Presentation of SMC^2 at BISP7

Introduction and State Space ModelsQuick reminder on Sequential Monte Carlo

Particle Markov Chain Monte CarloSMC2

SMC2: A sequential Monte Carlo algorithm withparticle Markov chain Monte Carlo updates

N. CHOPIN1, P.E. JACOB2, & O. PAPASPILIOPOULOS3

BISP7 – September, 2011

1ENSAE-CREST2CREST & Universite Paris Dauphine, funded by AXA research3Universitat Pompeu Fabra

N. CHOPIN, P.E. JACOB, & O. PAPASPILIOPOULOS SMC2 1/ 16



Sequential Monte Carlo for filtering

Suppose we are interested in pθ(xT |y1:T ), for a given θ.

General idea

Sample recursively from pθ(xt |y1:t) to pθ(xt+1|y1:t+1).

After the SMC run, we can approximate the likelihood:

ZT (θ) = p(y1:T |θ) =

(T∏

t=2

p(yt |y1:t−1, θ)

)p(y1|θ)

with an unbiased estimate ZNxT (θ).




Sequential Monte Carlo Samplers

Same kind of method but to perform bayesian inference:

p(θ|y1:T )

General idea

Sample recursively from p(θ|y1:t) to p(θ|y1:t+1).

MCMC moves to diversify the particles.

Requires the ability to compute point-wise p(yt |y1:t−1, θ).




Idealized Metropolis–Hastings for SSM

Motivation

Bayesian parameter inference in state space models:

p(θ|y1:T )

If only. . .

. . . we could compute p(θ|y1:T ) ∝ p(θ)p(y1:T |θ), we could run aMH algorithm.




Valid Metropolis–Hastings for SSM

Plug in estimates

We have ZNxT (θ) ≈ p(y1:T |θ) by running a SMC filter, and we can

try to run a MH algorithm using the estimate instead of the rightlikelihood.

Particle MCMC

This is called Particle Marginal Metropolis-Hastings, by Andrieu,Doucet and Holenstein.




Our contribution. . .

. . . was to use the same method to get a valid SMC sampler forstate space models.

Foreseen benefits

to sample more efficiently from the posterior distributionp(θ|y1:T ),

to sample sequentially from p(θ|y1), p(θ|y1, y2), . . . p(θ|y1:T ).

and it turns out, it allows even a bit more.




Valid SMC sampler for SSM

Plug in estimates

Similarly to PMCMC methods, we want to replace

p(yt |y1:t−1, θ)

with an unbiased estimate, and see what happens.

SMC everywhere

We associate Nx x-particles to each of the Nθ θ-particles,

these are used to get estimates of the incremental likelihoodsfor each θ-particle.




Side benefits

Evidence

SMC2 provides an estimate of the “evidence”:

p(y1:t) =t∏

s=1

p(ys |y1:s−1)

Automatic tuning

θ-particles are moved with adaptive particle MCMC steps,

the number of Nx particles can be dynamically increased ifneed be.




Numerical illustrations: Stochastic Volatility

Time

Obs

erva

tions

−4

−2

0

2

100 200 300 400 500 600 700

Figure: The S&P 500 data from 03/01/2005 to 21/12/2007.





Stochastic Volatility model

Observations (“log returns”):

yt = µ+ βvt + v1/2t εt , εt ∼ N (0, 1)

Hidden states: the “actual volatility” (vt), a process thatdepends on another process, the “spot volatility” (zt).

All these processes are parameterized by θ ∈ (µ, β, ξ, ω2, λ).





µ

Den

sity

0

2

4

6

8

T = 250

−1.0 −0.5 0.0 0.5 1.0

T = 500

−1.0 −0.5 0.0 0.5 1.0

T = 750

−1.0 −0.5 0.0 0.5 1.0

T = 1000

−1.0 −0.5 0.0 0.5 1.0

Figure: Concentration of the posterior distribution for parameter µ.





Model comparison

For the same problem there could be various models that we wantto compare. Here:

the “basic” previous model,

a similar model with more factors (= more hidden states),

a similar model with more factors and “leverage” (= differentlikelihood function with more parameters).





Time

Squ

ared

obs

erva

tions

5

10

15

20

100 200 300 400 500 600 700

(a)

Iterations

Evi

denc

e co

mpa

red

to th

e on

e fa

ctor

mod

el

−2

0

2

4

100 200 300 400 500 600 700

variableMulti factor without leverageMulti factor with leverage

(b)

Figure: Left: observations; right: log-evidence relative to the basic model.




Conclusion

A powerful framework

The SMC2 framework allows to obtain various quantities ofinterest, especially for sequential analysis.

It extends the PMCMC framework introduced by Andrieu,Doucet and Holenstein.

A python package is available:

http://code.google.com/p/py-smc2/.


http://code.google.com/p/py-smc2/



Bibliography

SMC2: A sequential Monte Carlo algorithm with particle Markovchain Monte Carlo updates, N. Chopin, P.E. Jacob, O.Papaspiliopoulos, submitted, available on arXiv.Main references:

Particle Markov Chain Monte Carlo methods, C. Andrieu, A.Doucet, R. Holenstein, JRSS B., 2010, 72(3):269–342

The pseudo-marginal approach for efficient computation, C.Andrieu, G.O. Roberts, Ann. Statist., 2009, 37, 697–725

Random weight particle filtering of continuous time processes,P. Fearnhead, O. Papaspiliopoulos, G.O. Roberts, A. Stuart,JRSS B., 2010, 72:497–513

Feynman-Kac Formulae, P. Del Moral, Springer


Presentation of SMC^2 at BISP7

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