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This paper deals with the Bayesian analysis of graphical models of marginal independence for three way contingency tables. Each marginal independence model corresponds to a particular factorization of the cell probabilities and a conjugate analysis based on Dirichlet prior can be performed. We illustrate a comprehensive Bayesian analysis of such models, involving suitable choices of prior parameters, estimation, model determination, as well as the allied computational issues. The posterior distributions of the marginal log-linear parameters is indirectly obtained using simple Monte Carlo schemes. The methodology is illustrated using two real data sets.

We propose a conjugate and conditional conjugate Bayesian analysis of models of marginal independence with a bi-directed graph representation. We work with Markov equivalent directed acyclic graphs (DAGs) obtained using the same vertex set with the addition of some latent vertices when required. The DAG equivalent model is characterised by a minimal set of marginal and conditional probability parameters. This allows us to use compatible prior distributions based on products of Dirichlet distributions. For models with DAG representation on the same vertex set, the posterior distribution and the marginal likelihood is analytically available, while for the remaining ones a data augmentation scheme introducing additional latent variables is required. For the latter, we estimate the marginal likelihood using Chib’s (1995) estimator. Additional implementation details including identifiability of such models is discussed. For all models, we also provide methodology for the computation of the posterior distributions of the marginal log-linear parameters based on a simple transformation of the simulated values of the probability parameters. We illustrate our method using a popular 4-way dataset.

We present a Bayesian approach for the analysis of rating data when a scaling component is taken into account. Model-based probability effect measures for comparing distributions of several groups, adjusted for explanatory variables affecting both location and scale components, are computed. Markov Chain Monte Carlo techniques are implemented to obtain parameter estimates and the mentioned measures. An analysis on students’ evaluation of a university orientation service is carried out to assess the performance of the method and make more valuable the decision making process of university players (stakeholders).

This article deals with ordinal effect measures in the cup models. They are mixture models for ordinal data with an uncertainty component, where the Uniform distribution is used to model indecision and the standard cumulative model is employed for the analysis of evaluation. We present probability-based measures for comparing clusters on ratings, while adjusting for other explanatory variables, and discuss marginal effects to address the interpretation of the results on the extreme categories of a cyber risk scale.

In this paper we propose a novel Bayesian methodology for Value-at-Risk computation based on parametric Product Partition Models. Value-at-Risk is a standard tool for measuring and controlling the market risk of an asset or portfolio, and is also required for regulatory purposes. Its popularity is partly due to the fact that it is an easily understood measure of risk. The use of Product Partition Models allows us to remain in a Normal setting even in the presence of outlying points, and to obtain a closed-form expression for Value-at-Risk computation. We present and compare two different scenarios: a product partition structure on the vector of means and a product partition structure on the vector of variances. We apply our methodology to an Italian stock market data set from Mib30. The numerical results clearly show that Product Partition Models can be successfully exploited in order to quantify market risk exposure. The obtained Value-at-Risk estimates are in full agreement with Maximum Likelihood approaches, but our methodology provides richer information about the clustering structure of the data and the presence of outlying points. © 2012 Copyright Taylor and Francis Group, LLC.

This contribution deals with effect measures for covariates in ordinal data models to address the interpretation of the results on the extreme categories of the scales, evaluate possible response styles, and motivate collapsing of extreme categories. It provides a simpler interpretation of the influence of the covariates on the probability of the response categories both in standard cumulative link models under the proportional odds assumption and in the recent extension of the Combination of Uncertainty and Preference of the respondents models, the mixture models introduced to account for uncertainty in rating systems. The article shows by means of marginal effect measures that the effects of the covariates are underestimated when the uncertainty component is neglected. Visualization tools for the effect of covariates are proposed, and measures of relative size and partial effect based on rates of change are evaluated by the use of real data sets.