**Recent advances in Bayesian modelling based on exponential and conjugate families of distributions**

Organizer: **Eduardo Gutiérrez Peña,** IIMAS-UNAM, Mexico

Exponential families are an important class of probability models that occur, in one form or another, as part of more complex models widely used in applied statistics (such as generalized linear models, hierarchical models and dynamic models). On the other hand, conjugate families play an important role in the Bayesian approach to parametric inference. One of the main features of such families is that they are closed under sampling, but a conjugate family often provides prior distributions which are tractable in various other respects. In this session we shall discuss some recent advances in statistical modelling and analysis based on exponential and conjugate families of distributions.

**General dependence structures for exponential family models****Luis Enrique Nieto-Barajas,** ITAM, Mexico

We introduce general dependence structures on a set of random variables. These include order-p autoregressive-type structures, seasonal, periodic and spatial dependencies. The marginal distribution can be any member of the exponential family, with strict stationarity as a special case. Dependence is induced via exchangeable latent variables whose conditional distributions are members of the corresponding conjugate family.

**Bayesian robustness revisited****Manuel Mendoza,** ITAM, México

The idea of studying the performance of inferential procedures when some of the assumptions on the underlying probabilistic model are relaxed has been part of statistical research for many years. In particular, if the original model is not able to accommodate atypical observations, the search for robust procedures relies on the use of overdispersion models. In the context of Bayesian parametric inference, the underlying probabilistic model jointly describes the uncertainty regarding the data and the parameter. This model can be represented as the product the sampling model for the data and the prior distribution for the parameter. Thus, we can consider overdispersion models to replace the sampling model, but we can also do the same with the prior distribution. There are many contributions in the literature describing robustness studies for either the sampling model or the prior distribution. One of the issues there is the choice of the overdispersion models. Here, we explore the use of predictive distributions as overdispersion models. We do this is the context of exponential and conjugate families of distributions, and discuss the idea for both the sampling model and the prior distribution. We show that, under certain conditions, this approach makes a robustness study with respect to the prior equivalent to a robustness study with respect to the sampling model, and vice versa.

**Families of multivariate distributions derived from conjugate exponential family models****Eduardo Gutiérrez-Peña,** IIMAS-UNAM, Mexico

Predictive distributions derived from exponential family models with conjugate priors are not necessarily exponential family models themselves, but contain the generating exponential family as a limiting case. Observations that are i.i.d. under the generating exponential family are only exchangeable under the corresponding predictive distributions, and hence are identically distributed but not necessarily independent. We use related ideas to derived families of multivariate "predictive" distributions whose marginal distributions belong to the same family but are not equal. These families can be regarded as robust versions of the original exponential families and may replace them as sampling models for data analysis.

**Dirichlet process mixtures of von Mises distributions****Alberto Contreras-Cristán,** IIMAS-UNAM, Mexico

The von Mises distribution is one of the most important models for circular data. It features a number of useful properties; for instance, it is a conditional offset distribution, it is the solution of a maximum entropy problem with moment restrictions, and it is a member of a two-parameter regular exponential family. On the other hand, Bayesian semiparametric models are popular due to their flexibility and robustness. In particular, assigning a Dirichlet process prior to some of the parameters of a circular model in order to build a Dirichlet process mixture has been studied with good results. In this work we discuss Dirichlet process mixtures of von Mises distributions, where the mixing contemplates both the mean direction and the concentration parameter. Posterior samples are obtained via an MCMC scheme which considers the use of Guttorp and Lockhart's conjugate prior for the parameters of von Mises model, and a recently proposed sampling scheme for the full conditional distribution of the concentration parameter arising from the conjugacy property.