Particle MCMC methods for parameter inference in state space models

An intro to particle methods for parameterinference in state-space models

PhD course FMS020F–NAMS002 “Statistical inference for partiallyobserved stochastic processes”, Lund University

http://goo.gl/sX8vU9

Umberto PicchiniCentre for Mathematical Sciences,

Lund Universitywww.maths.lth.se/matstat/staff/umberto/

Umberto Picchini (umberto@maths.lth.se)

This lecture will explore possibilities offered by particle filters, alsoknown as Sequential Monte Carlo (SMC) methods, when applied toparameter estimation problems.

We will not give a thorough introduction to SMC methods.

Instead, we will introduce the most basic (popular) SMC method withthe main goal of performing inference for θ, a vector of unknownparameters entering a state space / Hidden Markov model.

Results anticipation

Thanks to recent and astonishing results, we show how it is possibleto construct a practical method for exact Bayesian inference on theparameters of a state-space model.

Acknowledgements (early enough, not to forget...)

Part of the presented material, slides, MATLAB codes etc. has beenkindly shared by:

Magnus Wiktorsson (Matematikcentrum, Lund);

Fredrik Lindsten (Automatic Control, Linköping).

Working framework

We consider hereafter the (important) case where data are modelledaccording to an Hidden Markov Model (HMM), often denotedstate-space model (SSM).

(In recent literature the terms SSM and HMM have been usedinterchangeably. We do the same here, though elsewhere it issometimes assumed that HMMs→discrete space andSSM →continuous space)

A state-space model refers to a class of probabilistic graphical modelsthat describes the probabilistic dependence between latent statevariables and the observed measurements of a dynamical system.

Our system of interest is {Xt}t>0. This is a latent/unobservablestochastic process.

(Unavailable) values of the system on a discrete time grid{0, 1, ..., T} are X0:T = {X0, ..., XT }

(we consider integer times t for ease of notation).

Actual observations (data) are noisy y1:T = {y1, ..., yT }

That is at time t each yt is a noisy/corrupted version of the truestate of the system Xt.

{Xt}t>0 is assumed a Markovian stochastic process.

Observations yt are assumed conditionally independent given thecorresponding latent state Xt.

We are going to formalize this in next slide.

SSMs as “graphical models”

Graphically:

"!# "!# "!#

Yt−1 Yt Yt+1

Xt−1 Xt Xt+1- - - -

... ... (Markov chain)

(Observations)

Yt|Xt = xt ∼ p(yt|xt) (Observation density)

Xt+1|Xt = xt ∼ p(xt+1|xt) (Transition density)

X0 ∼ π(x0) (Initial distribution)

Example: Gaussian random walk

A most trivial example (linear, Gaussian SSM).

xt = xt−1 + ξt, ξt∼iidN(0, q2)

yt = xt + et, et ∼iid N(0, r2)

Therefore

p(xt|xt−1) = N(xt; xt−1, q2)

p(yt|xt) = N(yt; xt, r2)

Notation: here and in the following N(µ,σ2) is the Gaussiandistribution with mean µ and variance σ2. N(x;µ,σ2) is theevaluation at x of the pdf of a N(µ,σ2) distribution.

Notation reminder

Always remember X0 is NOT the state for the first observational time,that one is X1. Instead X0 is the (typically unknown) system’s initialstate for process {Xt}.

X0 can be set to be a deterministic constant (as in the example we aregoing to discuss soon).

Markov property of hidden states

Briefly: Xt (and actually the whole future Xt+1, Xt+2,...) given Xt−1 isindependent from anything that has happened before time t − 1:

p(xt|x0:t−1, y1:t−1) = p(xt|xt−1), t = 1, ..., T

Also, past is independent of the future given the present:

p(xt−1|xt:T , yt:T) = p(xt−1|xt)

Conditional independence of measurements

The current measurement yt given xt is conditionally independent ofthe measurement and state histories:

p(yt|x0:t, y1:t−1) = p(yt|xt).

