20
25
30
10 dimensional run
co
rre
latio
n
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
10 dimensional acf
0 200 400 600 800 1000
02
04
06
0
20 dimensional run
co
rre
latio
n
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
20 dimensional acf
0 200 400 600 800 1000
01
00
20
03
00
40
0100 dimensional run
co
rre
latio
n
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
100 dimensional acf
Sample runs of log(u) and ACFs for log(u) (Roberts
& Rosenthal, 1999)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
The Hammersley-Clifford theorem
Hammersley-Clifford theorem
An illustration that conditionals determine the joint distribution
Theorem
If the joint density g(y1, y2) have conditional distributionsg1(y1|y2) and g2(y2|y1), then
g(y1, y2) =g2(y2|y1)∫
g2(v|y1)/g1(y1|v) dv.
[Hammersley & Clifford, circa 1970]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
The Gibbs Sampler
The Hammersley-Clifford theorem
General HC decomposition
Under the positivity condition, the joint distribution g satisfies
g(y1, . . . , yp) ∝p∏j=1
g`j (y`j |y`1 , . . . , y`j−1, y′`j+1
, . . . , y′`p)
g`j (y′`j|y`1 , . . . , y`j−1
, y′`j+1, . . . , y′`p)
for every permutation ` on {1, 2, . . . , p} and every y′ ∈ Y .
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Sequential importance sampling
Computational issues in Bayesianstatistics
The Metropolis-Hastings Algorithm
The Gibbs Sampler
Population Monte Carlo
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Importance sampling (revisited)
basic importance
Approximation of integrals
I =
∫h(x)π(x)dx
by unbiased estimators
I =1
n
n∑i=1
%ih(xi)
when
x1, . . . , xniid∼ q(x) and %i
def=
π(xi)
q(xi)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Iterated importance sampling
As in Markov Chain Monte Carlo (MCMC) algorithms,introduction of a temporal dimension :
x(t)i ∼ qt(x|x
(t−1)i ) i = 1, . . . , n, t = 1, . . .
and
It =1
n
n∑i=1
%(t)i h(x
(t)i )
is still unbiased for
%(t)i =
πt(x(t)i )
qt(x(t)i |x
(t−1)i )
, i = 1, . . . , n
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Fundamental importance equality
Preservation of unbiasedness
E[h(X(t))
π(X(t))
qt(X(t)|X(t−1))
]
=
∫h(x)
π(x)
qt(x|y)qt(x|y) g(y) dx dy
=
∫h(x)π(x) dx
for any distribution g on X(t−1)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Sequential variance decomposition
Furthermore,
var(It
)=
1
n2
n∑i=1
var(%
(t)i h(x
(t)i )),
if var(%
(t)i
)exists, because the x
(t)i ’s are conditionally uncorrelated
Note
This decomposition is still valid for correlated [in i] x(t)i ’s when
incorporating weights %(t)i
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Simulation of a population
The importance distribution of the sample (a.k.a. particles) x(t)
qt(x(t)|x(t−1))
can depend on the previous sample x(t−1) in any possible way aslong as marginal distributions
qit(x) =
∫qt(x
(t)) dx(t)−i
can be expressed to build importance weights
%it =π(x
(t)i )
qit(x(t)i )
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Special case of the product proposal
If
qt(x(t)|x(t−1)) =
n∏i=1
qit(x(t)i |x
(t−1))
[Independent proposals]then
var(It
)=
1
n2
n∑i=1
var(%
(t)i h(x
(t)i )),
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Validation
skip validation
E[%
(t)i h(X
(t)i ) %
(t)j h(X
(t)j )]
=
∫h(xi)
π(xi)
qit(xi|x(t−1))
π(xj)
qjt(xj |x(t−1))h(xj)
qit(xi|x(t−1)) qjt(xj |x(t−1)) dxi dxj g(x(t−1))dx(t−1)
= Eπ [h(X)]2
whatever the distribution g on x(t−1)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Self-normalised version
In general, π is unscaled and the weight
%(t)i ∝
π(x(t)i )
qit(x(t)i )
, i = 1, . . . , n ,
is scaled so that ∑i
%(t)i = 1
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Self-normalised version properties
I Loss of the unbiasedness property and the variancedecomposition
I Normalising constant can be estimated by
$t =1
tn
t∑τ=1
n∑i=1
π(x(τ)i )
qiτ (x(τ)i )
I Variance decomposition (approximately) recovered if $t−1 isused instead
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Sampling importance resampling
Importance sampling from g can also produce samples from thetarget π
[Rubin, 1987]
Theorem (Bootstraped importance sampling)
If a sample (x?i )1≤i≤m is derived from the weighted sample(xi, %i)1≤i≤n by multinomial sampling with weights %i, then
x?i ∼ π(x)
Note
Obviously, the x?i ’s are not iid
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Sampling importance resampling
Importance sampling from g can also produce samples from thetarget π
[Rubin, 1987]
Theorem (Bootstraped importance sampling)
If a sample (x?i )1≤i≤m is derived from the weighted sample(xi, %i)1≤i≤n by multinomial sampling with weights %i, then
x?i ∼ π(x)
Note
Obviously, the x?i ’s are not iid
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Sampling importance resampling
Importance sampling from g can also produce samples from thetarget π
[Rubin, 1987]
Theorem (Bootstraped importance sampling)
If a sample (x?i )1≤i≤m is derived from the weighted sample(xi, %i)1≤i≤n by multinomial sampling with weights %i, then
x?i ∼ π(x)
Note
Obviously, the x?i ’s are not iid
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Iterated sampling importance resampling
This principle can be extended to iterated importance sampling:After each iteration, resampling produces a sample from π
[Again, not iid!]
