scalable Metropolis-Hastings, nested Monte Carlo, and normalising flows

Over a sunny if quarantined Sunday, I started reading the PhD dissertation of Rob Cornish, Oxford University, as I am the external member of his viva committee. Ending up in a highly pleasant afternoon discussing this thesis over a (remote) viva yesterday. (If bemoaning a lost opportunity to visit Oxford!) The introduction to the viva was most helpful and set the results within the different time and geographical zones of the Ph.D since Rob had to switch from one group of advisors in Engineering to another group in Statistics. Plus an encompassing prospective discussion, expressing pessimism at exact MCMC for complex models and looking forward further advances in probabilistic programming.

Made of three papers, the thesis includes this ICML 2019 [remember the era when there were conferences?!] paper on scalable Metropolis-Hastings, by Rob Cornish, Paul Vanetti, Alexandre Bouchard-Côté, Georges Deligiannidis, and Arnaud Doucet, which I commented last year. Which achieves a remarkable and paradoxical O(1/√n) cost per iteration, provided (global) lower bounds are found on the (local) Metropolis-Hastings acceptance probabilities since they allow for Poisson thinning à la Devroye (1986) and  second order Taylor expansions constructed for all components of the target, with the third order derivatives providing bounds. However, the variability of the acceptance probability gets higher, which induces a longer but still manageable if the concentration of the posterior is in tune with the Bernstein von Mises asymptotics. I had not paid enough attention in my first read at the strong theoretical justification for the method, relying on the convergence of MAP estimates in well- and (some) mis-specified settings. Now, I would have liked to see the paper dealing with a more complex problem that logistic regression.

The second paper in the thesis is an ICML 2018 proceeding by Tom Rainforth, Robert Cornish, Hongseok Yang, Andrew Warrington, and Frank Wood, which considers Monte Carlo problems involving several nested expectations in a non-linear manner, meaning that (a) several levels of Monte Carlo approximations are required, with associated asymptotics, and (b) the resulting overall estimator is biased. This includes common doubly intractable posteriors, obviously, as well as (Bayesian) design and control problems. [And it has nothing to do with nested sampling.] The resolution chosen by the authors is strictly plug-in, in that they replace each level in the nesting with a Monte Carlo substitute and do not attempt to reduce the bias. Which means a wide range of solutions (other than the plug-in one) could have been investigated, including bootstrap maybe. For instance, Bayesian design is presented as an application of the approach, but since it relies on the log-evidence, there exist several versions for estimating (unbiasedly) this log-evidence. Similarly, the Forsythe-von Neumann technique applies to arbitrary transforms of a primary integral. The central discussion dwells on the optimal choice of the volume of simulations at each level, optimal in terms of asymptotic MSE. Or rather asymptotic bound on the MSE. The interesting result being that the outer expectation requires the square of the number of simulations for the other expectations. Which all need converge to infinity. A trick in finding an estimator for a polynomial transform reminded me of the SAME algorithm in that it duplicated the simulations as many times as the highest power of the polynomial. (The ‘Og briefly reported on this paper… four years ago.)

The third and last part of the thesis is a proposal [to appear in ICML 20] on relaxing bijectivity constraints in normalising flows with continuously index flows. (Or CIF. As Rob made a joke about this cleaning brand, let me add (?) to that joke by mentioning that looking at CIF and bijections is less dangerous in a Trump cum COVID era at CIF and injections!) With Anthony Caterini, George Deligiannidis and Arnaud Doucet as co-authors. I am much less familiar with this area and hence a wee bit puzzled at the purpose of removing what I understand to be an appealing side of normalising flows, namely to produce a manageable representation of density functions as a combination of bijective and differentiable functions of a baseline random vector, like a standard Normal vector. The argument made in the paper is that imposing this representation of the density imposes a constraint on the topology of its support since said support is homeomorphic to the support of the baseline random vector. While the supporting theoretical argument is a mathematical theorem that shows the Lipschitz bound on the transform should be infinity in the case the supports are topologically different, these arguments may be overly theoretical when faced with the practical implications of the replacement strategy. I somewhat miss its overall strength given that the whole point seems to be in approximating a density function, based on a finite sample.

Viva in Toronto (not really!)

This was the second viva of the week, for the thesis of Madeleine Thompson, but as it was in Toronto, I took part in it by a phone connection. This was rather ineffective as the connection was rather poor and I could not follow most of the questions… I had previously read (and commented) two papers,  Slice Sampling with Adaptive Multivariate Steps: The Shrinking-Rank Method, and Graphical Comparison of MCMC Performance,  co-written by Madeleine so I was well-aware of a part of the contents of the thesis, which I read in toto a few weeks ago. It was an interesting thesis with diversified threads in the various chapter, but I found frustrating to be unable to fully take part in the thesis debate… In retrospect, I should have flown to Toronto from Manchester yesterday or abstained from taking part in the viva!

Viva and talk in Lancaster [back]

Both viva and talk went on well (even though I was a bit too tired to give a good talk, I fear!), with interesting outcomes in both cases. The viva lasted over two hours with an exciting exchange over the increase in overall error linked with the increase in dimension and over handling HMMs with four parameters to calibrate in parallel. At some point I got confused with Dennis’ result that

which I though was contradicting my favourite example of the non-central chi-square domination of the regular normal mean, namely that

under squared error loss. (This is Example 3.35 in The Bayesian Choice.) I had completely forgotten that the Jeffreys’ priors associated with both posterior expectations were different! The above equality is thus not invalidated by this example. It is further quite interesting in that it shows the posterior expectation is a sort of weak sufficient statistics for the estimation of the parameter, even though I remain in favour of using more summary statistics in ABC than a posterior expectation or a pseudo-MLE. In any case, the discussion of the corresponding Read Paper at the Royal Statistical Society next December 14 promises to be interesting and well-attended… Overall, the trip was quite pleasant (nice hotel, nice run in the countryside, where I took the attached pictures) and profitable, with discussions with Paul Fearnhead gearing me towards taking advantage of my colleagues’ expertise on indirect inference at CREST.