The Markov property on {Xt} and the conditional independence on{Yt} are the key ingredients for defining an HMM or SSM.

In general {Xt} and {Yt} can be either continuous– or discrete–valuedstochastic processes. However in the following we assume {Xt} and{Yt} to be defined on continuous spaces.

Our goal

In all previous slides we haven’t made explicit reference to thepresence of unknown constants that we wish to estimate.

For example in the Gaussian random walk example the unknownswould be the variances (q2, r2) (and perhaps the initial state x0, insome situation).

In that case we would like to estimate θ = (q2, r2) using observationsy1:T . More in general...

Main goal

We introduce general methods for SSM producing inference for thevector of parameters θ. We will be particularly interested in Bayesianinference.

Of course θ can contain all sorts of unknowns, not just variances.Umberto Picchini (umberto@maths.lth.se)

A quick look into the final goal

p(y1:T |θ) is the likelihood of the measurements conditionally onθ.π(θ) is the prior density of θ (we always assumecontinuous-valued parameters). It encloses knowledge about θbefore we “see” our current data y1:T .Bayes theorem gives the posterior distribution:

π(θ|y1:T) =p(y1:T |θ)π(θ)

p(y1:T)∝ p(y1:T |θ)π(θ)

inference based on π(θ|y1:T) is called Bayesian inference.p(y1:T) is the marginal likelihood (evidence), independent of θ.Goal: calculate (sample draws from) π(θ|y1:T).Remark: θ is a random quantity in the Bayesian framework.

As you know... (remember we set some background requirements fortaking this course) Bayesian inference can rarely be performedwithout using some Monte Carlo sampling.

Except for simple cases (e.g. data y1, ..., yT independently distributedfrom some member of the exponential family) it is usually impossibleto write the likelihood p(y1:T |θ) in closed form.

Since early 90’s MCMC (Markov chain Monte Carlo) has opened thepossibility to perform Bayesian inference in practice.

Are we obsessed with Bayesian inference? NO! However itsometimes offers the easiest way to deal with complex, non-trivialmodels. Some good reads

The Bayesian approach actually opens the possibility for a surprisingresult (see later on...).

The likelihood function for SSMs

First of all, notice that in the Bayesian framework, since θ israndom, we do not simply write the likelihood function aspθ(y1:T) nor p(y1:T ; θ), but we must condition on θ and writep(y1:T |θ).

In a SSM data are not independent, they are only conditionallyindependent→ complication!:

p(y1:T |θ) = p(y1|θ)

T∏t=2

p(yt|y1:t−1, θ) =?

We don’t have a closed for expression for the product above becausewe do not know how to calculate p(yt|y1:t−1, θ).

Let’s see why.Umberto Picchini (umberto@maths.lth.se)

In a SSM the observed process is assumed to depend on the latentMarkov process {Xt}: we can write

p(y1:T |θ) =

∫p(y1:T , x0:T |θ)dx0:T =

∫p(y1:T |x0:T , θ)︸︷︷︸

use cond. indep.

× p(x0:T |θ)︸︷︷︸use Markovianity

∫ T∏t=1

p(yt|xt, θ)×{

p(x0|θ)

T∏t=1

p(xt|xt−1, θ)}

Problems

The expression above is a (T + 1)-dimensional integral /

For most (nontrivial) models, transition densities p(xt|xt−1) areunknown /

Special cases

Despite the analytic difficulties, find approximations for thelikelihood function is possible (we’ll consider some approachsoon).

In some simple cases, closed form solutions do exist: forexample when the SSM is linear and Gaussian (see the Gaussianrandom walk example) then the classic Kalman filter gives theexact likelihood.

In the SSM literature important (Gaussian) approximations aregiven by the extended and unscented Kalman filters. However,approximations offered by particle filters (a.k.a. sequentialMonte Carlo) are presently the state-of-art for general non-linearnon-Gaussian SSM.

What if we had p(y1:T |θ)...