Incentive
Use previous sample(s) to learn about π and q
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Iterated sampling importance resampling
This principle can be extended to iterated importance sampling:After each iteration, resampling produces a sample from π
[Again, not iid!]
Incentive
Use previous sample(s) to learn about π and q
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Generic Population Monte Carlo
Algorithm (Population Monte Carlo Algorithm)
For t = 1, . . . , T
For i = 1, . . . , n,
1. Select the generating distribution qit(·)2. Generate x
(t)i ∼ qit(x)
3. Compute %(t)i = π(x
(t)i )/qit(x
(t)i )
Normalise the %(t)i ’s into %
(t)i ’s
Generate Ji,t ∼M((%(t)i )1≤i≤N ) and set xi,t = x
(t)Ji,t
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
D-kernels in competition
A general adaptive construction:
Construct qi,t as a mixture of D different transition kernels
depending on x(t−1)i
qi,t =
D∑`=1
pt,`K`(x(t−1)i , x),
D∑`=1
pt,` = 1 ,
and adapt the weights pt,`.
Darwinian example
Take pt,` proportional to the survival rate of the points
(a.k.a. particles) x(t)i generated from K`
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
D-kernels in competition
A general adaptive construction:
Construct qi,t as a mixture of D different transition kernels
depending on x(t−1)i
qi,t =
D∑`=1
pt,`K`(x(t−1)i , x),
D∑`=1
pt,` = 1 ,
and adapt the weights pt,`.
Darwinian example
Take pt,` proportional to the survival rate of the points
(a.k.a. particles) x(t)i generated from K`
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Implementation
Algorithm (D-kernel PMC)
For t = 1, . . . , T
generate (Ki,t)1≤i≤N ∼M ((pt,k)1≤k≤D)
for 1 ≤ i ≤ N , generate
xi,t ∼ KKi,t(x)
compute and renormalize the importance weights ωi,t
generate (Ji,t)1≤i≤N ∼M ((ωi,t)1≤i≤N )
take xi,t = xJi,t,t and pt+1,d =∑N
i=1 ωi,tId(Ki,t)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Links with particle filters
I Sequential setting where π = πt changes with t: PopulationMonte Carlo also adapts to this case
I Can be traced back all the way to Hammersley and Morton(1954) and the self-avoiding random walk problem
I Gilks and Berzuini (2001) produce iterated samples with (SIR)resampling steps, and add an MCMC step: this step must usea πt invariant kernel
I Chopin (2001) uses iterated importance sampling to handlelarge datasets: this is a special case of PMC where the qit’sare the posterior distributions associated with a portion kt ofthe observed dataset
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Links with particle filters (2)
I Rubinstein and Kroese’s (2004) cross-entropy method isparameterised importance sampling targeted at rare events
I Stavropoulos and Titterington’s (1999) smooth bootstrap andWarnes’ (2001) kernel coupler use nonparametric kernels onthe previous importance sample to build an improvedproposal: this is a special case of PMC
I West (1992) mixture approximation is a precursor of smoothbootstrap
I Mengersen and Robert (2002) “pinball sampler” is an MCMCattempt at population sampling
I Del Moral, Doucet and Jasra (2006, JRSS B) sequentialMonte Carlo samplers also relates to PMC, with a Markoviandependence on the past sample x(t) but (limited) stationarityconstraints
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Things can go wrong
Unexpected behaviour of the mixture weights when the number ofparticles increases
N∑i=1
ωi,tIKi,t=d−→P1
D
Conclusion
At each iteration, every weight converges to 1/D:the algorithm fails to learn from experience!!
