Bayesian hierarchical models naturally lend themselves to coordinate-wise computational algorithms, such as Gibbs sampling and coordinate-ascent variational inference (CAVI). Despite popularity and good empirical performances, our quantitative understanding of their convergence properties is still very limited compared to, e.g., gradient-based methods. In this talk we consider two examples: Gibbs Sampling for classical (i.e. two-level nested) hierarchical models and CAVI for crossed effect models. We show that, under favourable circumstances, the total computational cost of these schemes scales linearly with the number of observations and of parameters in the model, thus providing state-of-the-art performances. In the first example, our analysis relies on Bayesian asymptotics and applies to general likelihoods. In the second, we focus on Gaussian likelihood and exploit connections to random graph theory to provide a sharp high-dimensional analysis. Some results of independent interest relating the convergence of CAVI to the accuracy of the corresponding variational approximation are also discussed.

Statistical hypotheses are translations of scientific hypotheses into statements about one or more distributions, often concerning their centre. Tests that assess statistical hypotheses of centre implicitly assume a specific centre, e.g., the mean or median. Yet, scientific hypotheses do not always specify a particular centre. This ambiguity leaves the possibility for a gap between scientific theory and statistical practice that can lead to rejection of a true null. In the face of replicability crises in many scientific disciplines, significant results of this kind are concerning. Rather than testing a single centre, this paper proposes testing a family of plausible centres, such as that induced by the Huber loss function (the Huber family). Each centre in the family generates a testing problem, and the resulting family of hypotheses constitutes a familial hypothesis. A Bayesian nonparametric procedure is devised to test familial hypotheses, enabled by a novel pathwise optimization routine to fit the Huber family. The favourable properties of the new test are demonstrated theoretically and experimentally. Two examples from psychology serve as real-world case studies. The talk will cover joint work with Ryan Thompson (Postdoctoral Researcher, University of New South Wales), Mario Peruggia (Ohio State University) and Steve MacEachern (Ohio State University). A current version of the paper is available at https://arxiv.org/abs/2202.12540.

This talk seeks to illustrate how the (natural) use of probability models in Bayesian inference brings remarkable opportunities, and also potential pitfalls unless used carefully. In a first application we discuss strategies to obtain more accurate inference, and also more interpretable, by integrating infirmation in external data. We show an application to graphical models and network data. We also discuss that integrating external data allows pushing the theoretical conditions under which one can recover sparse signals from data. In a second application we discuss testing local null hypothesis via semi-parametric models. We discuss an important pitfall: when (unavoidably) the model is misspecified, there's a serious type I error inflation that becomes worse as n grows. We show how the issue can be easily fixed by tweaking the model into using orthogonal bases that allow discountinuities.

Bayesian computation remains onerous at scale for inference under many discrete-valued stochastic process-based models, while these models remain ubiquitous across biology and public health. In this talk, we will explore how one can construct computationally efficient approximations to the gradient of the data likelihood under continuous-time Markov chain (CTMC) models with respect to their high-dimensional parameters. CTMCs underpin the most popular models for learning about how rapidly evolving pathogens change over time and space to give rise to human infection, and the dimensionality of these problems are daunting. With these approximations in hand, a new variant of Hamiltonian Monte Carlo (HMC) becomes tractable to explore the parameter posterior, and we bound the approximation error using several small tricks from matrix analysis. This new sampling approach enables the introduction of a novel random-effects CTMC model that captures biological realism previously missing. Applied to the analysis of early SARS-CoV-2 genomes, the random-effects remove bias in inference of the location and timing of the pathogen's split-over into humans, while the approximate-gradient-based machinery is over an order of magnitude more time efficient than conventional sampling approaches.

Spatio-temporal counts of infectious disease cases often contain an excess of zeros. With existing zero-inflated count models applied to such data, it is difficult to quantify space-time heterogeneity in the effects of disease spread between areas. Also, existing methods do not allow for separate dynamics to affect the reemergence and persistence of the disease. As an alternative, we develop a new zero-state coupled Markov switching negative binomial model, under which the disease switches between periods of presence and absence in each area through a series of partially hidden non-homogeneous Markov chains coupled between neighbouring locations. When the disease is present, an autoregressive negative binomial model generates the cases, with a possible zero representing the disease being undetected. We fit different versions of the proposed model to the weekly number of cases of dengue fever across the districts of Rio de Janeiro. We also fit an extension of the model to weekly cases of Zika across municipalities of Colombia. Finally, we propose a hurdle version of the zero-state coupled Markov switching negative binomial model and compare the models when analyzing the first epidemic of Chikungunya experienced in Rio de Janeiro between 2015 and 2016. This talk comprises joint work with Dirk Douwes-Schultz (Biostatistics PhD Candidate, McGill University), Laís P. Freitas (Post-Doctoral Fellow, Université de Montréal) and Mingchi Xu (Biostatistics PhD Candidate, McGill University).