Ideally if we had p(y1:T |θ) we could use Metropolis-Hastings (MH) tosample from the posterior π(θ|y1:T). At iteration r + 1 we have:

1 current value is θr, propose a new θ∗ ∼ q(θ∗|θr), e.g.θ∗ ∼ N(θr,Σθ) for some covariance matrix Σθ.

2 draw u ∼ U(0, 1)accept θ∗ if u < min(1, A), where

A =p(y1:T |θ

p(y1:T |θr)× π(θ

π(θr)× q(θr|θ

q(θ∗|θr)

then set θr+1 := θ∗ and p(y1:T |θr+1) := p(y1:T |θ∗). Otherwise,

reject θ∗, set θr+1 := θr. Set r := r + 1, go to 1 and repeat.

By repeating steps 1-2 as much as wanted we are guaranteed that, bydiscarding a “long enough” number of iterations (burnin), theremaining draws form

a Markov chain (hence dependent values) having π(θ|y1:T) astheir stationary distribution.

Therefore if you have produced R = R1 + R2 iterations ofMetropolis-Hastings, where R1 is a sufficiently long burnin, forscalar θ you can then plot the histogram of the last R2 drawsθR1+1, ..., θR−R1 . Such histogram gives the density π(θ|y1:T), upto a Monte Carlo error induced by using a finite R2.

for a vector valued θ ∈ Rp create p separate histograms of thedraws pertaining each component of θ. Such histogramsrepresent the posterior marginals π(θj|y), j = 1, ..., p.

Brush up MH with a simple example

A simple toy problem not related to state-space modelling.

Data: n = 1, 000 observations i.i.d yj ∼ N(3, 1).

Now assume µ = E(yj) unknown. Estimate µ.

Assume for example a “wide” prior: µ ∼ N(2, 22) (do not look atdata!)

Gaussian random walk to propose µ∗ = µr + 0.1 · ξ, withξ ∼ N(0, 1). Notice Gaussian proposals are symmetric, henceq(µ∗|µr) = q(µr|µ

∗) therefore

A =p(y1:n|µ

p(y1:n|µr)× π(µ

π(µr)

Start far at µ = 10. Produce 10,000 iterations.

MCMC iterations0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

True value µ = 3-1 0 1 2 3 4 5 6

14posterior π(µ|y) after burnin

posteriorprior

The posterior on the right plot discards the initial 500 draws (burnin).

Remark

Notice Metropolis-Hastings is not an optimization algorithm.

Unlike in maximum likelihood estimation here we are not trying toconverge towards some mode.

What we want is to explore thoroughly (i.e. sample from) π(θ|data),including its tails.

This way we can directly assess the uncertainty about θ by looking atπ(θ|data), instead of having to resort to asymptotic argumentsregarding the sampling distribution of θ̂n when n→∞ as inML-theory.

Example: a nonlinear SSM

{xt = 0.5xt−1 + 25 xt−1

(1+x2t−1)

+ 8 cos(1.2(t − 1)) + vt,

yt = 0.05x2t + et,

with a deterministic x0 = 0, measurements at times t = 1, 2, ..., 100.

vt ∼ N(0, q), et ∼ N(0, r) all independent.

Data generated with q = 0.1 and r = 1.

time0 10 20 30 40 50 60 70 80 90 100

Priors: q ∼ InvGamma(0.01, 0.01), r ∼ InvGamma(0.01, 0.01)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.4Prior density InvGamma(0.01,0.01)

Recall true values are q = 0.1 and r = 1. The chosen priors coverboth.

Results anticipation...

Perform an MCMC (how about the likelihood? we’ll see this later).

R = 10, 000 iterations (generous burnin = 3, 000 iterations).

Proposals: qnew ∼ N(qold, 0.22), and similarly for rnew.

Start far away: initial values qinit = 1, rinit = 0.1.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1.5Trace plot of p(q | y

1:T), acc. rate 1.103015e+01 percent

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

3Trace plot of p(r | y

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

20Empirical PDF of p(q | y

1:T) after burnin

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

3Empirical PDF of p(r | y

1:T) after burnin

How do we deal with general SSM for which the exact likelihoodp(y1:T |θ) is unavailable? That’s the case of the previous example.

