Bayesian Nonparametrics, Robustness and Fréchet Classes
Connections between Bayesian nonparametric inference and the classes of probability measures with known marginals (Fréchet classes) are investigated. In particular, a class of Dirichlet processes with parameters proportional to probability measures in a Fréchet class will be considered. Relations between two statistical characters are then studied, by presenting a new, Bayesian interpretation of the Kolmogorov-Smirnov test statistics, by looking for upper and lower bounds on Bayes estimators, under squared loss function, of some indices, like covariance and concordance, and by considering positive dependence and ``optimal'' estimators of distribution functions according to some criteria (e.g. posterior regret). It should be pointed out that, in general, bounds are found when considering Dirichlet processes whose parameter is either one of the prior Fréchet bounds or one obtained by updating them after observing a sample. Properties of some classes of probability measures are investigated too, along with their relations. In general, computing upper and lower bounds in a suitable larger class turns out to be easier than in the true class, whereas it is shown that the bounds on the two classes coincide. It is proved that, given a probability measure chosen from a Dirichlet process, then it follows that the conditional probabilities, conditioned on a given subset, are still realisations of a suitable Dirichlet process. It is therefore possible to find Bayes estimators, under squared loss function, of conditional probabilities and upper and lower bounds on them. It is shown that dilation occurs when comparing Bayes estimators of set probabilities and conditional set probabilities, for both prior and posterior Dirichlet processes. A new, nonparametric approach to Bayesian robustness is introduced, considering some distances between Dirichlet processes, the distance between distributions of set probabilities and the range spanned either by the probability of some subsets in the space of the probability measures or by some optimal estimators of the means or the distribution. Prior and posterior quantities are compared, too. Finally, an application of some results to earthquakes occurring in an Italian area called Sannio Matese is shown.