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Is there a clear and precise explanation of why minimising the variance of the weights in SIS with respect to a proposal ensures that the samples generated from the empirical distribution induced by the normalised weights will be closer to the posterior/target distribution? Also is there a reference which proves something about the distribution (mass function) induced by SIS in relation to the target/posterior density? Like random variables sampled from SIS with increasing particle counts converge in distribution to random variables sampled from the target? --Apologies for abuse of notation, or any mis-use of terminology. It just seems like most results for SIS / SIR are provided in terms consistency with respect to arbitrary test functions, instead of the actual samples generated under the normalised weights. Most of my understanding stems from~https://www.cs.ubc.ca/~arnaud/doucet_johansen_tutorialPF.pdf, so if I missed something obvious here let me know, and thank you in advance!

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Is there a clear and precise explanation of why minimising the variance of the weights in SIS with respect to a proposal ensures that the samples generated from the empirical distribution induced by the normalised weights will be closer to the posterior/target distribution?

I tend to think of this problem in terms of the effective sample size (ESS). Quoting from Monte Carlo Strategies in Scientific Computing, Liu (2008) (pdf) pp. 34-36

Importance sampling suggests estimating $\mu = E_\pi\{h(x)\}$ by first generating independent samples $x^{(1)}, \dots , x^{(m)}$ from an easy-to-sample trial distribution, $g( \thinspace )$, and then correcting the bias by incorporating the importance weight $w^{(j)} \propto \pi(x^{(j)})/g(x^{(j)})$

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A useful "rule of thumb" is to use the effective sample size (ESS) to measure how different the trial distribution is from the target distribution. Suppose $m$ independent samples are generated from $g(x)$; then, the ESS of this method is defined as $$ESS(m) = \frac{m}{1 + \textrm{var}_g(w(x))}$$

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This can be interpreted as that the $m$ weighted samples is worth of $m / \{1 + \textrm{var}_g [w(x)]\}$ i.i.d. samples drawn from the target distribution.

There's much more detail in the book.

Also is there a reference which proves something about the distribution (mass function) induced by SIS in relation to the target/posterior density? Like random variables sampled from SIS with increasing particle counts converge in distribution to random variables sampled from the target?

If you're referring to the SIS algorithm on p. 11 of Doucet and Johanson's tutorial, with no resampling, then the distribution of the particles will just be sampled from $q_n(x_{1:n})$, regardless of the target/posterior distribution. SIS with a massive number of particles might still be useful in estimating an expectation in cases where the error decreases with the number of particles.

Once resampling is introduced to make SMC then you get the results given in Section 3.6 of the tutorial

Comparing (26) to (37), we see that the SMC variance expression has replaced the importance distribution $q_n(x_{1:n})$ in the SIS variance with the importance distributions $\pi_{k−1} (x_{1:k−1}) q_k (x_k| x_{1:k−1})$ obtained after the resampling step at time $k-1$.

The authors give two citations for the results about the density in section 3.6.

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