Miguel A. Gómez-Villegas

A MATRIX VARIATE GENERALIZATION OF THE POWER EXPONENTIAL FAMILY OF DISTRIBUTIONS

ABSTRACT This paper proposes a matrix variate generalization of the power exponential distribution family, which can be useful in generalizing statistical procedures in multivariate analysis and in designing robust alternatives to them. An example is added to show an application of the generalization.

Multivariate Exponential Power Distributions as Mixtures of Normal Distributions with Bayesian Applications

This paper shows that a multivariate exponential power distribution is a scale mixture of normal distributions, with respect to a probability distribution function, when its kurtosis parameter belongs to the interval (0, 1]. The corresponding mixing probability distribution function is presented. This result is used to design, through a Bayesian hierarchical model, an algorithm to generate samples of the posterior distribution; this is applied to a problem of quantitative genetics.

Reconciling Bayesian and frequentist evidence in the point null testing problem

For the point null hypothesis testing problem it is shown that, in some situations, the classical evidence againstH0, expressed in terms of the p-value, is in the range of Bayesian measures of evidence. In these situations, it is therefore possible to reconcile measures of evidence between Bayesian and frequentist approaches. More specifically, for the class of unimodal, symmetric and nonincreasing prior distributions, it is shown that the infimum of the posterior probability ofH0 is numerically equal to thep value. The discrepancy which appears in the literature dedicated to this subject until now, is due to the form of the mixed distribution and not due to its use as a prior.

Sensitivity Analysis in Gaussian Bayesian Networks Using a Divergence Measure

This article develops a method for computing the sensitivity analysis in a Gaussian Bayesian network. The measure presented is based on the Kullback–Leibler divergence and is useful to evaluate the impact of prior changes over the posterior marginal density of the target variable in the network. We find that some changes do not disturb the posterior marginal density of interest. Finally, we describe a method to compare different sensitivity measures obtained depending on where the inaccuracy was. An example is used to illustrate the concepts and methods presented.

ε-contaminated priors in testing point null hypothesis: a procedure to determine the prior probability

In this paper the problem of testing a point null hypothesis from the Bayesian perspective and the relation between this and the classical approach is studied. A procedure to determine the mixed prior distribution is introduced and a justification for this construction based on a measure of discrepancy is given. Then, we compare a lower bound for the posterior probability, when the prior is in the class of [var epsilon]-contaminated distributions, of the point null hypothesis with the p-value.

Bayes factor in testing precise hypotheses

Substitution of a mixed prior distribution by a continuous one for the point null hypothesis testing problem is discussed. Conditions are established in order to approximate the Bayes factors for the two problems. Besides, trough this approximation an assignation of priorprobabilities is suggested.

Bayesian Analysis of Contingency Tables

ABSTRACT The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.

The stochastic control of process capability indices

In manufacturing science, process capability indices play a role analogous to economic indices in government statistics. The existing capability indices are passive devices whose main role is to retroactively monitor process capability. The have been developed under the restrictive assumption of process stability, and the procedures for using them are based on ad hoc rules. Using the normative point of view for decision making, it can be shown that some of the indices are, at best, convoluted special cases of a more general strategy; they can be justified only under special assumptions, and the manner in which they are currently used could lead to incoherent actions. The available process capability indices should therefore be abandoned and replaced by procedures that are normative, and also proactive with respect to both, prediction and control. An approach towards achieving this goal is proposed.

The effect of block parameter perturbations in Gaussian Bayesian networks: Sensitivity and robustness

In this work we study the effects of model inaccuracies on the description of a Gaussian Bayesian network with a set of variables of interest and a set of evidential variables. Using the Kullback-Leibler divergence measure, we compare the output of two different networks after evidence propagation: the original network, and a network with perturbations representing uncertainties in the quantitative parameters. We describe two methods for analyzing the sensitivity and robustness of a Gaussian Bayesian network on this basis. In the sensitivity analysis, different expressions are obtained depending on which set of parameters is considered inaccurate. This fact makes it possible to determine the set of parameters that most strongly disturbs the network output. If all of the divergences are small, we can conclude that the network output is insensitive to the proposed perturbations. The robustness analysis is similar, but considers all potential uncertainties jointly. It thus yields only one divergence, which can be used to confirm the overall sensitivity of the network. Some practical examples of this method are provided, including a complex, real-world problem.

A Bayesian decision procedure for testing multiple hypotheses in DNA microarray experiments

Abstract DNA microarray experiments require the use of multiple hypothesis testing procedures because thousands of hypotheses are simultaneously tested. We deal with this problem from a Bayesian decision theory perspective. We propose a decision criterion based on an estimation of the number of false null hypotheses (FNH), taking as an error measure the proportion of the posterior expected number of false positives with respect to the estimated number of true null hypotheses. The methodology is applied to a Gaussian model when testing bilateral hypotheses. The procedure is illustrated with both simulated and real data examples and the results are compared to those obtained by the Bayes rule when an additive loss function is considered for each joint action and the generalized loss 0–1 function for each individual action. Our procedure significantly reduced the percentage of false negatives whereas the percentage of false positives remains at an acceptable level.