For example, the previously analysed model is nonlinear in xt andtherefore the exact Kalman filter can’t be used.

General Monte Carlo integrationRecall our likelihood function:

p(y1:T |θ) = p(y1|θ)

T∏t=2

p(yt |y1:t−1, θ)

We can write p(yt |y1:t−1, θ) as

p(yt |y1:t−1, θ) =∫

p(yt |xt, θ)p(xt |y1:t−1, θ)dxt = E(p(yt |xt, θ))

Use Monte Carlo integration: generate N draws from p(xt |y1:t−1, θ), then invoke thelaw of large numbers.

produce N independent draws xit ∼ p(xt |y1:t−1, θ), i = 1, ..., N

for each xit compute p(yt |xi

t, θ)and by LLN we have

N∑i=1

p(yt |xit, θ)→ E(p(yt |xt, θ)), N →∞

error term is O(N−1/2) regardless the dimension of xt ,

SMC and the bootstrap filter

But how to generate “good” draws (particles) xit ∼ p(xt|y1:t−1, θ)?

Here “good” means that we want particles such that the values ofp(yt|xi

t, θ) are not negligible (“explain” a large fraction of theintegrand).

For SSM, sequential Monte Carlo (SMC) is the winning strategy.

We will NOT give a thorough introduction to SMC methods. We onlyuse a few notions to solve our parameter inference problem.

Importance sampling

Let’s remove for a moment the dependence on θ.

p(yt|y1:t−1) =

∫p(yt|x0:t)p(x0:t|y1:t−1)dx0:t

∫p(yt|xt)p(xt|xt−1)p(x0:t−1|y1:t−1)dx0:t

∫p(yt|xt)p(xt|xt−1)p(x0:t−1|y1:t−1)

h(x0:t|y1:t)h(x0:t|y1:t)dx0:t

where h(·) is an arbitrary (positive) density function called“importance density”. Choose one easy to simulate from.

1 simulate N samples: xi0:t ∼ h(x0:t|y1:t), i = 1, ..., N

2 construct importance weights wit =

p(yt|xit)p(xi

t|xit−1)p(xi

0:t−1|y1:t−1)

h(xi0:t|y1:t)

3 p(yt|y1:t−1) = E(p(yt|xt)) ≈ 1N

∑Ni=1 wi

Importance Sampling

Importance Sampling is an appealing and revolutionary idea forMonte Carlo integration.

However generating at each time a “cloud of particles” xi0:t is not

really computationally appealing, and it’s not clear how to do so.

We need something enabling a sort of sequential mechanism, as tincreases.

Sequential Importance Sampling

When h(·) is chosen in an intelligent way, an important property is theone that allows sequential update of weights. After some derivationon the board, we have (see p. 121 in Särkkä1 and p. 5 in Cappe et al.2)

wit ∝

p(yt|xit)p(x

it−1)

h(xit|xi

0:t−1, y1:t)wi

t−1.

However, the dependence of wt on wt−1 is a curse (with remedy) asdescribed in next slide.

1Särkkä, available here.2Cappe, Godsill and Moulines. Available here.

The particle degeneracy problem

Particle degeneracy occurs when at time t all but one of theimportance weights wi

t are close to zero. This implies a poorapproximation to p(yt|y1:t−1).

Notice that when a particle gets a zero weight (or a “small” positiveweight that your computer sets to zero→ numerical underflow) thatparticle is doomed! Since

wit ∝

p(yt|xit)p(x

it−1)

h(xit|xi

0:t−1, y1:t)wi

t−1.

if for a given i we have wit−1 = 0 particle i will have zero weight for

all subsequent times.

The particle degeneracy problem

For example, suppose a new data point yt is encountered and thisis an outlier.

Then most particles will end up far away from yt and theirweights will be ≈ 0.

Those particles will die.

