In these slides I presented the SMC^2 method (see the article here: http://arxiv.org/abs/1101.1528 ) to an audience of marine biogeochemistry people, emphasizing on the model evidence estimation aspect.

1. SMC2
: an algorithm for sequential analysis of
state-space models
Nicolas Chopin (ENSAE-CREST, Paris),
Pierre Jacob (National University of Singapore)
& Omiros Papaspiliopoulos (Univ. Pompeu Fabra, Barcelona)
BGC DA Symposium – May 2013
Pierre Jacob SMC2
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2. Outline
1 Monte Carlo for state-space models
2 SMC2
3 Complexity
4 Applicability for BGC models
Pierre Jacob SMC2
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3. Outline
1 Monte Carlo for state-space models
2 SMC2
3 Complexity
4 Applicability for BGC models
Pierre Jacob SMC2
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4. State-space models
y2
X2X0
y1
X1
...
... yT
XT
θ
Figure: Graph representation of a general state-space model.
Hidden process: initial distribution µθ, transition fθ.
Observations conditional upon the hidden process, from gθ.
Prior p on the parameter θ ∈ Θ.
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5. State-space models
Target distributions
Particle MCMC methods provide N-samples from:
p(θ, x1:T |y1:T )
SMC2 provides N-samples for all t ∈ [1, T] from:
p(θ, x1:t|y1:t)
Exact approximation
Convergence of the sample distribution to the distribution of
interest.
Pierre Jacob SMC2
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6. State-space models
Challenge of the model evidence
Bayes rule yields
p(θ | y1:t) =
p(θ)p(y1:t | θ)
p(y1:t)
where the evidence is
p(y1:t) =
Θ
p(θ)p(y1:t | θ)dθ
=
Θ
p(θ)
Xt+1
p(y1:t | x0:t, θ)p(x0:t | θ) dθ
=
Θ
p(θ)
Xt+1
µθ(x0)
t
k=1
fθ(xk | xk−1)gθ(yk | xk) dθ
⇒ integral of dimension dim(Θ) × dim(X) × (t + 1). . . !
Pierre Jacob SMC2
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7. Sequential Monte Carlo for filtering
If we were interested in pθ(x1:T |y1:T ), for a given θ. . .
Particle Filter
Input:
model (satisfying the requirements), dataset y1:T ,
number of particles Nx, possibly other parameters.
Output:
Nx-samples from p(x1:t | y1:t, θ) for all t ∈ [1, T],
likelihood estimates ˆZNx
t (θ) ≈ p(y1:t | θ) for all t ∈ [1, T].
Pierre Jacob SMC2
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8. Particle Markov Chain Monte Carlo
If we are now interested in p(x1:T , θ|y1:T ). . .
Particle Marginal Metropolis–Hastings
Input:
model (satisfying the requirements), dataset y1:T ,
number of iterations M, number of particles Nx, possibly
other parameters.
Output:
M-samples from p(θ, x1:T | y1:T ),
evidence estimates could be obtained based on the Markov
chain (Chib’s method, thermodynamics integration. . . ).
Pierre Jacob SMC2
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9. Sequential Monte Carlo Samplers
Similar to Particle Filter, in the context of Bayesian inference on
static (non-dynamical) problems:
p(θ|y1:T )
(Neal 2001, Chopin 2004, Del Moral, Doucet & Jasra 2006. . . )
SMC Sampler
Input:
model (satisfying the requirements), dataset y1:T ,
number of particles Nθ, possibly other parameters.
Output:
Nθ-samples from p(θ | y1:t) for all t ∈ [1, T],
evidence estimates ˆENθ
t ≈ p(y1:t) for all t ∈ [1, T].
Pierre Jacob SMC2
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10. Outline
1 Monte Carlo for state-space models
2 SMC2
3 Complexity
4 Applicability for BGC models
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11. Motivation
A valid SMC sampler for state-space models.
Foreseen benefits compared to PMCMC
Sample sequentially from
p(θ, x1|y1), p(θ, x1:2|y1:2), . . . , p(θ, x1:T |y1:T ),
to estimate the model evidence.
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12. Valid SMC sampler for SSM
Plugging estimates of the incremental likelihood
Similarly to PMCMC replacing likelihoods p(y1:T | θ) by estimates,
we can replace incremental likelihoods p(yt|y1:t−1, θ) by estimates.
θ-particles and x-particles
We associate Nx x-particles to each of the Nθ θ-particles.
These provide estimates of the incremental likelihoods for
each θ-particle.
Whenever we need to rejuvenate θ-particles, PMCMC steps.
Pierre Jacob SMC2
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13. Summary of the vanilla SMC2
sampler
Input:
model (satisfying the requirements), dataset y1:T ,
numbers of particles Nθ, Nx, other algorithmic parameters.
Output:
Nθ-samples from p(θ | y1:t) for all t ∈ [1, T],
Nx-samples from p(x1:t | y1:t, θ)
for all t ∈ [1, T] and for each θ in the Nθ-sample,
evidence estimates ˆENθ
t ≈ p(y1:t) for all t ∈ [1, T].
Sequential but not online!
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14. SMC2
y2
X2X0
y1
X1
...
... yT
XT
Θ
Pierre Jacob SMC2
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15. SMC2
y2
X2X0
y1
X1
...
