**M**y friends and coauthors Matthieu Gerber and Randal Douc have just arXived a massive paper on online approximate Bayesian learning, namely the handling of the posterior distribution on the parameters of a state-space model, which remains a challenge to this day… Starting from the iterated batch importance sampling (IBIS) algorithm of Nicolas (Chopin, 2002) which he introduced in his PhD thesis. The online (“by online we mean that the memory and computational requirement to process each observation is finite and bounded uniformly in t”) method they construct is guaranteed for the approximate posterior to converge to the (pseudo-)true value of the parameter as the sample size grows to infinity, where the sequence of approximations is a Cesaro mixture of initial approximations with Gaussian or t priors, AMIS like. (I am somewhat uncertain about the notion of a sequence of priors used in this setup. Another funny feature is the necessity to consider a fat tail t prior from time to time in this sequence!) The sequence is in turn approximated by a particle filter. The computational cost of this IBIS is roughly in O(NT), depending on the regeneration rate.

## Archive for approximate MCMC

## online approximate Bayesian learning

Posted in Statistics with tags AMIS, approximate MCMC, Bayesian learning, consistency, convergence, ibis, sequential Monte Carlo, state space model on September 25, 2020 by xi'an## is there such a thing as optimal subsampling?

Posted in Books, Statistics, University life with tags approximate MCMC, burnin, Charles Stein, convergence diagnostics, Hilbert space, Lotka-Volterra, ODEs, principal points, pseudo-marginal MCMC, reproducible kernel Hilbert space, secretary problem, Stein thinning, subsampling, thinning on June 12, 2020 by xi'an**T**his idea of optimal thinnin and burnin has been around since the early days of the MCMC revolution and did not come up with a definite answer. For instance, from a pure estimation perspective, subsampling always increases the variance of the resulting estimator. My personal approach is to ignore both burnin and thinnin and rather waste time on running several copies of the code to check for potential discrepancies and get a crude notion of the variability. And to refuse to answer to questions like is 5000 iterations long enough for burnin?

A recent arXival by Riabiz et al. readdresses the issue. In particular concerning this notion that the variance of the subsampled version is higher: this only applies to a deterministic subsampling, as opposed to an MCMC-based subsampling (although this intricacy only makes the problem harder!). I however fail to understand the argument in favour of subsampling based on storage issues (p.4), as a dynamic storage of the running mean for all quantities of interest does not cost anything if the integrand is not particularly demanding. I also disagree at the pessimistic view that the asymptotic variance of the MCMC estimate is hard to estimate: papers by Flegal, Hobert, Jones, Vat and others have rather clearly shown how batch means can produce converging estimates of this asymptotic variance.

“We do not to attempt to solve a continuous optimisation problem for selection of the next point[in the sample].Such optimisation problems are fundamentally difficult and can at best be approximately solved. Instead, we exactly solve the discrete optimisation problem of selecting a suitable element from a supplied MCMC output.”

One definitely positive aspect of the paper is that the (thinning) method is called Stein thinning, in connection with *Stein’s discrepancy*, and this honours Charles Stein. The method looks at the optimal subsample, with optimality defined in terms of minimising Stein’s discrepancy from the true target over a reproducible kernel Hilbert space. And then over a subsample to minimise the distance from the empirical distribution to the theoretical distribution. The kernel (11) is based on the gradient of the target log density and the solution is determined by greedy algorithms that determine which next entry to add to the empirical distribution. Which is of complexity *O(nm ^{2})* if the subsample is of size

*m*. Some entries may appear more than once and the burnin step could be automatically included as (relatively) unlikely values are never selected (at least this was my heuristic understanding). While the theoretical backup for the construct is present and backed by earlier papers of some of the authors, I do wonder at the use of the most rudimentary representation of an approximation to the target when smoother versions could have been chosen and optimised on the same ground. And I am also surprised at the dependence of both estimators and discrepancies on the choice of the (sort-of) covariance matrix in the inner kernel, as the ODE examples provided in the paper (see, e.g., Figure 7). (As an aside and at a shallow level, the approach also reminded me of the principal points of my late friend Bernhard Flury…) Storage of all MCMC simulations for a later post-processing is of course costly in terms of storage, at O(nm). Unless a “secretary problem” approach can be proposed to get sequential. Another possible alternate would be to consider directly the chain of the accepted values (à la vanilla Rao-Blackwellisation). Overall, since the stopping criterion is based on a fixed sample size, and hence depends on the sub-efficiency of evaluating the mass of different modes, I am unsure the method is anything but what-you-get-is-what-you-see, i.e. prone to get misled by a poor exploration of the complete support of the target.

“This paper focuses on nonuniform subsampling and shows that it is more efficiency than uniform subsampling.”

Two weeks later, Guanyu Hu and Hai Ying Wang arXived their Most Likely Optimal Subsampled Markov Chain Monte Carlo, in what I first thought as an answer to the above! But both actually have little in common as this second paper considers subsampling on the data, rather than the MCMC output, towards producing scalable algorithms. Building upon Bardenet et al. (2014) and Korattikara et al. (2014). Replacing thus the log-likelihood with a random sub-sampled version and deriving the sample size based on a large deviation inequality. By a Cauchy-Schwartz inequality, the authors find sampling probabilities proportional to the individual log-likelihooods. Which depend on the running value of the MCMC’ed parameters. And thus replaced with the values at a fixed parameter, with cost O(n) but only once, but no so much optimal. (The large deviation inequality therein is only concerned with an approximation to the log-likelihood, without examining the long term impact on the convergence of the approximate Markov chain as this is no longer pseudo-marginal MCMC. For instance, both current and prospective log-likelihoods are re-estimated at each iteration. The paper compares with uniform sampling on toy examples, to demonstrate a smaller estimation error for the statistical problem, rather than convergence to the true posterior.)

## 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.