SMC2 Algorithm for State Space Models

Introduction and State Space Models
Quick reminder on Sequential Monte Carlo
Particle Markov Chain Monte Carlo
SMC2

SMC2 : A sequential Monte Carlo algorithm with
particle Markov chain Monte Carlo updates

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

BISP7 – September, 2011

1
ENSAE-CREST
2
CREST & Universit´ Paris Dauphine, funded by AXA research
e
3
Universitat Pompeu Fabra
N. CHOPIN, P.E. JACOB, & O. PAPASPILIOPOULOS SMC2 1/ 16

SMC2

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:
T
ZT (θ) = p(y1:T |θ) = p(yt |y1:t−1 , θ) p(y1 |θ)
t=2

ˆN
with an unbiased estimate ZT x (θ).


SMC2

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).


SMC2

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 a
MH algorithm.


SMC2

Valid Metropolis–Hastings for SSM

Plug in estimates
ˆN
We have ZT x (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 right
likelihood.

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


SMC2

Our contribution. . .

. . . was to use the same method to get a valid SMC sampler for
state space models.
Foreseen beneﬁts
to sample more eﬃciently from the posterior distribution
p(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.


SMC2

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 likelihoods
for each θ-particle.


SMC2

Side beneﬁts

Evidence
SMC2 provides an estimate of the “evidence”:
t
p(y1:t ) = p(ys |y1:s−1 )
s=1

Automatic tuning
θ-particles are moved with adaptive particle MCMC steps,
the number of Nx particles can be dynamically increased if
need be.


SMC2

Numerical illustrations: Stochastic Volatility

2

Observations

0

−2

−4

100 200 300 400 500 600 700
Time

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


SMC2


Stochastic Volatility model
Observations (“log returns”):
1/2
yt = µ + βvt + vt t, t ∼ N (0, 1)

Hidden states: the “actual volatility” (vt ), a process that
depends on another process, the “spot volatility” (zt ).
All these processes are parameterized by θ ∈ (µ, β, ξ, ω 2 , λ).


SMC2


T = 250 T = 500 T = 750 T = 1000
8

6
Density

4

2

0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
µ

Figure: Concentration of the posterior distribution for parameter µ.


SMC2


Model comparison
For the same problem there could be various models that we want
to compare. Here:
the “basic” previous model,
a similar model with more factors (= more hidden states),
a similar model with more factors and “leverage” (= diﬀerent
likelihood function with more parameters).


SMC2


Evidence compared to the one factor model
variable
20 Multi factor without leverage
4 Multi factor with leverage
Squared observations

15

2

10

0

5

−2

100 200 300 400 500 600 700 100 200 300 400 500 600 700
Time Iterations

(a) (b)

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


SMC2

Conclusion

A powerful framework
The SMC2 framework allows to obtain various quantities of
interest, 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/.


SMC2

Bibliography

SMC2 : A sequential Monte Carlo algorithm with particle Markov
chain 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


SMC2 Algorithm for State Space Models

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