Created: July 14, 2021
Modified: July 14, 2021

nested SMC

This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.

Christian Naesseth, Fredrik Lindsten, Thomas Schon (2015): http://proceedings.mlr.press/v37/naesseth15.html
The main idea:
- In an SMC sampler, we'd like to propose from the actual target distribution at each step.
- We presumably can't do that directly, but we can run an 'inner' sampler that generates (weighted) proposals.
- It turns out to be valid to just use those.
The best thing about this paper is the definition of a 'properly weighted sample'. A pair of random variables $(X, W)$ following a joint measure $\gamma$ is 'properly weighted' for a distribution $\pi(x)$ if $E_\gamma[W f(X)] = E_{\pi}[f(x)]$ . This seems very useful for analyzing importance-sampling algorithms.
For a given X it's sufficient to have an unbiased estimate $\tilde W$ of the weight W. You can see this easily by staging the expectation: $E[ f(X) \tilde W] = E[f(X) E[\tilde W | f(X)]] = E[f(X) W]$ .
The idea is to use an inner sampler to construct a properly weighted sample for the proposal distribution, which we set to the (local) target distribution. This has juice when that distribution is itself structured. For example, on a 2D grid, we could run SMC along the sequence of vector-valued row distributions, and use an inner SMC across columns to propose from the current target distribution on rows. This will work a lot better than sampling a giant vector from an unstructured proposal, since it can use structure to bypass the curse of dimensionality.

nested SMC

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