Discussion: Population SMC
SMC or MCMC – is it still a question?
Rong Chen
Department of Statistics
Rutgers University
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SMC and MCMC
Misconceptions:
• SMC is used only for state-space models, HMM and other
types of dynamic systems
– A sequence of target distributions with increasing dimension
• MCMC is (the only solution) for fixed dimensional problems
– A single distribution
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We claim:
• SMC is a valid alternative to MCMC
• SMC can be a (possibly more) powerful tool in solving fixdimensional problems
• MCMC is a special case of SMC
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Change
The dynamic system framework for SMC (Liu and Chen 1998)
• A sequence of distributions π1(x1), . . . , πt(xt), . . .
• Increasing dimension xt = (xt−1, xt)
• Sequential importance sampling:
(j)
– generate xt
(j)
conditioned on xt−1
(j)
(j)
(j)
– append: xt = (xt−1, xt )
– update weight
What do we do with one single target distribution π(x)?
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Approach (I): The Growth Principle
Decompose a complex problem into a sequence of simpler problems.
• Target distribution π(x) where x = (x1, . . . , xN )
• Let xt = (x1, . . . , xt) = (xt−1, xt)
• Define a sequence of intermediate distributions πt(xt).
• Moving from πt−1(xt−1) to πt(xt) is simple.
• Moving from πt−1(xt−1) to πt(xt) is smooth.
Z
πt(xt−1, xt)dxt ≈ πt−1(xt−1)
• πN (xN ) = π(x)
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Approach (II): Fixed dimension with augmentation
• Target distribution π(x), for x ∈ Ω
• Let xt = (x1, . . . , xt) ∈ Ωt where xi ∈ Ω.
• Construct a sequence of intermediate (joint) distributions
πt(x1, . . . , xt)
on
Ωt,
often through a transition rule πt−1 → πt.
• The low-dimensional marginal distributions of the giant joint
distribution πn(x1, . . . , xn) is in some way related to the target
distribution π(x).
– e.g. πn(x1) = π(x) or πn(xn) = π(x)
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Arnaud’s approach:
• starting with a relatively trivial π1(x1), where x1 ∈ Ω
• augment: πt(x1, . . . , xt), t = 1, . . . , n.
• eventually, at t = n, the marginal distribution of the last component πn(xn) is the target distribution π(x)
(j)
• inference using the samples of the last component xn
Chris’ approach:
• starting with true π1(x1) = π(x).
• augment: πt(x1, . . . , xt) so the marginal distribution of the last
component πt(xt) is the target distribution π(x)
• inference using samples of all components.
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MCMC as a special case of SMC:
MCMC:
• Target distribution π(x) where x ∈ Ω.
• A Markov chain x1, . . . , xt, . . ., with a transition kernel k and
each xt ∈ Ω.
• x1 ∼ π1
• (x1, . . . , xt) ∼ πt(x1, . . . , xt) where
πt(xt) = πt−1(xt−1)k(xt | xt−1)
• (hopefully) after some N , the marginal distribution of the last
component xt, πt(xt) is close to π(x), for t > N .
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MCMC as a special case of SMC:
• A dynamic system through augmentation
• The sequence of intermediate distributions is built based on a
special type of kernel to ensure that the marginal distributions
of the t-th component converge to target distribution
• Trial distribution is the same as kernel (often through rejection method, as in M-H)
• Does not use importance weights but uses burn-in
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Open questions:
Caution: flexibility comes with more ways of making mistakes
• Design the sequence of intermediate distributions πt(xt)
• Construct efficient trial distributions
• Take advantage of MCMC-like moves
• Avoid the problems often encountered with MCMC algorithms.
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