finding our way in the dark

The paper Finding our Way in the Dark: Approximate MCMC for Approximate Bayesian Methods by Evgeny Levi and (my friend) Radu Craiu, recently got published in Bayesian Analysis. The central motivation for their work is that both ABC and synthetic likelihood are costly methods when the data is large and does not allow for smaller summaries. That is, when summaries S of smaller dimension cannot be directly simulated. The idea is to try to estimate

since this is the substitute for the likelihood used for ABC. (A related idea is to build an approximate and conditional [on θ] distribution on the distance, idea with which Doc. Stoehr and I played a wee bit without getting anything definitely interesting!) This is a one-dimensional object, hence non-parametric estimates could be considered… For instance using k-nearest neighbour methods (which were already linked with ABC by Gérard Biau and co-authors.) A random forest could also be used (?). Or neural nets. The method still requires a full simulation of new datasets, so I wonder at the gain unless the replacement of the naïve indicator with h(θ) brings clear improvement to the approximation. Hence much fewer simulations. The ESS reduction is definitely improved, esp. since the CPU cost is higher. Could this be associated with the recourse to independent proposals?

In a sence, Bayesian synthetic likelihood does not convey the same appeal, since is a bit more of a tough cookie: approximating the mean and variance is multidimensional. (BSL is always more expensive!)

As a side remark, the authors use two chains in parallel to simplify convergence proofs, as we did a while ago with AMIS!

conditioning on insufficient statistics in Bayesian regression

“…the prior distribution, the loss function, and the likelihood or sampling density (…) a healthy skepticism encourages us to question each of them”

A paper by John Lewis, Steven MacEachern, and Yoonkyung Lee has recently appeared in Bayesian Analysis. Starting with the great motivation of a misspecified model requiring the use of a (thus necessarily) insufficient statistic and moving to their central concern of simulating the posterior based on that statistic.

Model misspecification remains understudied from a B perspective and this paper is thus most welcome in addressing the issue. However, when reading through, one of my criticisms is in defining misspecification as equivalent to outliers in the sample. An outlier model is an easy case of misspecification, in the end, since the original model remains meaningful. (Why should there be “good” versus “bad” data) Furthermore, adding a non-parametric component for the unspecified part of the data would sound like a “more Bayesian” alternative. Unrelated, I also idly wondered at whether or not normalising flows could be used in this instance..

The problem in selecting a T (Darjeeling of course!) is not really discussed there, while each choice of a statistic T leads to a different signification to what misspecified means and suggests a comparison with Bayesian empirical likelihood.

“Acceptance rates of this [ABC] algorithm can be intolerably low”

Erm, this is not really the issue with ABC, is it?! Especially when the tolerance is induced by the simulations themselves.

When I reached the MCMC (Gibbs?) part of the paper, I first wondered at its relevance for the mispecification issues before realising it had become the focus of the paper. Now, simulating the observations conditional on a value of the summary statistic T is a true challenge. I remember for instance George Casella mentioning it in association with a Student’s t sample in the 1990’s and Kerrie and I having an unsuccessful attempt at it in the same period. Persi Diaconis has written several papers on the problem and I am thus surprised at the dearth of references here, like the rather recent Byrne and Girolami (2013), Florens and Simoni (2015), or Bornn et al. (2019). In the present case, the  linear model assumed as the true model has the exceptional feature that it leads to a feasible transform of an unconstrained simulation into a simulation with fixed statistics, with no measure theoretic worries if not free from considerable efforts to establish the operation is truly valid… And, while simulating (θ,y) makes perfect sense in an insufficient setting, the cost is then precisely the same as when running a vanilla ABC. Which brings us to the natural comparison with ABC. While taking ε=0 may sound as optimal for being “exact”, it is not from an ABC perspective since the convergence rate of the (summary) statistic should be roughly the one of the tolerance (Fearnhead and Liu, Frazier et al., 2018).

