Archive for convergence diagnostics

is there such a thing as optimal subsampling?

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on June 12, 2020 by xi'an

This 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(nm2) 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.)

assessing MCMC convergence

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on June 6, 2019 by xi'an

When MCMC became mainstream in the 1990’s, there was a flurry of proposals to check, assess, and even guarantee convergence to the stationary distribution, as discussed in our MCMC book. Along with Chantal Guihenneuc and Kerrie Mengersen, we also maintained for a while a reviewww webpage categorising theses. Niloy Biswas and Pierre Jacob have recently posted a paper where they propose the use of couplings (and unbiased MCMC) towards deriving bounds on different metrics between the target and the current distribution of the Markov chain. Two chains are created from a given kernel and coupled with a lag of L, meaning that after a while, the two chains become one with a time difference of L. (The supplementary material contains many details on how to induce coupling.) The distance to the target can then be bounded by a sum of distances between the two chains until they merge. The above picture from the paper is a comparison a Polya-Urn sampler with several HMC samplers for a logistic target (not involving the Pima Indian dataset!). The larger the lag L the more accurate the bound. But the larger the lag the more expensive the assessment of how many steps are needed to convergence. Especially when considering that the evaluation requires restarting the chains from scratch and rerunning until they couple again, rather than continuing one run which can only brings the chain closer to stationarity and to being distributed from the target. I thus wonder at the possibility of some Rao-Blackwellisation of the simulations used in this assessment (while realising once more than assessing convergence almost inevitably requires another order of magnitude than convergence itself!). Without a clear idea of how to do it… For instance, keeping the values of the chain(s) at the time of coupling is not directly helpful to create a sample from the target since they are not distributed from that target.

[Pierre also wrote a blog post about the paper on Statisfaction that is definitely much clearer and pedagogical than the above.]

revisiting the Gelman-Rubin diagnostic

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on January 23, 2019 by xi'an

Just before Xmas, Dootika Vats (Warwick) and Christina Knudson arXived a paper on a re-evaluation of the ultra-popular 1992 Gelman and Rubin MCMC convergence diagnostic. Which compares within-variance and between-variance on parallel chains started from hopefully dispersed initial values. Or equivalently an under-estimating and an over-estimating estimate of the MCMC average. In this paper, the authors take advantage of the variance estimators developed by Galin Jones, James Flegal, Dootika Vats and co-authors, which are batch mean estimators consistently estimating the asymptotic variance. They also discuss the choice of a cut-off on the ratio R of variance estimates, i.e., how close to one need it be? By relating R to the effective sample size (for which we also have reservations), which gives another way of calibrating the cut-off. The main conclusion of the study is that the recommended 1.1 bound is too large for a reasonable proximity to the true value of the Bayes estimator (Disclaimer: The above ABCruise header is unrelated with the paper, apart from its use of the Titanic dataset!)

In fact, I have other difficulties than setting the cut-off point with the original scheme as a way to assess MCMC convergence or lack thereof, among which

  1. its dependence on the parameterisation of the chain and on the estimation of a specific target function
  2. its dependence on the starting distribution which makes the time to convergence not absolutely meaningful
  3. the confusion between getting to stationarity and exploring the whole target
  4. its missing the option to resort to subsampling schemes to attain pseudo-independence or scale time to convergence (albeit see 3. above)
  5. a potential bias brought by the stopping rule.

calibrating approximate credible sets

Posted in Books, Statistics with tags , , , , , , , on October 26, 2018 by xi'an

Earlier this week, Jeong Eun Lee, Geoff Nicholls, and Robin Ryder arXived a paper on the calibration of approximate Bayesian credible intervals. (Warning: all three authors are good friends of mine!) They start from the core observation that dates back to Monahan and Boos (1992) of exchangeability between θ being generated from the prior and φ being generated from the posterior associated with one observation generated from the prior predictive. (There is no name for this distribution, other than the prior, that is!) A setting amenable to ABC considerations! Actually, Prangle et al. (2014) relies on this property for assessing the ABC error, while pointing out that the test for exchangeability is not fool-proof since it works equally for two generations from the prior.

“The diagnostic tools we have described cannot be “fooled” in quite the same way checks based on the exchangeability can be.”

The paper thus proposes methods for computing the coverage [under the true posterior] of a credible set computed using an approximate posterior. (I had to fire up a few neurons to realise this was the right perspective, rather than the reverse!) A first solution to approximate the exact coverage of the approximate credible set is to use logistic regression, instead of the exact coverage, based on some summary statistics [not necessarily in an ABC framework]. And a simulation outcome that the parameter [simulated from the prior] at the source of the simulated data is within the credible set. Another approach is to use importance sampling when simulating from the pseudo-posterior. However this sounds dangerously close to resorting to an harmonic mean estimate, since the importance weight is the inverse of the approximate likelihood function. Not that anything unseemly transpires from the simulations.

 

Statistical rethinking [book review]

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , on April 6, 2016 by xi'an

Statistical Rethinking: A Bayesian Course with Examples in R and Stan is a new book by Richard McElreath that CRC Press sent me for review in CHANCE. While the book was already discussed on Andrew’s blog three months ago, and [rightly so!] enthusiastically recommended by Rasmus Bååth on Amazon, here are the reasons why I am quite impressed by Statistical Rethinking!

“Make no mistake: you will wreck Prague eventually.” (p.10)

While the book has a lot in common with Bayesian Data Analysis, from being in the same CRC series to adopting a pragmatic and weakly informative approach to Bayesian analysis, to supporting the use of STAN, it also nicely develops its own ecosystem and idiosyncrasies, with a noticeable Jaynesian bent. To start with, I like the highly personal style with clear attempts to make the concepts memorable for students by resorting to external concepts. The best example is the call to the myth of the golem in the first chapter, which McElreath uses as an warning for the use of statistical models (which almost are anagrams to golems!). Golems and models [and robots, another concept invented in Prague!] are man-made devices that strive to accomplish the goal set to them without heeding the consequences of their actions. This first chapter of Statistical Rethinking is setting the ground for the rest of the book and gets quite philosophical (albeit in a readable way!) as a result. In particular, there is a most coherent call against hypothesis testing, which by itself justifies the title of the book. Continue reading