Guido Consonni

A Bayesian Method for Combining Results from Several Binomial Experiments

Abstract The problem of combining information related to I binomial experiments, each having a distinct probability of success θ i , is considered. Instead of using a standard exchangeable prior for θ; = (θ1, …, θ I ), we propose a more flexible distribution that takes into account various degrees of similarity among the θ i s. Using ideas developed by Malec and Sedransk, we consider a partition g of the experiments and take the θ i s belonging to the same partition subset to be exchangeable and the θ i s belonging to distinct subsets to be independent. Next we perform Bayesian inference on θ; conditional on g. Of course, one is typically uncertain about which partition to use, and so a prior distribution is assigned on a set of plausible partitions g. The final inference on θ; is obtained by combining the conditional inferences according to the posterior distribution of g. The methodology adopted in this article offers a wide flexibility in structuring the dependence among the θ i s. This allows the ...

Conjugate Priors for Exponential Families Having Quadratic Variance Functions

Abstract Consider a natural exponential family parameterized by θ. It is well known that the standard conjugate prior on θ is characterized by a condition of posterior linearity for the expectation of the model mean parameter μ. Often, however, this family is not parameterized in terms of θ but rather in terms of a more usual parameter, such at the mean μ. The main question we address is: Under what conditions does a standard conjugate prior on μ induce a linear posterior expectation on μ itself? We prove that essentially this happens iff the exponential family has quadratic variance function. A consequence of this result is that the standard conjugate on μ coincides with the prior on μ induced by the standard conjugate on θ iff the variance function is quadratic. The rest of the article covers more specific issues related to conjugate priors for exponential families. In particular, we analyze the monotonicity of the expected posterior variance for μ with respect to the sample size and the hyperparameter ...

Mean-field variational approximate Bayesian inference for latent variable models

The ill-posed nature of missing variable models offers a challenging testing ground for new computational techniques. This is the case for the mean-field variational Bayesian inference. The behavior of this approach in the setting of the Bayesian probit model is illustrated. It is shown that the mean-field variational method always underestimates the posterior variance and, that, for small sample sizes, the mean-field variational approximation to the posterior location could be poor.

Exponential and bayesian conjugate families: Review and extensions

SummaryThe notion of a conjugate family of distributions plays a very important role in the Bayesian approach to parametric inference. One of the main features of such a family is that it is closed under sampling, but a conjugate family often provides prior distributions which are tractable in various other respects. This paper is concerned with the properties of conjugate families for exponential family models. Special attention is given to the class of natural exponential families having a quadratic variance function, for which the theory is particularly fruitful. Several classes of conjugate families have been considered in the literature and here we describe some of their most interesting features. Relationships between such classes are also discussed. Our aim is to provide a unified approach to the theory of conjugate families for exponential family likelihoods. An important aspect of the theory concerns reparameterisations of the exponential family under consideration. We briefly review the concept of a conjugate parameterisation, which provides further insight into many of the properties discussed throughout the paper. Finally, further implications of these results for Bayesian conjugate analysis of exponential families are investigated.

Bayesian analysis of extreme values by mixture modeling

Modeling of extreme values in the presence of heterogeneity is still a relatively unexplored area. We consider losses pertaining to several related categories. For each category, we view exceedances over a given threshold as generated by a Poisson process whose intensity is regulated by a specific location, shape and scale parameter. Using a Bayesian approach, we develop a hierarchical mixture prior, with an unknown number of components, for each of the above parameters. Computations are performed using Reversible Jump MCMC. Our model accounts for possible grouping effects and takes advantage of the similarity across categories, both for estimation and prediction purposes. Some guidance on the specification of the prior distribution is provided, together with an assessment of inferential robustness. The method is illustrated throughout using a data set on large claims against a well-known insurance company over a 15-year period.

Compatibility of Prior Specifications Across Linear Models

Bayesian model comparison requires the specification of a prior distribution on the parameter space of each candidate model. In this connection two concerns arise: on the one hand the elicitation task rapidly becomes prohibitive as the number of models increases; on the other hand numerous prior specifications can only exacerbate the well-known sensitivity to prior assignments, thus producing less dependable conclusions. Within the subjective framework, both difficulties can be counteracted by linking priors across models in order to achieve simplification and compatibility; we discuss links with related objective approaches. Given an encompassing, or full, model together with a prior on its parameter space, we review and summarize a few procedures for deriving priors under a submodel, namely marginalization, conditioning, and Kullback--Leibler projection. These techniques are illustrated and discussed with reference to variable selection in linear models adopting a conventional

Conditionally Reducible Natural Exponential Families and Enriched Conjugate Priors

g

Tests Based on Intrinsic Priors for the Equality of Two Correlated Proportions

-prior; comparisons with existing standard approaches are provided. Finally, the relative merits of each procedure are evaluated through simulated and real data sets.

Testing Hardy–Weinberg equilibrium: An objective Bayesian analysis

Consider a standard conjugate family of prior distributions for a vector-parameter indexing an exponential family. Two distinct model parameterizations may well lead to standard conjugate families which are not consistent, i.e. one family cannot be derived from the other by the usual change-of-variable technique. This raises the problem of finding suitable parameterizations that may lead to enriched conjugate families which are more flexible than the traditional ones. The previous remark motivates the definition of a new property for an exponential family, named conditional reducibility. Features of conditionally-reducible natural exponential families are investigated thoroughly. In particular, we relate this new property to the notion of cut, and show that conditionally-reducible families admit a reparameterization in terms of a vector having likelihood-independent components. A general methodology to obtain enriched conjugate distributions for conditionally-reducible families is described in detail, generalizing previous works and more recent contributions in the area. The theory is illustrated with reference to natural exponential families having simple quadratic variance function.

Objective Bayesian search of Gaussian directed acyclic graphical models for ordered variables with non-local priors

Correlated proportions arise in longitudinal (panel) studies. A typical example is the “opinion swing“ problem: “Has the proportion of people favoring a politician changed after his recent speech to the nation on TV?“ Because the same group of individuals is interviewed before and after the speech, the two proportions are correlated. A natural null hypothesis to be tested is whether the corresponding population proportions are equal. A standard Bayesian approach to this problem has already been considered in the literature, based on a Dirichlet prior for the cell probabilities of the underlying 2 × 2 table under the alternative hypothesis, together with an induced prior under the null. With a lack of specific prior information, a diffuse (e.g., uniform) distribution may be used. We claim that this approach is not satisfactory, because in a testing problem one should make sure that the prior under the alternative is adequately centered around the region specified by the null, in order to obtain a fairer comparison between the two hypotheses, especially when the data are in reasonable agreement with the null. Following an intrinsic prior methodology, we develop two strategies for the construction of a collection of objective priors increasingly peaked around the null. We provide a simple interpretation of their structure in terms of weighted imaginary sample scenarios. We illustrate our method by means of three examples, carrying out sensitivity analysis and providing comparison with existing results.