Sandra Fortini

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.

Concentration functions and Bayesian robustness

Abstract The concentration function, extending the classical notion of Lorenz curve, is well suited for comparing probability measures. Such a feature can be useful in different issues in Bayesian robustness, when a probability measure is deemed a baseline to be compared with other measures by means of their functional forms. Neighbourhood classes Γ of probability measures, including well-known ones, can be defined through the concentration function and both prior and posterior expectations of given functions of the unknown parameter are studied. The ranges of such expectations over Γ can be found, restricting the search among the extremal measures in Γ. The concentration function can be also used as a criterion to assess posterior robustness, when considering sensitivity to changes in the likelihood and the prior.

A characterization for mixtures of semi-Markov processes

Mixtures of recurrent semi-Markov processes are characterized through a partial exchangeability condition of the array of successor states and holding times. A stronger invariance condition on the joint law of successor states and holding times leads to mixtures of Markov laws.

Spherical symmetry: An elementary justification

The present paper includes characterizations of the conditions of spherical symmetry and of centered spherical symmetry. These characterizations provide an empirical justification for the above mentioned conditions of symmetry.

Predictive characterization of mixtures of Markov chains

Predictive constructions are a powerful way of characterizing the probability law of stochastic processes with certain forms of invariance, such as exchangeability or Markov exchangeability. When de Finetti-like representation theorems are available, the predictive characterization implicitly defines the prior distribution, starting from assumptions on the observables; moreover, it often helps designing efficient computational strategies. In this paper we give necessary and sufficient conditions on the sequence of predictive distributions such that they characterize a Markov exchangeable probability law for a discrete valued process X. Under recurrence, Markov exchangeable processes are mixtures of Markov chains. Thus, our results help checking when a predictive scheme characterizes a prior for Bayesian inference on the unknown transition matrix of a Markov chain. Our predictive conditions are in some sense minimal sufficient conditions for Markov exchangeability; we also provide predictive conditions for recurrence. We illustrate their application in relevant examples from the literature and in novel constructions.

Concentration function and sensitivity to the prior

In robust bayesian analysis, ranges of quantities of interest (e. g. posterior means) are usually considered when the prior probability measure varies in a class Γ. Such quantities describe the variation of just one aspect of the posterior measure. The concentration function describes changes in the posterior probability measure more globally, detecting differences in probability concentration and providing, simultaneously, bounds on the posterior probability of all measurable subsets. In this paper, we present a novel use of the concentration function, and two concentration indices, to study such posterior changes for a general class Γ, restricting then our attention to some ∈-contamination classes of priors.

On the Use of the Concentration Function in Bayesian Robustness

We present applications of the concentration function in both global and local sensitivity analyses, along with its connection with Choquet capacities.

Concentration function and coefficients of divergence for signed measures

Comparisons among probability measures are rather frequent in many statistical problems and they are sometimes performed through the coefficients of divergence or the concentration functions with respect to a reference measure. Extending the notion of Lorenz-Gini curve, the concentration function studies the discrepancy between two probability measures Π and Π0.