accronyms [CDT lectures]

This week, I gave a short and introductory course in Warwick for the CDT (PhD) students on my perceived connections between reverse logistic regression à la Geyer and GANS, among other things. The first attempt was cancelled in 2020 due to the pandemic, the second one in 2021 was on-line and thus offered little possibilities for interactions. Preparing for this third attempt made me read more papers on some statistical analyses of GANs and WGANs, which was more satisfactory [for me] even though I could not get into the technical details…

distributed evidence

Alexander Buchholz (who did his PhD at CREST with Nicolas Chopin), Daniel Ahfock, and my friend Sylvia Richardson published a great paper on the distributed computation of Bayesian evidence in Bayesian Analysis. The setting is one of distributed data from several sources with no communication between them, which relates to consensus Monte Carlo even though model choice has not been particularly studied from that perspective. The authors operate under the assumption of conditionally conjugate models, i.e., the existence of a data augmentation scheme into an exponential family so that conjugate priors can be used. For a division of the data into S blocks, the fundamental identity in the paper is

where α is the normalising constant of the sub-prior exp{log[p(θ)]/S} and the other terms are associated with this prior. Under the conditionally conjugate assumption, the integral can be approximated based on the latent variables. Most interestingly, the associated variance is directly connected with the variance of

under the joint:

“The variance of the ratio measures the quality of the product of the conditional sub-posterior as an importance sample proposal distribution.”

Assuming this variance is finite (which is likely). An approximate alternative is proposed, namely to replace the exact sub-posterior with a Normal distribution, as in consensus Monte Carlo, which should obviously require some consideration as to which parameterisation of the model produces the “most normal” (or the least abnormal!) posterior. And ensures a finite variance in the importance sampling approximation (as ensured by the strong bounds in Proposition 5). A problem shared by the bridgesampling package.

“…if the error that comes from MCMC sampling is relatively small and that the shard sizes are large enough so that the quality of the subposterior normal approximation is reasonable, our suggested approach will result in good approximations of the full data set marginal likelihood.”

The resulting approximation can also be handy in conjunction with reversible jump MCMC, in the sense that RJMCMC algorithms can be run in parallel on different chunks or shards of the entire dataset. Although the computing gain may be reduced by the need for separate approximations.

NCE, VAEs, GANs & even ABC…

As I was preparing my (new) lectures for a PhD short course “at” Warwick (meaning on Teams!), I read a few surveys and other papers on all these acronyms. It included the massive Guttmann and Hyvärinen 2012 NCE JMLR paper,  Goodfellow’s NIPS 2016 tutorial on GANs, and  Kingma and Welling 2019 introduction to VAEs. Which I found a wee bit on the light side, maybe missing the fundamentals of the notion… As well as the pretty helpful 2019 survey on normalising flows by Papamakarios et al., although missing on the (statistical) density estimation side.  And also a nice (2017) survey of GANs by Shakir Mohamed and Balaji Lakshminarayanan with a somewhat statistical spirit, even though convergence issues are not again not covered. But misspecification is there. And the many connections between ABC and GANs, if definitely missing on the uncertainty aspects. While Deep Learning by Goodfellow, Bengio and Courville adresses both the normalising constant (or partition function) and GANs, it was somehow not deep enough (!) to use for the course, offering only a few pages on NCE, VAEs and GANs. (And also missing on the statistical references addressing the issue, incl. [or excl.]  Geyer, 1994.) Overall, the infinite variations offered on GANs leave me uncertain about their statistical relevance, as it is unclear how good the regularisation therein is for handling overfitting and consistent estimation. (And if I spot another decomposition of the Kullback-Leibler divergence, I may start crying…)

PhD studenships at Warwick

There is an exciting opening for several PhD positions at Warwick, in the departments of Statistics and of Mathematics, as part of the Centre for Doctoral Training in Mathematics and Statistics newly created by the University. CDT studentships are funded for four years and funding is open to students from the European Union without restrictions. (No Brexit!) Funding includes a stipend at UK/RI rates and tuition fees at UK/EU rates. Applications are made via the University of Warwick Online Application Portal and should be made  as quickly as possible since the funding will be allocated on a first come first serve basis. For more details, contact the CDT director, Martyn Plummer. I cannot but strongly encourage interested students to apply as this is a great opportunity to start a research career in a fantastic department!

distributed posteriors

Another presentation by our OxWaSP students introduced me to the notion of distributed posteriors, following a 2018 paper by Botond Szabó and Harry van Zanten. Which corresponds to the construction of posteriors when conducting a divide & conquer strategy. The authors show that an adaptation of the prior to the division of the sample is necessary to recover the (minimax) convergence rate obtained in the non-distributed case. This is somewhat annoying, except that the adaptation amounts to take the original prior to the power 1/m, when m is the number of divisions. They further show that when the regularity (parameter) of the model is unknown, the optimal rate cannot be recovered unless stronger assumptions are made on the non-zero parameters of the model.

“First of all, we show that depending on the communication budget, it might be advantageous to group local machines and let different groups work on different aspects of the high-dimensional object of interest. Secondly, we show that it is possible to have adaptation in communication restricted distributed settings, i.e. to have data-driven tuning that automatically achieves the correct bias-variance trade-off.”

I find the paper of considerable interest for scalable MCMC methods, even though the setting may happen to sound too formal, because the study incorporates parallel computing constraints. (Although I did not investigate the more theoretical aspects of the paper.)