Eventually, for a time horizon T which is long enough, all butone particle will have zero weight.

This means that the methodology is very fragile and particledegeneracy has affected SMC methods for decades.

The Resampling idea

A life saving solution is to use resampling with replacement (Gordonet al.3).

1 interpret w̃it as the probability to sample xi

t from the weighted set{xi

t, w̃it, i = 1, ..., N}, with w̃i

t := wit/∑

i wit.

2 draw N times with replacement from the weighted set. Replacethe old particles with the new ones {x̃1

t , ..., x̃Nt }.

3 Reset weights wit = 1/N (the resampling has destroyed the

information on “how” we reached time t).

Since resampling is done with replacement, a particle with a largeweight is likely to be drawn multiple times. Particles with very smallweights are not likely to be drawn at all. Nice!

3Gordon, Salmond and Smith. IEEE Proceedings F. 140(2) 1993.Umberto Picchini (umberto@maths.lth.se)

Sequential Importance Sampling Re-sampling (SISR)

We are now in the position to introduce a method that samples fromh(·), so that these samples can be used as if they were fromp(x0:t|y1:t−1) provided these are appropriately re-weighted.

1 Sample x1t , ..., xN

t ∼ h(·) (same as before)

2 compute weights wit (same as before)

3 normalize weights w̃it = wi

t/∑N

i=1 wit

4 We have a discrete probability distribution {xit, w̃i

t}, i = 1, ..., N

5 Resample N times with replacement from the set {x1t , ..., xN

having associate probabilities {w̃1t , ..., w̃N

t }, to generate anothersample {x̃1

t , ..., x̃Nt }.

6 {x̃1t , ..., x̃N

t } is an approximate sample from p(x0:t|y1:t−1).

Sketch of Proof on page 111 in Wilkinson’s book.

The next animation illustrates the concept of sequential importancesampling resampling with N = 5 particles.

Light blue: observed trajectory (data)

dark blue: forward simulation of the latent process {Xt}

pink balls: particles xit

green balls: selected particles x̃it from resampling

red curve: density p(yt|xt)

Bootstrap filter

This is the method we will actually use to ease Bayesian inference. Thebootstrap filter is a simple example of SISR.

Choose h(xit|x

i0:t−1, y1:t) := p(xi

t|xit−1)

Recall in SISR after performing resampling reset all weights→ wi

t−1 = 1N

wit ∝

p(yt|xit)p(x

it−1)

h(xit|xi

0:t−1, y1:t)wi

p(yt|xit) ∝ p(yt|xi

t), i = 1, ..., N

This is a SISR strategy therefore we still have that xit ≈ p(x0:t|y1:t−1)

Bootstrap filter in detail

1 t = 0 (initialize) xi0 ∼ π(x0), assign w̃i

0 = 1/N, i = 1, ..., N

2 at the current t assume we have the weighted particles {xit, w̃i

3 from the current sample, resample with replacement N times toobtain {x̃i

t, i = 1, ..., N}.

4 From your model propagate forward xit+1 ∼ p(xt+1|x̃i

t),i = 1, ..., N.

5 Compute wit+1 = p(yt+1|xi

t+1) and normalise weightsw̃i

t+1 = wit+1/∑N

i=1 wit+1

6 set t := t + 1 and if t < T go to step 2.

So the bootstrap filter by et al. (1993)4 easily provide what we need!

p̂(yt|y1:t−1) =1N

∑Ni=1 wi

Finally a likelihood approximation:

p̂(y1:T) = p̂(y1)

T∏t=2

p̂(yt|y1:t−1)

Put back θ in the notation so to obtain:

approximate maximum likelihood

θmle = argmaxθp̂(y1:T ; θ)

exact Bayesian inference by using p̂(y1:T |θ) insideMetropolis-Hastings.why exact?? let’s check it after the example...

