Francesco Bertolino

Default Bayesian analysis of the Behrens–Fisher problem

Abstract In the Bayesian approach, the Behrens–Fisher problem has been posed as one of estimation for the difference of two means. No Bayesian solution to the Behrens–Fisher testing problem has yet been given due, perhaps, to the fact that the conventional priors used are improper. While default Bayesian analysis can be carried out for estimation purposes, it poses difficulties for testing problems. This paper generates sensible intrinsic and fractional prior distributions for the Behrens–Fisher testing problem from the improper priors commonly used for estimation. It allows us to compute the Bayes factor to compare the null and the alternative hypotheses. This default procedure of model selection is compared with a frequentist test and the Bayesian information criterion. We find discrepancy in the sense that frequentist and Bayesian information criterion reject the null hypothesis for data, that the Bayes factor for intrinsic or fractional priors do not.

Bayesian model selection approach to analysis of variance under heteroscedasticity

The classical Bayesian approach to analysis of variance assumes the homoscedastic condition and uses conventional uniform priors on the location parameters and on the logarithm of the common scale. The problem has been developed as one of estimation of location parameters. We argue that this does not lead to an appropriate Bayesian solution. A solution based on a Bayesian model selection procedure is proposed. Our development is in the general heteroscedastic setting in which a frequentist exact test does not exist. The Bayes factor involved uses intrinsic and fractional priors which are used instead of the usual default prior distributions for which the Bayes factor is not well defined. The behaviour of these Bayes factors is compared with the Bayesian information criterion of Schwarz and the frequentist asymptotic approximations of Welch and Brown and Forsythe.

Multiple Bayes Factors for Testing Hypotheses

Abstract Partial and multiple Bayes factors are introduced to obtain pairwise comparisons of hypotheses in a statistical experiment with a partition on the parameter space. Robust Bayesian analyses are performed by introducing suitable classes of priors and by calculating lower and upper bounds of Bayes factors and posterior probabilities. Classes of intuitively meaningful priors are introduced, including unimodal densities without the constraint of symmetry for the case of precise hypotheses. Procedures for the corresponding optimizations are specified, and examples are given.

Bayesian Inference Under Partial Prior Information

Partial prior information on the marginal distribution of an observable random variable is considered. When this information is incorporated into the statistical analysis of an assumed parametric model, the posterior inference is typically non-robust so that no inferential conclusion is obtained. To overcome this difficulty a method based on the standard default prior associated to the model and an intrinsic procedure is proposed. Posterior robustness of the resulting inferences is analysed and some illustrative examples are provided. Copyright 2003 Board of the Foundation of the Scandinavian Journal of Statistics..

A strategy analysis for genetic association studies with known inbreeding

BackgroundAssociation studies consist in identifying the genetic variants which are related to a specific disease through the use of statistical multiple hypothesis testing or segregation analysis in pedigrees. This type of studies has been very successful in the case of Mendelian monogenic disorders while it has been less successful in identifying genetic variants related to complex diseases where the insurgence depends on the interactions between different genes and the environment. The current technology allows to genotype more than a million of markers and this number has been rapidly increasing in the last years with the imputation based on templates sets and whole genome sequencing. This type of data introduces a great amount of noise in the statistical analysis and usually requires a great number of samples. Current methods seldom take into account gene-gene and gene-environment interactions which are fundamental especially in complex diseases. In this paper we propose to use a non-parametric additive model to detect the genetic variants related to diseases which accounts for interactions of unknown order. Although this is not new to the current literature, we show that in an isolated population, where the most related subjects share also most of their genetic code, the use of additive models may be improved if the available genealogical tree is taken into account. Specifically, we form a sample of cases and controls with the highest inbreeding by means of the Hungarian method, and estimate the set of genes/environmental variables, associated with the disease, by means of Random Forest.ResultsWe have evidence, from statistical theory, simulations and two applications, that we build a suitable procedure to eliminate stratification between cases and controls and that it also has enough precision in identifying genetic variants responsible for a disease. This procedure has been successfully used for the beta-thalassemia, which is a well known Mendelian disease, and also to the common asthma where we have identified candidate genes that underlie to the susceptibility of the asthma. Some of such candidate genes have been also found related to common asthma in the current literature.ConclusionsThe data analysis approach, based on selecting the most related cases and controls along with the Random Forest model, is a powerful tool for detecting genetic variants associated to a disease in isolated populations. Moreover, this method provides also a prediction model that has accuracy in estimating the unknown disease status and that can be generally used to build kit tests for a wide class of Mendelian diseases.

Unscaled Bayes factors for multiple hypothesis testing in microarray experiments.

Multiple hypothesis testing collects a series of techniques usually based on p-values as a summary of the available evidence from many statistical tests. In hypothesis testing, under a Bayesian perspective, the evidence for a specified hypothesis against an alternative, conditionally on data, is given by the Bayes factor. In this study, we approach multiple hypothesis testing based on both Bayes factors and p-values, regarding multiple hypothesis testing as a multiple model selection problem. To obtain the Bayes factors we assume default priors that are typically improper. In this case, the Bayes factor is usually undetermined due to the ratio of prior pseudo-constants. We show that ignoring prior pseudo-constants leads to unscaled Bayes factor which do not invalidate the inferential procedure in multiple hypothesis testing, because they are used within a comparative scheme. In fact, using partial information from the p-values, we are able to approximate the sampling null distribution of the unscaled Bayes factor and use it within Efrons multiple testing procedure. The simulation study suggests that under normal sampling model and even with small sample sizes, our approach provides false positive and false negative proportions that are less than other common multiple hypothesis testing approaches based only on p-values. The proposed procedure is illustrated in two simulation studies, and the advantages of its use are showed in the analysis of two microarray experiments.

Bayesian model selection methods for nonnested models

In the Bayesian approach to model selection and prediction, the posterior probability of each model under consideration must be computed. In the presence of weak prior information we need using default or automatic priors, that are typically improper, for the parameters of the models. However this leads to ill-defined posterior probabilities.