Approximate Bayesian Inferences: Numerical experiences when measuring the loss of information (original) (raw)

Bayesian evaluation of non-admissible conditioning: the case of Fisher test

1998

We rst analyse the general problem of admissible conditioning and next consider the evaluation of the loss of information when a non-admissible conditioning is used as an approximation of the exact posterior distribution. Considering the case of Fisher test, we evaluate from a Bayesian point of view how much information is lost when the sampling process for a 2x2 contingency table is analysed conditionally on the two margins. This loss of information due to non-admissible conditioning is evaluated for di erent sampling models and with respect to the entropy divergence and to the Hellinger distance between the exact and the approximate posterior distributions and with respect to relative risks based on a quadratic loss function. The numerical results obtained through simulation indicate that for a speci c range of parameters the loss of information increases with the sample size and decreases with the precision of the a priori distribution. Hence such an approximation is shown to be a non-asymptotic one.

EVALUATING APPROXIMATIONS OF BAYESIAN SOLUTIONS: THE CASE OF FISHER TEST

1993

In this paper, we evaluate from a Bayesian point of view how much information is lost when the sampling process for the 2x2 contingency table is speci ed conditionally on the two margins as in the exact test of Fisher. We rst analyse the general problem of admissible conditioning and next consider the evaluation of the loss of information when a nonadmissible conditioning is used for an approximation of the exact posterior distribution. Turning to the Fisher test, three di erent sampling models are considered and three comparisons are designed between the exact and the approximate posterior distributions. The numerical results obtained through simulation indicate that for a speci c range of parameters the loss of information increases with the sample size and decreases with the precision of the a priori distribution.

Objective Bayesian Analysis of Contingency Tables

2002

The statistical analysis of contingency tables is typically carried out with a hypothesis test. In the Bayesian paradigm, default priors for hypothesis tests are typically improper, and cannot be used. Although such priors are available, and proper, for testing contingency tables, we show that for testing independence they can be greatly improved on by so-called intrinsic priors. We also argue that because there is no realistic situation that corresponds to the case of conditioning on both margins of a contingency table, the proper analysis of an a × b contingency table should only condition on either the table total or on only one of the margins. The posterior probabilities from the intrinsic priors provide reasonable answers in these cases. Examples using simulated and real data are given.

Bayesian inference for contingency tables with given marginals

Journal of the Italian Statistical Society, 1995

In this paper we introduce a class of prior distributions for contingency tables with given marginals. We are interested in the structure of concordance/discordance of such tables. There is actually a minor limitation in that the marginals are required to assume only rational values. We do argue, though, that this is not a serious drawback for all applicatory purposes. The posterior and predictive distributions given an Msample are computed. Examples of Bayesian estimates of some classical indices of concordance are also given. Moreover, we show how to use simulation in order to overcome some difficulties which arise in the computation of the posterior distribution.

ε-Contaminated priors in contingency tables

TEST, 2008

The r × s table is used for discussing different approaches to statistical inference. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. We consider testing equality of proportions of independent multinomial distributions when the common proportions are known. A lower bound of the posterior probabilities of the null hypothesis is calculated with respect to a mixture of point prior on the null and a ε−contaminated prior on the proportions under the alternative. The resulting Bayes tests are compared numerically to Pearson's χ 2 in a number of examples. For the examined examples the lower bound and the p-value can be made close. The obtained results are generalized when the common proportions vector under the null is unknown or with known functional form.

Tighter estimates for the posteriors of imprecise prior and conditional probabilities

IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 1996

White [13] and Snow [lo] have presented approximate characterizations for the posterior probabilities when the priors and conditionals are specified through linear constraint systems. This paper extends their results by developing alternative linear inequality approximations for posterior probabilities, with a particular focus on partial information expressed in terms of (i) bounds on the components of probability vectors and (ii) hounds on ratios of probabilities.

Bayesian Analysis of Contingency Tables

Communications in Statistics - Theory and Methods, 2005

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.

Bayesian Inference Under Partial Prior Information

Scandinavian Journal of Statistics, 2003

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

Efficient and accurate approximate Bayesian inference with an application to insurance data

Computational Statistics & Data Analysis, 2008

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