**D**espite the evidence brought by ABC on the inefficiency of culling in massive proportions the British Isles badger population against bovine tuberculosis, the [sorry excuse for a] United Kingdom government has permitted a massive expansion of badger culling, with up to 64,000 animals likely to be killed this autumn… Since the cows are the primary vectors of the disease, what about starting with these captive animals?!

## Archive for University of Warwick

## unimaginable scale culling

Posted in Books, pictures, Statistics, Travel with tags ABC, Approximate Bayesian computation, Badger Trust, badgers, Britain, British Wildlife Centre, cows, culling, Gareth Roberts, Nature, TB, tuberculosis, United Kingdom, University of Warwick on September 17, 2019 by xi'an## likelihood-free inference by ratio estimation

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags ABC, Austria, Bayesian lasso, Charlie Geyer, conflict of interest, Cross Validation, discussion, estimating constants, likelihood-free methods, logistic regression, nonparametrics, normalising constant, Salzburg, SimStat2019, summer, University of Warwick on September 9, 2019 by xi'an

“This approach for posterior estimation with generative models mirrors the approach of Gutmann and Hyvärinen (2012) for the estimation of unnormalised models. The main difference is that here we classify between two simulated data sets while Gutmann and Hyvärinen (2012) classified between the observed data and simulated reference data.”

**A** 2018 arXiv posting by Owen Thomas et al. (including my colleague at Warwick, Rito Dutta, *CoI warning!*) about estimating the likelihood (and the posterior) when it is intractable. Likelihood-free but not ABC, since the ratio likelihood to marginal is estimated in a non- or semi-parametric (and biased) way. Following Geyer’s 1994 fabulous estimate of an unknown normalising constant via logistic regression, the current paper which I read in preparation for my discussion in the ABC optimal design in Salzburg uses probabilistic classification and an exponential family representation of the ratio. Opposing data from the density and data from the marginal, assuming both can be readily produced. The logistic regression minimizing the asymptotic classification error is the logistic transform of the log-ratio. For a finite (double) sample, this minimization thus leads to an empirical version of the ratio. Or to a smooth version if the log-ratio is represented as a convex combination of summary statistics, turning the approximation into an exponential family, which is a clever way to buckle the buckle towards ABC notions. And synthetic likelihood. Although with a difference in estimating the exponential family parameters β(θ) by minimizing the classification error, parameters that are indeed conditional on the parameter θ. Actually the paper introduces a further penalisation or regularisation term on those parameters β(θ), which could have been processed by Bayesian Lasso instead. This step is essentially dirving the selection of the summaries, except that it is for each value of the parameter θ, at the expense of a X-validation step. This is quite an original approach, as far as I can tell, but I wonder at the link with more standard density estimation methods, in particular in terms of the precision of the resulting estimate (and the speed of convergence with the sample size, if convergence there is).

## unbiased product of expectations

Posted in Books, Statistics, University life with tags ABC, g-and-k distributions, importance sampling, latent variable models, NP-complete problem, permanent of a matrix, pseudo-marginal MCMC, recycling, University of Warwick on August 5, 2019 by xi'an**W**hile I was not involved in any way, or even aware of this research, Anthony Lee, Simone Tiberi, and Giacomo Zanella have an incoming paper in Biometrika, and which was partly written while all three authors were at the University of Warwick. The purpose is to design an efficient manner to approximate the product of n unidimensional expectations (or integrals) all computed against the same reference density. Which is not a real constraint. A neat remark that motivates the method in the paper is that an improved estimator can be connected with the permanent of the n x N matrix A made of the values of the n functions computed at N different simulations from the reference density. And involves N!/ (N-n)! terms rather than N to the power n. Since it is NP-hard to compute, a manageable alternative uses random draws from constrained permutations that are reasonably easy to simulate. Especially since, given that the estimator recycles most of the particles, it requires a much smaller version of N. Essentially N=O(n) with this scenario, instead of O(n²) with the basic Monte Carlo solution, towards a similar variance.