4Gordon, Salmond and Smith. IEEE Proceedings F. 140(2) 1993.Umberto Picchini (umberto@maths.lth.se)

Back to the nonlinear SSM example

We can now comment on how we obtained previously shown results(reproposed here). We used the bootstrap filter with N = 500 particles andR = 10, 000 MCMC iterations.

forward propagation: xt+1 = 0.5xt + 25 xt(1+x2

t )+ 8 cos(1.2t) + vt+1

wit+1 = N(0.05(xi

t+1)2, r)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1.5Trace plot of p(q | y

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

3Trace plot of p(r | y

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

20Empirical PDF of p(q | y

1:T) after burnin

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

3Empirical PDF of p(r | y

1:T) after burnin

Numerical issues with particle degeneracy

When coding your algorithms try to consider the following beforenormalizing weights

code unnormalised weights on log-scale: e.g. whenwi

t = N(yt|xit) the exp() in the Gaussian pdf will likely produce

an underflow (wit = 0) for xi

t far from yt.Solution: reason in terms of logw instead of w.

However afterwards we necessarily have to go back tow:=exp(logw) then normalize and the above might still notbe enough.Solution: subtract the maximum (log)weight from each(log)weight, e.g. set logw:=logw-max(logw). This istotally ok, the importance of particles is unaffected, weights areonly scaled by the same constant c=max(logw).

Safe (log)likelihood computation

As usual, better compute the (approximate) loglikelihood instead ofthe likelihood:

log p̂(y1:T |θ) =

T∑t=1

(− log N + log

N∑i=1

)At time t set ct = max{log wi

t, i = 1, ..., N} and setwi∗

t := exp(log wit − ct)

Once all the wi∗t are available compute

log∑N

i=1 wit = ct + log

∑Ni=1 wi∗

(the latter follows from wit = wi∗

t exp{ct} which you don’t need toevaluate)

Incredible result

Quite astonishingly Andrieu and Roberts5 proved that using an unbiasedestimate of the likelihood function into the MCMC routine is sufficient toobtain exact Bayesian inference!

That is using the acceptance ratio

A =p̂(y1:T |θ

p̂(y1:T |θ)× π(θ

π(θ)× q(θ|θ∗)

q(θ∗|θ)

will return a Markov chain with stationary distribution π(θ|y1:T) regardlessthe finite number N of particles used to approximate the likelihood!.

The good news is that E(p̂(y1:T |θ)) = p(y1:T |θ) with p̂(y1:T |θ) obtained viaSMC.

5Andrieu and Roberts (2009), Annals of Statistics, 37(2) 697–725.Umberto Picchini (umberto@maths.lth.se)

The previous result will be considered, in my opinion, one of the mostimportant statistical results of the XXI century.

In fact, it offers an “exact-approximate” approach, where because ofcomputing limitations we can only produce N <∞ particles, while still bereassured to obtain exact (Bayesian) inference under minor assumptions.

But let’s give a rapid (technically informal) look at why it works.

Key result: unbiasedness (del Moral 20046)

We have that

E(p̂(y1:T |θ)) =

∫p̂(y1:T |θ, ξ)p(ξ)dξ = p(y1:T |θ)

with ξ ∼ p(ξ) vector of all random variates generated during SMC (both topropagate forward the state and to perform particles resampling).

6Easier to look at Pitt, Silva, Giordani, Kohn. J. Econometrics 171, 2012.Umberto Picchini (umberto@maths.lth.se)

To prove the exactness of the approach we look at the (easier and lessgeneral) argument in sec. 2.2 of Pitt, Silva, Giordani, Kohn. J.Econometrics 171, 2012.

To simplify the notation take y := y1:T .

π̂(θ, ξ|y) approximate joint posterior of (θ, ξ) obtained via SMC

π̂(θ, ξ|y) =p̂(y|θ, ξ)p(ξ)π(θ)

(notice ξ and θ are assumed a-priori independent)

Notice we put p(y) not p̂(y) at the denominator: this followsfrom the unbiasedeness assumption as we obtain∫ ∫

p̂(y|θ, ξ)p(ξ)π(θ)dξdθ =∫π(θ){

∫p̂(y|θ, ξ)p(ξ)dξ}dθ =∫

π(θ)p(y|θ)dθ = p(y).

