**I**n the latest September issue of JASA I received a few days ago, I spotted a review paper by Jaewoo Park & Murali Haran on intractable normalising constants Z(θ). There have been many proposals for solving this problem as well as several surveys, some conferences and even a book. The current survey focus on MCMC solutions, from auxiliary variable approaches to likelihood approximation algorithms (albeit without ABC entries, even though the 2006 auxiliary variable solutions of Møller et al. et of Murray et al. do simulate pseudo-observations and hence…). This includes the MCMC approximations to auxiliary sampling proposed by Faming Liang and co-authors across several papers. And the paper Yves Atchadé, Nicolas Lartillot and I wrote ten years ago on an adaptive MCMC targeting Z(θ) and using stochastic approximation à la Wang-Landau. Park & Haran stress the relevance of using sufficient statistics in this approach towards fighting computational costs, which makes me wonder if an ABC version could be envisioned. The paper also includes pseudo-marginal techniques like Russian Roulette (once spelled Roullette) and noisy MCMC as proposed in Alquier et al. (2016). These methods are compared on three examples: (1) the Ising model, (2) a social network model, the Florentine business dataset used in our original paper, and a larger one where most methods prove too costly, and (3) an attraction-repulsion point process model. In conclusion, an interesting survey, taking care to spell out the calibration requirements and the theoretical validation, if of course depending on the chosen benchmarks.

## Archive for noisy MCMC

## Bayesian inference with intractable normalizing functions

Posted in Books, Statistics with tags adaptive MCMC methods, American Statistical Association, auxiliary variable, benchmark, doubly intractable problems, importance sampling, Ising model, JASA, MCMC algorithms, noisy MCMC, normalising constant, Russian roulette on December 13, 2018 by xi'an## approximations of Markov Chains [another garden of forking paths]

Posted in Books, Mountains, pictures, Statistics, University life with tags approximate MCMC, computational budget, Doeblin's condition, Markov chain Monte Carlo, MCMskv, minimaxity, Monte Carlo Statistical Methods, noisy MCMC, total variation, uniform ergodicity, uniform geometric ergodicity on March 15, 2016 by xi'an**J**ames Johndrow and co-authors from Duke wrote a paper on approximate MCMC that was arXived last August and that I missed. David Dunson‘s talk at MCMski made me aware of it. The paper studies the impact of replacing a valid kernel with a close approximation. Which is a central issue for many usages of MCMC in complex models, as exemplified by the large number of talks on that topic at MCMski.

“All of our bounds improve with the MCMC sample path length at the expected rate in t.”

A major constraint in the paper is Doeblin’s condition, which implies uniform geometric ergodicity. Not only it is a constraint on the Markov kernel but it is also one for the Markov operator in that it may prove impossible to… prove. The second constraint is that the approximate Markov kernel is close enough to the original, which sounds reasonable. Even though one can always worry that the total variation norm is too weak a norm to mean much. For instance, I presume with some confidence that this does not prevent the approximate Markov kernel from not being ergodic, e.g., not irreducible, not absolutely continuous wrt the target, null recurrent or transient. Actually, the assumption is stronger in that there exists a *collection* of approximations for all small enough values ε of the total variation distance. (*Small enough* meaning ε is much smaller than the complement α to 1 of the one step distance between the Markov kernel and the target. With poor kernels, the approximation must thus be *very* good.) This is less realistic than assuming the availability of one single approximation associated with an existing but undetermined distance ε. (For instance, the three examples of Section 3 in the paper show the existence of approximations achieving a certain distance ε, without providing a constructive determination of such approximations.) Under those assumptions, the average of the sequence of Markov moves according to the approximate kernel converges to the target in total variation (and in expectation for bounded functions). With sharp bounds on those distances. I am still a bit worried at the absence of conditions for the approximation to be ergodic.

“…for relatively short path lengths, there should exist a range of values for which aMCMC offers better performance in the compminimax sense.”

The paper also includes computational cost into the picture. Introducing the notion of compminimax error, which is the smallest (total variation) distance among all approximations at a given computational budget. Quite an interesting, innovative, and relevant notion that may however end up being too formal for practical use. And that does not include the time required to construct and calibrate the approximations.

## at CIRM [#2]

Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags approximate MCMC, Cap Morgiou, CIRM, clustering, EM algorithm, functional programming languages, improper priors, Jeffreys-Lindley paradox, La Grande Candelle, Luminy, Marseille, Monte Carlo Statistical Methods, noisy MCMC, pGibbs, pMCMC, q-vague convergence, trail running, variable selection, yeast on March 2, 2016 by xi'an**S**ylvia Richardson gave a great talk yesterday on clustering applied to variable selection, which first raised [in me] a usual worry of the lack of background model for clustering. But the way she used this notion meant there was an infinite Dirichlet process mixture model behind. This is quite novel [at least for me!] in that it addresses the covariates and not the observations themselves. I still wonder at the meaning of the cluster as, if I understood properly, the dependent variable is not involved in the clustering. Check her R package PReMiuM for a practical implementation of the approach. Later, Adeline Samson showed us the results of using pMCM versus particle Gibbs for diffusion processes where (a) pMCMC was behaving much worse than particle Gibbs and (b) EM required very few particles and Metropolis-Hastings steps to achieve convergence, when compared with posterior approximations.

Today Pierre Druilhet explained to the audience of the summer school his measure theoretic approach [I discussed a while ago] to the limit of proper priors via *q-vague* convergence, with the paradoxical phenomenon that a Be(n⁻¹,n⁻¹) converges to a sum of two Dirac masses when the parameter space is [0,1] but to Haldane’s prior when the space is (0,1)! He also explained why the Jeffreys-Lindley paradox vanishes when considering different measures [with an illustration that came from my Statistica Sinica 1993 paper]. Pierre concluded with the above opposition between two Bayesian paradigms, a [sort of] tale of two sigma [fields]! Not that I necessarily agree with the first paradigm that priors are supposed to have generated the actual parameter. If only because it mechanistically excludes all improper priors…

Darren Wilkinson talked about yeast, which is orders of magnitude more exciting than it sounds, because this is Bayesian big data analysis in action! With significant (and hence impressive) results based on stochastic dynamic models. And massive variable selection techniques. Scala, Haskell, Frege, OCaml were [functional] languages he mentioned that I had never heard of before! And Daniel Rudolf concluded the [intense] second day of this Bayesian week at CIRM with a description of his convergence results for (rather controlled) noisy MCMC algorithms.