Another paper on astro ph today about parallel tempering for exploring the gravitational wave signal posterior, this one by Farr et al. But, I’m going to overlook their missed trick on the possible use of RLR/biased sampling for evidence estimation during tempering, which would genuinely allow them to compare against NS and MultiNest (which are doing both inference *and* marginal likelihood calculation). Instead I’m going to share an observation about tempering. While most often we think of tempering in terms of a softening of the posterior via an inverse temperature parameter there is another way to soften the posterior that can also be thought of in the same sense: tempering via partial data likelihoods. The method is exemplified in the SMC case by Chopin’s 2002 paper (he calls these “partial posteriors”), and it can easily be used to assist mixing between modes across parallel MCMC chains (and again for marginal likelihood estimation).

Perhaps its most important use remains that of exploring “big data” posteriors in the SMC case, but I’ve found it rather useful for my “small data” problems too!

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“tempering via partial data likelihoods”

The main advantage there is that while exploring the more “priory” distributions, your likelihood evaluations might be faster. The disadvantage is there’s nothing special about the sequence of distributions, unlike in Nested Sampling.

I wouldn’t go as far as to say there’s nothing special about them: for iid data introduced in equal sized increments the proportional reduction in posterior ‘volume’ goes also in equal increments according to the expected information (which solves the problem of choosing a temperature sequence in, e.g., MC^3)!