The exact (unavailable) posterior of θ is

π(θ|y) =p(y|θ)π(θ)

therefore the marginal likelihood (evidence) is

p(y) =p(y|θ)π(θ)π(θ|y)

π̂(θ, ξ|y) =p̂(y|θ, ξ)p(ξ)π(θ)

=π(θ|y)p̂(y|θ, ξ)p(ξ)��π(θ)

p(y|θ)��π(θ)

Now, we know that applying an MCMC targeting π̂(θ, ξ|y) thendiscarding the output pertaining to ξ corresponds to integrate-out ξfrom the posterior∫

π̂(θ, ξ|y)dξ =π(θ|y)p(y|θ)

∫p̂(y|θ, ξ)p(ξ)dξ︸︷︷︸E(p̂(y|θ))=p(y|θ)

= π(θ|y)

We are thus performing a pseudo-marginal approach: “marginal”because we disregard ξ; pseudo because we use p̂(·) not p(·).

Therefore we proved that, using MCMC on an (artificially)augmented posterior, then discard from the output all the randomvariates created during SMC returns exact Bayesian inference.

Notice that discarding the ξ is something that we naturally do inMetropolis-Hastings hence nothing strange is happening here. The ξare just instrumental, uninteresting, variates independent of θ andindependent of {Xt}.

One more example

Let’s study the stochastic Ricker model.{yt ∼ Pois(φNt)

Nt = r · Nt−1e−Nt−1+et , et ∼ N(0,σ2)

Nt is the size of a population at t

r is the growth rate

φ is a scale parameter

et environmental noise

So this example shows a observational model for yt given by adiscrete distribution.

Classic Kalman based filtering can’t accommodate discreteness.

We can use the R package pomp to fit this model.

pomp supports a number of inference methods of interest to us:

particle marginal methods

iterated filtering

approximate Bayesian computation (ABC)

synthetic likelihoods

...and more.

Essentially a user who’s interested in fitting a SSM can benefit from awell tested platform, and experiment with a number of possibilities.

I have never used it used it until 4 days ago (!). But now I have aworking example (also as an optional exercise!).

Here are 50 observations from the Ricker model at times (1, 2, ..., 50)generated with (r,σ,φ) = (44.7, 0.3, 10) and starting size N0 = 7 att = 0.

0 10 20 30 40 50

myobservedricker

You can also notice the path of the (latent) process Nt as well as the(not so interesting) realizations of et.

The pomp pmcmc function runs a pseudo-marginal MCMC. Assumewe are only interested in (r,φ) and keep σ = 0.3 constant (“known”).Here we use 500 particles with 5,000 MCMC iterations. We start at(r,φ) = (7.4, 5). We assumed flat (improper) priors.

0 1000 2000 3000 4000 5000

PMCMC iteration

0 1000 2000 3000 4000 5000

PMCMC iteration

PMCMC convergence diagnostics

We used a non-adaptive Gaussian random walk (variances are keptfixed). You might get better results with an adaptive version.

A topic set as an (optional) exercise is to have a thought at why themethod fails at estimating parameters of a nearly deterministic(smaller σ) stochastic Ricker model.

Furthermore, for Nt deterministic (σ = 0), where particle filters arenot applicable (nor needed), exact likelihood calculation is alsochallenging.

A great coverage of this issue is in Fasiolo et al. (2015)arXiv:1411.4564 comparing particle marginal methods, approximateBayesian computation, iterated filtering and more. Very muchrecommended read.

Conclusions

We have outlined a powerful methodology for exact Bayesianinference, regardless the number of particles N

a too small N will have a negative impact on chain mixing→many rejections, sticky chain.

the methodology is perfectly suited for state-space modelling.

the methodology is “plug-and-play” as besides knowing theobservations density p(yt|xt) nothing else is required.

in fact we only need the ability to forward simulate xt given xt−1which can be performed numerically from the state model.

the above implies that knowledge of the transition densityexpression p(xt|xt−1) is not required. BINGO!

