Giovanni Petris

On mixtures of distributions of Markov chains

Abstract Let X be a chain with discrete state space I , and V be the matrix of entries V i , n , where V i , n denotes the position of the process immediately after the n th visit to i . We prove that the law of X is a mixture of laws of Markov chains if and only if the distribution of V is invariant under finite permutations within rows (i.e., the V i , n s are partially exchangeable in the sense of de Finetti). We also prove that an analogous statement holds true for mixtures of laws of Markov chains with a general state space and atomic kernels. Going back to the discrete case, we analyze the relationships between partial exchangeability of V and Markov exchangeability in the sense of Diaconis and Freedman. The main statement is that the former is stronger than the latter, but the two are equivalent under the assumption of recurrence. Combination of this equivalence with the aforesaid representation theorem gives the Diaconis and Freedman basic result for mixtures of Markov chains.

Kriging for Hilbert-space valued random fields

We develop a comprehensive framework for linear spatial prediction in Hilbert spaces. We explore the problem of Best Linear Unbiased (BLU) prediction in Hilbert spaces through an original point of view, based on a new Operatorial definition of Kriging. We ground our developments on the theory of Gaussian processes in function spaces and on the associated notion of measurable linear transformation. We prove that our new setting allows (a) to derive an explicit solution to the problem of Operatorial Ordinary Kriging, and (b) to establish the relation of our novel predictor with the key concept of conditional expectation of a Gaussian measure. Our new theory is posed as a unifying theory for Kriging, which is shown to include the Kriging predictors proposed in the literature on Functional Data through the notion of finite-dimensional approximations. Our original viewpoint to Kriging offers new relevant insights for the geostatistical analysis of either finite- or infinite-dimensional georeferenced dataset.

Some critical aspects of the use of exchangeability in statistics

This expository paper provides a framework for analysing de Finettis representation theorem for exchangeable finitely additive probabilities. Such an analysis is justified by reasoning of statistical nature, since it is shown that the abandonment of the axiom of σ-additivity has some noteworthy consequences on the common interpretation of the Bayesian paradigm. The usual (strong) fromulation of de Finettis theorem is deduced from the finitely additive (weak) formulation, and it is used to solve the problem of stating the existence of a stochastic process, with given finite-dimensional probability distributions, whose sample paths are probability distributions. It is of importance, in particular, to specify prior distributions for nonparametric inferential problems in a Bayesian setting.

Sexually Violent Predators: Toward Reasonable Estimates of Recidivism Base Rates

The sexual recidivism rate of sex offenders is a controversial issue. Perhaps as controversial is the sexual recidivism rate of the select group of sex offenders who are examined pursuant to sexually violent predator (SVP) statutes. At present, reliable estimates of SVP recidivism are unavailable. We propose that reasonable estimates of SVP recidivism can be reached by considering three available pieces of data: (i) a likely recidivism rate of the general population of sex offenders; (ii) procedures typically followed by jurisdictions that civilly commit sex offenders; and (iii) classification accuracy of procedures. Although sexual recidivism rates vary across jurisdictions, the results of our analyses suggest sex offenders referred for examination pursuant to SVP statutes recidivate at substantially higher rates than typical sex offenders. Our results further suggest that sex offenders recommended for commitment as SVPs recidivate at even greater rates than SVP respondents who are not recommended for commitment. We discuss practice and policy implications of these findings.