Parametric Modelling of Multivariate Count Data Using Probabilistic Graphical Models

Multivariate count data are defined as the number of items of different categories issued from sampling within a population, which individuals are grouped into categories. The analysis of multivariate count data is a recurrent and crucial issue in numerous modelling problems, particularly in the fields of biology and ecology (where the data can represent, for example, children counts associated with multitype branching processes), sociology and econometrics. We focus on I) Identifying categories that appear simultaneously, or on the contrary that are mutually exclusive. This is achieved by identifying conditional independence relationships between the variables; II)Building parsimonious parametric models consistent with these relationships; III) Characterising and testing the effects of covariates on the joint distribution of the counts. To achieve these goals, we propose an approach based on graphical probabilistic models, and more specifically partially directed acyclic graphs

A Markov Basis for Conditional Test of Common Diagonal Effect in Quasi-Independence Model for Square Contingency Tables

In two-way contingency tables we sometimes find that frequencies along the diagonal cells are relatively larger(or smaller) compared to off-diagonal cells, particularly in square tables with the common categories for the rows and the columns. In this case the quasi-independence model with an additional parameter for each of the diagonal cells is usually fitted to the data. A simpler model than the quasi-independence model is to assume a common additional parameter for all the diagonal cells. We consider testing the goodness of fit of the common diagonal effect by Markov chain Monte Carlo (MCMC) method. We derive an explicit form of a Markov basis for performing the conditional test of the common diagonal effect. Once a Markov basis is given, MCMC procedure can be easily implemented by techniques of algebraic statistics. We illustrate the procedure with some real data sets.Comment: 15 page

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of `conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.Comment: 38 pages, lots of picture

Disintegration and Bayesian Inversion via String Diagrams

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.Comment: Accepted for publication in Mathematical Structures in Computer Scienc

Food price policies and the distribution of body mass index: Theory and empirical evidence from France

This paper uses French food-expenditure data to examine the effect of the local prices of 23 food product categories on the distribution of Body Mass Index (BMI) in a sample of French adults. A dynamic choice model using standard assumptions in Physiology is developed. It is shown that the slope of the price-BMI relationship is affected by the individual's Physical Activity Level (PAL). When the latter is unobserved, identi cation of price effects at conditional quantiles of the BMI distribution requires quantile independence between PAL and the covariates, especially income. Using quantile regressions, unconditional BMI distributions can then be simulated for various price policies. In the preferred scenario, increasing the price of soft drinks, breaded proteins, deserts and pastries, snacks and ready-meals by 10%, and reducing the price of fruit and vegetables in brine by 10% would decrease the prevalence of overweight and obesity by 24% and 33% respectively. The fall in health care expenditures would represente up to 1.39% of total health care spendings in 2004.obesity ; overweight ; quantile regression ; food prices ; physical activity

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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