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Things can go wrong
Unexpected behaviour of the mixture weights when the number ofparticles increases
N∑i=1
ωi,tIKi,t=d−→P1
D
Conclusion
At each iteration, every weight converges to 1/D:the algorithm fails to learn from experience!!
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Saved by Rao-Blackwell!!
Modification: Rao-Blackwellisation (=conditioning)
Use the whole mixture in the importance weight:
ωi,t = π(xi,t)
D∑d=1
pt,dKd(xi,t−1, xi,t)
instead of
ωi,t =π(xi,t)
KKi,t(xi,t−1, xi,t)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Saved by Rao-Blackwell!!
Modification: Rao-Blackwellisation (=conditioning)
Use the whole mixture in the importance weight:
ωi,t = π(xi,t)
D∑d=1
pt,dKd(xi,t−1, xi,t)
instead of
ωi,t =π(xi,t)
KKi,t(xi,t−1, xi,t)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Adapted algorithm
Algorithm (Rao-Blackwellised D-kernel PMC)
At time t (t = 1, . . . , T ),
Generate(Ki,t)1≤i≤N
iid∼ M((pt,d)1≤d≤D);
Generate(xi,t)1≤i≤N
ind∼ KKi,t(xi,t−1, x)
and set ωi,t = π(xi,t)
/∑Dd=1 pt,dKd(xi,t−1, xi,t);
Generate(Ji,t)1≤i≤N
iid∼ M((ωi,t)1≤i≤N )
and set xi,t = xJi,t,t and pt+1,d =∑N
i=1 ωi,tpt,d.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Convergence properties
Theorem (LLN)
Under regularity assumptions, for h ∈ L1Π and for every t ≥ 1,
1
N
N∑k=1
ωi,th(xi,t)N→∞−→P Π(h)
andpt,d
N→∞−→P αtd
The limiting coefficients (αtd)1≤d≤D are defined recursively as
αtd = αt−1d
∫ (Kd(x, x
′)∑Dj=1 α
t−1j Kj(x, x′)
)Π⊗Π(dx, dx′).
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Recursion on the weights
Set F as
F (α) =
(αd
∫ [Kd(x, x
′)∑Dj=1 αjKj(x, x
′)
]Π⊗Π(dx, dx′)
)1≤d≤D
on the simplex
S =
{α = (α1, . . . , αD); ∀d ∈ {1, . . . , D}, αd ≥ 0 and
D∑d=1
αd = 1
}.
and define the sequence
αt+1 = F (αt)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Kullback divergence
Definition (Kullback divergence)
For α ∈ S,
KL(α) =
∫ [log
(π(x)π(x′)
π(x)∑D
d=1 αdKd(x, x′)
)]Π⊗Π(dx, dx′).
Kullback divergence between Π and the mixture.
Goal: Obtain the mixture closest to Π, i.e., that minimises KL(α)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Connection with RBDPMCA ??
Theorem
Under the assumption
∀d ∈ {1, . . . , D},−∞ <
∫log(Kd(x, x
′))Π⊗Π(dx, dx′) <∞
for every α ∈ SD,
KL(F (α)) ≤ KL(α).
Conclusion
The Kullback divergence decreases at every iteration of RBDPMCA
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Connection with RBDPMCA ??
Theorem
Under the assumption
∀d ∈ {1, . . . , D},−∞ <
∫log(Kd(x, x
′))Π⊗Π(dx, dx′) <∞
for every α ∈ SD,
KL(F (α)) ≤ KL(α).
Conclusion
The Kullback divergence decreases at every iteration of RBDPMCA
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
An integrated EM interpretation
skip interpretation
We have
αmin = arg minα∈S
KL(α) = arg maxα∈S
∫log pα(x)Π⊗Π(dx)
= arg maxα∈S
∫log
∫pα(x,K)dK Π⊗Π(dx)
for x = (x, x′) and K ∼M((αd)1≤d≤D). Then αt+1 = F (αt)means
αt+1 = arg maxα
∫∫Eαt(log pα(X,K)|X = x)Π⊗Π(dx)
andlimt→∞
αt = αmin
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Illustration
Example (A toy example)
Take the target
1/4N (−1, 0.3)(x) + 1/4N (0, 1)(x) + 1/2N (3, 2)(x)
and use 3 proposals: N (−1, 0.3), N (0, 1) and N (3, 2)[Surprise!!!]
Then
1 0.0500000 0.05000000 0.90000002 0.2605712 0.09970292 0.63972596 0.2740816 0.19160178 0.534316610 0.2989651 0.19200904 0.509025916 0.2651511 0.24129039 0.4935585
Weight evolution
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Illustration
Example (A toy example)
Take the target
1/4N (−1, 0.3)(x) + 1/4N (0, 1)(x) + 1/2N (3, 2)(x)
and use 3 proposals: N (−1, 0.3), N (0, 1) and N (3, 2)[Surprise!!!]
