A new survey on Bayesian inference with intractable normalising constants was posted on arXiv yesterday by Jaewoo Park and Murali Haran. A rather massive work of 58 pages, almost handy for a short course on the topic! In particular, it goes through the most common MCMC methods with a detailed description, followed by comments on components to be calibrated and the potential theoretical backup. This includes for instance the method of Liang et al. (2016) that I reviewed a few months ago. As well as the Wang-Landau technique we proposed with Yves Atchadé and Nicolas Lartillot. And the noisy MCMC of Alquier et al. (2016), also reviewed a few months ago. (The Russian Roulette solution is only mentioned very briefly as” computationally very expensive”. But still used in some illustrations. The whole area of pseudo-marginal MCMC is also missing from the picture.)
“…auxiliary variable approaches tend to be more efficient than likelihood approximation approaches, though efficiencies vary quite a bit…”
The authors distinguish between MCMC methods where the normalizing constant is approximated and those where it is omitted by an auxiliary representation. The survey also distinguishes between asymptotically exact and asymptotically inexact solutions. For instance, using a finite number of MCMC steps instead of the associated target results in an asymptotically inexact method. The question that remains open is what to do with the output, i.e., whether or not there is a way to correct for this error. In the illustration for the Ising model, the double Metropolis-Hastings version of Liang et al. (2010) achieves for instance massive computational gains, but also exhibits a persistent bias that would go undetected were it the sole method implemented. This aspect of approximate inference is not really explored in the paper, but constitutes a major issue for modern statistics (and machine learning as well, when inference is taken into account.)
In conclusion, this survey provides a serious exploration of recent MCMC methods. It begs for a second part involving particle filters, which have often proven to be faster and more efficient than MCMC methods, at least in state space models. In that regard, Nicolas Chopin and James Ridgway examined further techniques when calling to leave the Pima Indians [dataset] alone.