This framework offers many applications in latent variable models, including pseudo-marginal MCMC, of course, but also for ABC since the ABC posterior based on getting each simulated observation close enough from the corresponding actual observation fits this pattern (albeit the dependence on the chosen ordering of the data is an issue that can make the example somewhat artificial).

## a generalized representation of Bayesian inference

Posted in Books with tags approximate Bayesian inference, Bayesian decision theory, Bayesian robustness, Kullback-Leibler divergence, Likelihood Principle, University of Warwick, variational inference on July 5, 2019 by xi'an**J**eremias Knoblauch, Jack Jewson and Theodoros Damoulas, all affiliated with Warwick (hence a potentially biased reading!), arXived a paper on loss-based Bayesian inference that Jack discussed with me on my last visit to Warwick. As I was somewhat scared by the 61 pages, of which the 8 first pages are in NeurIPS style. The authors argue for a decision-theoretic approach to Bayesian inference that involves a loss over distributions and a divergence from the prior. For instance, when using the log-score as the loss and the Kullback-Leibler divergence, the regular posterior emerges, as shown by Arnold Zellner. Variational inference also falls under this hat. The argument for this generalization is that any form of loss can be used and still returns a distribution that is used to assess uncertainty about the parameter (of interest). In the axioms they produce for justifying the derivation of the optimal procedure, including cases where the posterior is restricted to a certain class, one [Axiom 4] generalizes the likelihood principle. Given the freedom brought by this general framework, plenty of fringe Bayes methods like standard variational Bayes can be seen as solutions to such a decision problem. Others like EP do not. Of interest to me are the potentials for this formal framework to encompass misspecification and likelihood-free settings, as well as for assessing priors, which is always a fishy issue. (The authors mention in addition the capacity to build related specific design Bayesian deep networks, of which I know nothing.) The obvious reaction of mine is one of facing an abundance of wealth (!) but encompassing approximate Bayesian solutions within a Bayesian framework remains an exciting prospect.

## O’Bayes 19/4

Posted in Books, pictures, Running, Statistics, Travel, University life with tags Bayesian model choice, Carl Friedrich Gauss, Coventry, Dickey-Savage ratio, factor analysis, improper priors, large p small n, nettles, O'Bayes 2019, Power-Expected-Posterior Priors, sent to Coventry, University of Warwick on July 4, 2019 by xi'anLast talks of the conference! With Rui Paulo (along with Gonzalo Garcia-Donato) considering the special case of factors when doing variable selection. Which is an interesting question that I had never considered, as at best I would remove all leves or keeping them all. Except that there may be misspecification in the factors as for instance when several levels have the same impact.With Michael Evans discussing a paper that he wrote for the conference! Following his own approach to statistical evidence. And including his reluctance to cover infinity (calling on Gauß for backup!) or continuity, and his call to falsify a Bayesian model by checking it can be contradicted by the data. His assumption that checking for prior is separable from checking for [sampling] model is debatable. (With another mention made of the Savage-Dickey ratio.)

And with Dimitris Fouskakis giving a wide ranging assessment [which Mark Steel (Warwick) called a PEP talk!] of power-expected-posterior priors, used with reference (and usually improper) priors. Which in retrospect would have suited better the beginning of the conference as it provided a background to several of the talks. Raising a question (from my perspective) on using the maximum likelihood estimator as a pseudo-sufficient statistic when this MLE is computed for the base (simplest) model. Maybe an ABC induced bias in this question as it would not work for ABC model choice.

Overall, I think the scientific outcomes of the conference were quite positive: a wide range of topics and perspectives, a reasonable and diverse attendance, especially when considering the heavy load of related conferences in the surrounding weeks (the “June fatigue”!), animated poster sessions. I am obviously not the one to assess the organisation of the conference! Things I forgot to do in this regard: organise transportation from Oxford to Warwick University, provide an attached room for in-pair research, insist on sustainability despite the imposed catering solution, facilitate sharing joint transportation to and from the Warwick campus, mention that tap water was potable, and… wear long pants when running in nettles.

