An Application of MCMC Methods and of Mixtures of Laws for the Determination of the Purity of a Product

The aim of this article is to illustrate practical issues that may arise when applying MCMC methods in a real case. MCMC methods are used over a model of mixture of distributions, in order to determine honnesty specifications about some kind of product. First, assuming a known number of components, parameters of each component are estimated using Gibbs sampling. Convergence and label-switching are taken into account. Then, determination of the number of components is mentioned, through model selection with Bayes Factors. Résumé Le but de cet article est d’illustrer des questions partiques qui peuvent survenir lorsque l’on applique des méthodes MCMC à un cas réel. Ces dernières sont utilisées sur un modèle de mélange de lois, afin de déterminer des spécifications d’honnêteté sur un certain type de produits. En travaillant un premier temps à nombre de composantes fixé, les paramètres de chaque composante sont estimés à l’aide de l’échantilloneur de Gibbs. La convergence et le label-switching sont pris en compte. Enfin, la détermination du nombre de composantes est envisagé, à l’aide des Facteurs de Bayes. 1. The Problem and its Modelization A particular kind of products, avalaible in stores almost everywhere, is subject to different kind of adulteration (or fraud). Some products are “pure” and thus have low glucose and xylose rates, while some other have been chemically modified (adjunction of extra-components), wich reduces their quality and increases their glucose and xylose rates. Given a set of 1002 observations of the couple (glucose rate, xylose rate), can we determine : (i) the number K of kinds of production, and their parameters (mean, standard deviation), (K − 1 different possible modifications, plus one for pure products), (ii) the proportion of each kind of production, and (iii) a way to detect wether a product