I am reading Importance Sampling and Necessary Sample Size: an Information Theory Approach.

Below is a quote from paragraph 3, section 3 of the article.

While [total variation distance] and [Hellinger distance] can be shown to be distances in P(X), [Kullback-Leibler divergence] and [χ2 divergence] are not. In particular, these latter divergences fail to be symmetric, a feature that makes them appealing for the analysis of importance sampling. Indeed, the very formulation of the method is built on an asymmetric premise (the absolute continuity of P with respect to Q). Moreover, it is well acknowledged that it is desirable that the proposal has heavier tails than the target —again an asymmetric requirement.

I do not understand how this claim (bolded quote) is justified by the two sentences following it.

I understand that importance sampling is essentially an application of the Radon-Nikodym derivative of the target with respect to the proposal, hence absolute continuity is required for the existence of such a density. However, I do not understand how these "asymmetries" are a "feature" rather than a defect.

My question is as the title reads, are there formal arguments for why an asymmetric f-divergence might be favourable to a symmetric one in analyzing importance sampling?


1 Answer 1


I finished reading the paper and can convince myself temporarily, via an example the author provided, that an asymmetric divergence can be beneficial in analyzing importance sampling.

Before jumping into the example, it is worth noting the goal of the paper is, at a high level, to derive necessary sample sizes (NSSs) for importance sampling.

In Section 5 of Importance Sampling and Necessary Sample Size: an Information Theory Approach, the author demonstrates how the total variation (TV) and Hellinger (H) distance can fail to produce a meaningful NSS. In one example, he considered a N(0, 25) target distribution, and a proposal N(0, 1) distribution.

The tails of the target distribution is heavier than the proposal's. Due to the symmetry of TV and H distances, these metrics are unable to distinguish the direction of tail discrepancies, resulting in the approximated NSS (under TV and H distances) to be less than 3 for both cases.

The benefit of asymmetric divergences can be further highlighted by reversing the setup, i.e., N(0,1) target and N(0,25) proposal. If we used TV and H distances, the approximated NSS will be the same for TV and H divergences. However, Kullback-Leibler and χ-square divergence will yield a lower NSS.

It would be interesting to see the effects of symmetry in divergences when considering sufficient sample sizes.


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