## O’Bayes 19/3.5

Posted in Books, pictures, Travel, University life with tags #betterposter, Beamer, O'Bayes 2019, poster session, University of Warwick on July 3, 2019 by xi'an

**A**mong the posters at the second poster session yesterday night, one by Judith ter Schure visually standing out by following the #betterposter design suggested by Mike Morrison a few months ago. Design on which I have ambivalent feelings. On the one hand, reducing the material on a poster is generally a good idea as they tend to be saturated and hard to read, especially in crowded conditions. Having the main idea or theorem immediately visible should indeed be a requirement, from immediately getting the point to starting from the result in explaining the advances in the corresponding work. But if this format becomes the standard, it will become harder to stand out! More fundamentally, this proposal may fall into the same abyss as powerpoint presentations, which is that insisting in making the contents simpler and sparser may reach the no-return point of no content [which was not the case of the above poster, let me hasten to state!]. Mathematical statistics poster may be automatically classified as too complicated for this #betterposter challenge as containing maths formulas! Or too many Greek letters as someone complained after one of my talks. And treating maths formulas as detail makes them even smaller than usual, which sounds like the opposite of the intended effect. (The issue is discussed on the betterposter blog, for a variety of opinions, mostly at odds with mine’s.)

## O’Bayes 19/3

Posted in Books, pictures, Statistics, Travel, University life with tags #betterposter, Beamer, BFF4, Coventry, Dickey-Savage ratio, dominating measure, frequency properties, minimaxity, O'Bayes 2019, poster session, SafeBayes, sent to Coventry, town of Warwick, uniform distribution, University of Coventry, University of Warwick on July 2, 2019 by xi'an**N**ancy Reid gave the first talk of the [Canada] day, in an impressive comparison of all approaches in statistics that involve a distribution of sorts on the parameter, connected with the presentation she gave at BFF4 in Harvard two years ago, including safe Bayes options this time. This was related to several (most?) of the talks at the conference, given the level of worry (!) about the choice of a prior distribution. But the main assessment of the methods still seemed to be centred on a frequentist notion of calibration, meaning that epistemic interpretations of probabilities and hence most of Bayesian answers were disqualified from the start.

In connection with Nancy’s focus, Peter Hoff’s talk also concentrated on frequency valid confidence intervals in (linear) hierarchical models. Using prior information or structure to build better and shrinkage-like confidence intervals at a given confidence level. But not in the decision-theoretic way adopted by George Casella, Bill Strawderman and others in the 1980’s. And also making me wonder at the relevance of contemplating a fixed coverage as a natural goal. Above, a side result shown by Peter that I did not know and which may prove useful for Monte Carlo simulation.

Jaeyong Lee worked on a complex model for banded matrices that starts with a regular Wishart prior on the unrestricted space of matrices, computes the posterior and then projects this distribution onto the constrained subspace. (There is a rather consequent literature on this subject, including works by David Dunson in the past decade of which I was unaware.) This is a smart demarginalisation idea but I wonder a wee bit at the notion as the constrained space has measure zero for the larger model. This could explain for the resulting posterior not being a true posterior for the constrained model in the sense that there is no prior over the constrained space that could return such a posterior. Another form of marginalisation paradox. The crux of the paper is however about constructing a functional form of minimaxity. In his discussion of the paper, Guido Consonni provided a representation of the post-processed posterior (P³) that involves the Dickey-Savage ratio, sort of, making me more convinced of the connection.

As a lighter aside, one item of local information I should definitely have broadcasted more loudly and long enough in advance to the conference participants is that the University of Warwick is not located in ye olde town of Warwick, where there is no university, but on the outskirts of the city of Coventry, but not to be confused with the University of Coventry. Located in Coventry.