## optimal mixture weights in multiple importance sampling

**M**ultiple importance sampling is back!!! I am always interested in this improvement upon regular importance sampling, even or especially after publishing a recent paper about our AMIS (for adaptive multiple importance sampling) algorithm, so I was quite eager to see what was in Hera He’s and Art Owen’s newly arXived paper. The paper is definitely exciting and set me on a new set of importance sampling improvements and experiments…

Some of the most interesting developments in the paper are that, (i) when using a collection of importance functions q_{i} with the same target p, every ratio q_{i}/p is a *control variate* function with expectation 1 [assuming each of the q_{i}‘s has a support smaller than the support of p]; (ii) the weights of a mixture of the q_{i}‘s can be chosen in an optimal way towards minimising the variance for a certain integrand; (iii) multiple importance sampling incorporates quite naturally stratified sampling, i.e. the q_{i}‘s may have disjoint supports; )iv) control variates contribute little, esp. when compared with the optimisation over the weights [which does not surprise me that much, given that the control variates have little correlation with the integrands]; (v) Veach’s (1997) seminal PhD thesis remains a driving force behind those results [and in getting Eric Veach an Academy Oscar in 2014!].

One extension that I would find of the uttermost interest deals with unscaled densities, both for p and the q_{i}‘s. In that case, the weights do not even sum up to a know value and I wonder at how much more difficult it is to analyse this realistic case. And unscaled densities led me to imagine using geometric mixtures instead. Or even harmonic mixtures! (Maybe not.)

Another one is more in tune with our adaptive multiple mixture paper. The paper works with *regret*, but one could also work with *remorse*! Besides the pun, this means that one could adapt the weights along iterations and even possible design new importance functions from the past outcome, i.e., be adaptive once again. He and Owen suggest mixing their approach with our adaptive sequential Monte Carlo model.

December 12, 2014 at 7:23 pm

Hi Christian,

Thanks for the mention. Nicolas Chopin earlier asked me about the asymptotic variance for the unnormalized case; from a brief look, I suspected that the mixture problem is convex. I probably was thinking of unnormalized p and normalized q’s because we have so much more control over q than p.

Geometric mixtures are intriguing if we can figure out a good way to sample from them. Harmonic ones put me in mind of Radford Neal’s comment on harmonic means for marginal likelihoods, but maybe this case would work out differently.

My earlier paper with Zhou had an example where the control variates helped a lot more than the ones in this paper do.

Darren: we did not mean to leave anything out. Please send a note to my Stanford email address describing the recent advances that you find most interesting.

-Art

December 12, 2014 at 2:27 am

It is good to see that Art is continuing to work in this area! and as many know on QMC methods also. Great that Cappé et al (2008) gets a mention too :-) but it is a little disappointing that they don’t talk about recent advances in the literature that much … (i.e time seems to have stopped in 2004!). I am just being nitpicky though.