“[The Borel Paradox] shows that the concept of a conditional probability with regard to an isolated given hypothesis whose probability equals 0 is inadmissible.” A. Колмого́ров (1933)

As a side note for measure-theoretic purists, the derivation of the conditional of y given T(y)=T⁰ is arbitrary since the event has probability zero (ie, the conditioning set is of measure zero). See the Borel-Kolmogorov paradox. The computations in the paper are undoubtedly correct, but this is only one arbitrary choice of a transform (or conditioning σ-algebra).

multilevel linear models, Gibbs samplers, and multigrid decompositions

A paper by Giacommo Zanella (formerly Warwick) and Gareth Roberts (Warwick) is about to appear in Bayesian Analysis and (still) open for discussion. It examines in great details the convergence properties of several Gibbs versions of the same hierarchical posterior for an ANOVA type linear model. Although this may sound like an old-timer opinion, I find it good to have Gibbs sampling back on track! And to have further attention to diagnose convergence! Also, even after all these years (!), it is always a surprise  for me to (re-)realise that different versions of Gibbs samplings may hugely differ in convergence properties.

At first, intuitively, I thought the options (1,0) (c) and (0,1) (d) should be similarly performing. But one is “more” hierarchical than the other. While the results exhibiting a theoretical ordering of these choices are impressive, I would suggest pursuing an random exploration of the various parameterisations in order to handle cases where an analytical ordering proves impossible. It would most likely produce a superior performance, as hinted at by Figure 4. (This alternative happens to be briefly mentioned in the Conclusion section.) The notion of choosing the optimal parameterisation at each step is indeed somewhat unrealistic in that the optimality zones exhibited in Figure 4 are unknown in a more general model than the Gaussian ANOVA model. Especially with a high number of parameters, parameterisations, and recombinations in the model (Section 7).

An idle question is about the extension to a more general hierarchical model where recentring is not feasible because of the non-linear nature of the parameters. Even though Gaussianity may not be such a restriction in that other exponential (if artificial) families keeping the ANOVA structure should work as well.

Theorem 1 is quite impressive and wide ranging. It also reminded (old) me of the interleaving properties and data augmentation versions of the early-day Gibbs. More to the point and to the current era, it offers more possibilities for coupling, parallelism, and increasing convergence. And for fighting dimension curses.

“in this context, imposing identifiability always improves the convergence properties of the Gibbs Sampler”

Another idle thought of mine is to wonder whether or not there is a limited number of reparameterisations. I think that by creating unidentifiable decompositions of (some) parameters, eg, μ=μ¹+μ²+.., one can unrestrictedly multiply the number of parameterisations. Instead of imposing hard identifiability constraints as in Section 4.2, my intuition was that this de-identification would increase the mixing behaviour but this somewhat clashes with the above (rigorous) statement from the authors. So I am proven wrong there!

Unless I missed something, I also wonder at different possible implementations of HMC depending on different parameterisations and whether or not the impact of parameterisation has been studied for HMC. (Which may be linked with Remark 2?)

Basque thesis defence [Bayes almost on the beach]

Yesterday morning I took part in a thesis defence (as a jury member) in the coastal city of Anglet, in the (French part of the) Basque Country. The PhD candidate was Sébastien Coube-Sisqueille, whom I did not know directly (although we had crossed paths at CIRM years ago and he had attended my MCMC course at ENSAE even more years ago). As it happened all other members of the committee, apart from Sébastien’s advisor, Benoît Liquet, were on Teams, being unable to travel to the Basque Country. Sébastien’s thesis is about MCMC strategies to accelerate convergence in spatial models represented as nearest neighbor Gaussian processes (NNGP), which relates to the earlier works of (X)XL on interweaving. (Unsurprisingly, the defence was successful and the candidate awarded his PhD!) Icing on the cake, I managed to take a dip in the Atlantic Ocean, before flying back to Paris for dinner, on a very warm afternoon (and slightly cooler water), thanks to Sébastien driving me to a nearby beach!

reproducibility check [Nature]

While reading the Nature article Swarm Learning, by Warnat-Herresthal et [many] al., which goes beyond federated learning by removing the need for a central coordinator, [if resorting to naïve averaging of the neural network parameters] I came across this reporting summary on the statistics checks made by the authors. With a specific box on Bayesian analysis and MCMC implementation!