... yT
XT
Θ
Pierre Jacob SMC2
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16. SMC2
y2
X2X0
y1
X1
...
... yT
XT
Θ
Pierre Jacob SMC2
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17. SMC2
y2
X2X0
y1
X1
...
... yT
XT
Θ
Pierre Jacob SMC2
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18. SMC2
y2
X2X0
y1
X1
...
... yT
XT
Θ
Pierre Jacob SMC2
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19. SMC2
y2
X2X0
y1
X1
...
... yT
XT
Θ
Pierre Jacob SMC2
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20. SMC2
y2
X2X0
y1
X1
...
... yT
XT
Θ
Pierre Jacob SMC2
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21. Outline
1 Monte Carlo for state-space models
2 SMC2
3 Complexity
4 Applicability for BGC models
Pierre Jacob SMC2
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22. Algorithmic complexity
Cost if resample move at each time step
A single move step at time t costs O (tNxNθ).
If move at every time, the total cost (up to t) becomes
O t2NxNθ .
If e.g. Nx increased linearly with t, the total cost would rise to
O t3Nθ .
With adaptive resampling. . .
. . . it is only essentially O t2Nθ . Why is that?
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23. Algorithmic complexity
iteration
computingtime(squarerootscale)
2500
10000
22500
40000
1000 2000 3000 4000 5000
Figure:
√
computing time vs iteration
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24. Algorithmic complexity
Computational effort
Most of the effort usually lies in drawing from the transition
fθ, or in evaluating the measurement gθ.
This can be done for all particles in parallel.
The remaining task is the resampling step.
Rethinking resampling in the particle filter on GPUs,
Lawrence Murray, PJ & Anthony Lee (submitted).
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25. Memory requirement
Vanilla version of SMC2
Only the latest θ-particles and the latest generation of x-particles
have to be stored, hence the cost is O(NθNx).
General version of SMC2 using particle trajectories
Naive cost of storing all the particles (x1:Nx
1:T ) for each θ-particle:
O(NθTNx).
More accurate cost of storing only the surviving trajectories:
O(NθT + NθNx log Nx).
(work in progress with Lawrence Murray & Sylvain Rubenthaler)
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26. Outline
1 Monte Carlo for state-space models
2 SMC2
3 Complexity
4 Applicability for BGC models
Pierre Jacob SMC2
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27. SMC2
on the PZ model
Phytoplankton–Zooplankton
log Y ∼ N(log P, τ2
)
αt ∼ N(µ, σ2
)
dP
dt
= αtP − cPZ
dZ
dt
= ecPZ − mlZ − mqZ2
θ = (µ, σ2
, τ2
)
or possibly θ = (µ, σ2
, τ2
, c, e, ml, mq)
L. Murray, E. Jones, J. Parslow (2012). On collapsed state-space
models and the particle marginal Metropolis-Hastings sampler.
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28. SMC2
on the PZ model
−1
0
1
2
3
0 20 40 60 80 100
Time
P
(a) Phytoplankton 90% credible interval of filtering distributions
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 20 40 60 80 100
Time
Z
(b) Zooplankton 90% credible interval of filtering distributions
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29. SMC2
on the PZ model
0
5
10
0 25 50 75 100
Time
Observations
(c) Simulated dataset
−600
−400
−200
0
0 25 50 75 100
Time
Logevidence
Model PZ PZW
(d) Log evidence p(y1:t | M) against time
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30. SMC2
on the PZ model
0
5
10
0 25 50 75 100
Time
Observations
(e) Simulated dataset
−100
0
100
200
0 25 50 75 100
Time
LogBayesfactor
(f) Log Bayes factor log(p(y1:t | PZW)/p(y1:t | PZ)) against time
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31. Ease of use
SMC2 is in the package LibBi.
libbi sample --target posterior --sampler smc2
--model-file PZ.bi --end-time 100.0 --nparticles 128
--nsamples 256. . .
and then enjoy the magic
--enable-cuda --enable-mpi. . .
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32. Remaining uncertainty
Error coming from numerical solvers for differential equations
(controlled but not taken into account in the results).
Pseudo-random number generators, assumed perfectly random
(recent work by Iain Murray and LLoyd Elliott).
Tractable measurement distribution requirement
(ABC methods relax this assumption at the cost of a bias).
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33. Towards automatic and scalable algorithms
Automatic calibration of Nx, Nθ, the proposal distribution.
Parallelization of the resampling step.
Scale to high-dimension problems using gradient estimates.
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34. Conclusion
The SMC2 framework allows to obtain various quantities of
interest for sequential analysis in state-space models.
It fits in the PMCMC framework introduced by
Andrieu, Doucet and Holenstein (2010).
SMC2 is already implemented in LibBi.
Sequential but not online.
Not practical for large spatial state-space models yet.
Pierre Jacob SMC2
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35. Bibliography
Main references:
Particle Markov Chain Monte Carlo methods, C. Andrieu, A.
Doucet, R. Holenstein
Sequential Monte Carlo samplers, P. Del Moral, A. Doucet, A.
Jasra
SMC2: an efficient algorithm for sequential
analysis of state-space models, N. Chopin, P. Jacob, O.
Papaspiliopoulos
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