Software

I’m not very much updated on all available software for parameterinference via SMC (tend to write my code from scratch). Somepossibilities are:

LiBBi (C++ template library)

Biips (C++ with interfaces to MATLAB/R)

demo("PMCMC") in the R package smfsb.

R package pomp

accompanying MATLAB code for the book by S. Särkkä. Codeavailable here.

the MATLAB code for the presented example is available at myGithub page.

Important issues

How to tune the number of particles?

Doucet, Pitt, and Kohn. Efficient implementation of Markovchain Monte Carlo when using an unbiased likelihood estimator.arXiv:1210.1871 (2012).

Pitt, dos Santos Silva, Giordani and Kohn. On some properties ofMarkov chain Monte Carlo simulation methods based on theparticle filter. Journal of Econometrics 171, no. 2 (2012):134-151.

Sherlock, Thiery, Roberts and Rosenthal. On the efficiency ofpseudo-marginal random walk Metropolis algorithms.arXiv:1309.7209 (2013).

Important issues

Comparison of resampling schemes.

Douc, Cappe and Moulines. Comparison of resampling schemesfor particle filtering.http://biblio.telecom-paristech.fr/cgi-bin/download.cgi?id=5755

Hol, Schön and Gustaffson. On resampling algorithms forparticle filters, http://people.isy.liu.se/rt/schon/Publications/HolSG2006.pdf

Appendix

Multinomial resampling

This is the simplest to explain (not the most efficient though) resamplingscheme.

At time t we wish to sample from a population of weighted particles(xi

t, w̃it, i = 1, ..., N). What we actually do is to sample N times particle

indeces with replacement from the population (i, w̃it, i = 1, ..., N). This will

be a sample of size N from a multinomial distribution.

Pick at particle from the “urn”, the larger its probability w̃it the more likely it

will be picked. Record its index i and put it back in the urn. Repeat for atotal of N times.

To code the sampling procedure we just need to recall the inverse transformmethod.For a generic random variable X, let FX be an invertible cdf. We can samplean x from FX using x := F−1

X (u), with u ∼ U(0, 1).

For example let’s start from a simple example of multinomialdistribution, the Bernoulli distribution.

X ∈ {0, 1} with p = P(X = 1), 1 − p = P(X = 0). Then

FX(x) =

0 x < 01 − p 0 6 x < 11 x > 1

Draw the “stair” represented by the plot of FX . Generate a u ∼ U(0, 1)and “hit the stair’s steps”. If 0 < u 6 1 − p then set x := 0 and ifu > 1 − p set x := 1.

For the multinomial case it is a simple generalization. Drop time t andset w̃i = pi. FX is a stair with N steps. Shoot a u ∼ U(0, 1) and returnindex i

i := min{

i ′ ∈ {1, ..., N}; (i ′∑

pi) − u > 0}

Cool reads on Bayesian methods (titles are linked)

You have an engineeristic/signal processing background: checkS. Särkkä “Bayesian Filtering and Smoothing” (free PDF fromthe author!)

You are a data-scientist: check K. Murphy “Machine Learning: aprobabilistic perspective”.

You are a theoretical statistician: C. Robert “The BayesianChoice”.

You are interested in bioinformatics/systems biology: check D.Wilkinson “Stochastic Modelling for Systems Biology, 2ed.”.

You are interested in inference for SDEs with applications to lifesciences: check the book by Wilkinson above and C. Fuchs“Inference for diffusion processes”.

Cool reads on Bayesian methods (titles are linked)

You are a computational statistician: check “Handbook ofMCMC”. Older (but excellent) titles are: J. Liu “Monte Carlostrategies in Scientific Computing” and Casella-Robert “MonteCarlo Statistical Methods”.

You want a practical hands-on and (almost) maths freeintroduction: check “The BUGS Book” and “Doing BayesianData Analysis”.

Particle MCMC methods for parameter inference in state space models

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