Then
1 0.0500000 0.05000000 0.90000002 0.2605712 0.09970292 0.63972596 0.2740816 0.19160178 0.534316610 0.2989651 0.19200904 0.509025916 0.2651511 0.24129039 0.4935585
Weight evolution
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
Target and mixture evolution
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
c© Learning scheme
The efficiency of the SNIS approximation depends on the choice ofQ, ranging from optimal
q(x) ∝ |h(x)−Π(h)|π(x)
to uselessvar ΠSNIS
Q,N (h) = +∞
Example (PMC=adaptive importance sampling)
Population Monte Carlo is producing a sequence of proposals Qtaiming at improving efficiency
Kull(π, qt) ≤ Kull(π, qt−1) or var ΠSNISQt,∞ (h) ≤ var ΠSNIS
Qt−1,∞(h)
[Cappe, Douc, Guillin, Marin, Robert, 04, 07a, 07b, 08]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
c© Learning scheme
The efficiency of the SNIS approximation depends on the choice ofQ, ranging from optimal
q(x) ∝ |h(x)−Π(h)|π(x)
to uselessvar ΠSNIS
Q,N (h) = +∞
Example (PMC=adaptive importance sampling)
Population Monte Carlo is producing a sequence of proposals Qtaiming at improving efficiency
Kull(π, qt) ≤ Kull(π, qt−1) or var ΠSNISQt,∞ (h) ≤ var ΠSNIS
Qt−1,∞(h)
[Cappe, Douc, Guillin, Marin, Robert, 04, 07a, 07b, 08]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
AMIS
Multiple Importance Sampling
Reycling: given several proposals Q1, . . . , QT , for 1 ≤ t ≤ Tgenerate an iid sample
xt1, . . . , xtN ∼ Qt
and estimate Π(h) by
ΠMISQ,N (h) = T−1
T∑t=1
N−1N∑i=1
h(xti)ωti
where
ωti 6=π(xti)
qt(xti)correct...
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
AMIS
Multiple Importance Sampling
Reycling: given several proposals Q1, . . . , QT , for 1 ≤ t ≤ Tgenerate an iid sample
xt1, . . . , xtN ∼ Qt
and estimate Π(h) by
ΠMISQ,N (h) = T−1
T∑t=1
N−1N∑i=1
h(xti)ωti
where
ωti =π(xti)
T−1∑T
`=1 q`(xti)
still correct!
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
AMIS
Mixture representation
Deterministic mixture correction of the weights proposed by Owenand Zhou (JASA, 2000)
I The corresponding estimator is still unbiased [if notself-normalised]
I All particles are on the same weighting scale rather than theirown
I Large variance proposals Qt do not take over
I Variance reduction thanks to weight stabilization & recycling
I [K.o.] removes the randomness in the component choice[=Rao-Blackwellisation]
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
AMIS
Global adaptation
Global Adaptation
At iteration t = 1, · · · , T ,
1. For 1 ≤ i ≤ N1, generate xti ∼ T3(µt−1, Σt−1)
2. Calculate the mixture importance weight of particle xti
ωti = π(xti) /δti
where
δti =
t−1∑l=0
qT (3)
(xti; µ
l, Σl)
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
AMIS
Backward reweighting
3. If t ≥ 2, actualize the weights of all past particles, xli1 ≤ l ≤ t− 1
ωli = π(xti) /δli
whereδli = δli + qT (3)
(xli; µ
t−1, Σt−1)
4. Compute IS estimates of target mean and variance µt and Σt,where
µtj =
t∑l=1
N1∑i=1
ωli(xj)li
/ t∑l=1
N1∑i=1
ωli . . .
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
AMIS
A toy example
−40 −20 0 20 40
−30
−20
−10
010
Banana shape benchmark: marginal distribution of (x1, x2) for the parameters
σ21 = 100 and b = 0.03. Contours represent 60% (red), 90% (black) and 99.9%
(blue) confidence regions in the marginal space.
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MCMC and likelihood-free methods Part/day I: Markov chain methods
Population Monte Carlo
AMIS
A toy example
●
●
●4e+
046e
+04
8e+
041e
+05
p=5
●
●
2000
040
000
6000
080
000
p=10
●
020
000
6000
0
p=20
Banana shape example: boxplots of 10 replicate ESSs for the AMIS scheme
(left) and the NOT-AMIS scheme (right) for p = 5, 10, 20.