## a case for Bayesian deep learnin

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , on September 30, 2020 by xi'an

Andrew Wilson wrote a piece about Bayesian deep learning last winter. Which I just read. It starts with the (posterior) predictive distribution being the core of Bayesian model evaluation or of model (epistemic) uncertainty.

“On the other hand, a flat prior may have a major effect on marginalization.”

Interesting sentence, as, from my viewpoint, using a flat prior is a no-no when running model evaluation since the marginal likelihood (or evidence) is no longer a probability density. (Check Lindley-Jeffreys’ paradox in this tribune.) The author then goes for an argument in favour of a Bayesian approach to deep neural networks for the reason that data cannot be informative on every parameter in the network, which should then be integrated out wrt a prior. He also draws a parallel between deep ensemble learning, where random initialisations produce different fits, with posterior distributions, although the equivalent to the prior distribution in an optimisation exercise is somewhat vague.

“…we do not need samples from a posterior, or even a faithful approximation to the posterior. We need to evaluate the posterior in places that will make the greatest contributions to the [posterior predictive].”

The paper also contains an interesting point distinguishing between priors over parameters and priors over functions, ony the later mattering for prediction. Which must be structured enough to compensate for the lack of data information about most aspects of the functions. The paper further discusses uninformative priors (over the parameters) in the O’Bayes sense as a default way to select priors. It is however unclear to me how this discussion accounts for the problems met in high dimensions by standard uninformative solutions. More aggressively penalising priors may be needed, as those found in high dimension variable selection. As in e.g. the 10⁷ dimensional space mentioned in the paper. Interesting read all in all!

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

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , on June 16, 2020 by xi'an

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.

## neural importance sampling

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , , , , on May 13, 2020 by xi'an

Dennis Prangle signaled this paper during his talk of last week, first of our ABC ‘minars now rechristened as The One World ABC Seminar to join the “One World xxx Seminar” franchise! The paper is written by Thomas Müller and co-authors, all from Disney research [hence the illustration], and we discussed it in our internal reading seminar at Dauphine. The authors propose to parameterise the importance sampling density via neural networks, just like Dennis is using auto-encoders. Starting with the goal of approximating

$\mathfrak I=\int_{\mathfrak D} f(x)\text{d}x$

(where they should assume f to be non-negative for the following), the authors aim at simulating from an approximation of f(x)/ℑ since this “ideal” pdf would give zero variance.

“Unfortunately, the above integral is often not solvable in closed form, necessitating its estimation with another Monte Carlo estimator.”

Among the discussed solutions, the Latent-Variable Model one is based on a pdf represented as a marginal. A mostly intractable integral, which the authors surprisingly seem to deem an issue as they do not mention the standard solution of simulating from the joint and using the conditional in the importance weight. (Or even more surprisingly and obviously wrongly see the latter as a biased approximation to the weight.)

“These “autoregressive flows” offer the desired exact evaluation of q(x;θ). Unfortunately, they generally only permit either efficient sample generation or efficient evaluation of q(x;θ), which makes them prohibitively expensive for our application to Mont Carlo integration.”

When presenting normalizing flows, namely the representation of the simulation output as the result of an invertible mapping of a standard (e.g., Gaussian or Uniform) random variable, x=h(u,θ), which can itself be decomposed into a composition of suchwise functions. And I am thus surprised this cannot be done in an efficient manner if transforms are well chosen…

“The key proposition of Dinh et al. (2014) is to focus on a specific class of mappings—referred to as coupling layers—that admit Jacobian matrices where determinants reduce to the product of diagonal terms.

Using a transform with a triangular Jacobian at each stage has the appeal of keeping the change of variable simple and allowing for non-linear transforms. Namely piecewise polynomials. When reading the one-blob (!) encoding , I am however uncertain the approach is more than the choice of a particular functional basis, as for instance wavelets (which may prove more costly to handle, granted!)

“Given that NICE scales well to high-dimensional problems…”

It is always unclear to me why almost every ML paper feels the urge to redefine & motivate the KL divergence. And to recall that it avoids bothering about the normalising constant. Looking at the variance of the MC estimator & seeking minimal values is praiseworthy, but only when the variance exists. What are the guarantees on the density estimate for this to happen? And where are the arguments for NICE scaling nicely to high dimensions? Interesting intrusion of path sampling, but is it of any use outside image analysis—I had forgotten Eric Veach’s original work was on light transport—?

## ABC webinar, first!

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , on April 13, 2020 by xi'an

The première of the ABC World Seminar last Thursday was most successful! It took place at the scheduled time, with no technical interruption and allowed 130⁺ participants from most of the World [sorry, West Coast friends!] to listen to the first speaker, Dennis Prangle,  presenting normalising flows and distilled importance sampling. And to answer questions. As I had already commented on the earlier version of his paper, I will not reproduce them here. In short, I remain uncertain, albeit not skeptical, about the notions of normalising flows and variational encoders for estimating densities, when perceived as a non-parametric estimator due to the large number of parameters it involves and wonder at the availability of convergence rates. Incidentally, I had forgotten at the remarkable link between KL distance & importance sampling variability. Adding to the to-read list Müller et al. (2018) on neural importance sampling.

## Julyan’s talk on priors in Bayesian neural networks [cancelled!]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on March 5, 2020 by xi'an

Next Friday, 13 March at 1:30p.m., Julyan Arbel, researcher at Inria Grenoble will give a All about that Bayes talk at CMLA, ENS Paris-Saclay (building D’Alembert, room Condorcet, Cachan, RER stop Bagneux) on

Understanding Priors in Bayesian Neural Networks at the Unit Level

We investigate deep Bayesian neural networks with Gaussian weight priors and a class of ReLU-like nonlinearities. Bayesian neural networks with Gaussian priors are well known to induce an L², “weight decay”, regularization. Our results characterize a more intricate regularization effect at the level of the unit activations. Our main result establishes that the induced prior distribution on the units before and after activation becomes increasingly heavy-tailed with the depth of the layer. We show that first layer units are Gaussian, second layer units are sub-exponential, and units in deeper layers are characterized by sub-Weibull distributions. Our results provide new theoretical insight on deep Bayesian neural networks, which we corroborate with simulation experiments.