Gabriella Salinetti

An overview of robust Bayesian analysis

SummaryRobust Bayesian analysis is the study of the sensitivity of Bayesian answers to uncertain inputs. This paper seeks to provide an overview of the subject, one that is accessible to statisticians outside the field. Recent developments in the area are also reviewed, though with very uneven emphasis.

On the convergence of closed-valued measurable multifunctions

In this paper we study the convergence almost everywhere and in measure of sequences of closed-valued multifunctions. We first give a number of criteria for the convergence of sequences of closed subsets. These results are used to obtain various characterizations for the convergence of measurable multifunctions. In particular we are interested in the convergence properties of (measurable) selections.

On the convergence in probability of random sets (measurable multifunctions)

It is shown that the convergence in probability of random sets introduced by Salinetti and Wets is consistent with the more standard definitions of convergence in probability, and is metric invariant.

Stability of Bayesian decisions

Abstract Stability of a Bayes decision is analysed with respect to small changes of the probability measure on the space of the states of nature. The problem leads to the study of continuity aspects of the infimum of the Bayes functional. These are approached through the epigraphical convergence of integral functionals as the minimal setting for convergence of infima.

Approximations of Bayes decision problems: the epigraphical approach

Solving Bayesian decision problems usually requires approximation procedures, all leading to study the convergence of the approximating infima. This aspect is analysed in the context of epigraphical convergence of integral functionals, as minimal context for convergence of infima. The results, applied to the Monte Carlo importance sampling, give a necessary and sufficient condition for convergence of the approximations of Bayes decision problems and sufficient conditions for a large class of Bayesian statistical decision problems.

Linearization Techniques in Bayesian Robustness

This paper deals with techniques which permit one to obtain the range of a posterior expectation through a sequence of linear optimizations. In the context of Bayesian robustness, the linearization algorithm plays a fundamental role. Its mathematical aspects and its connections with fractional programming procedures are reviewed and a few instances of its broad applicability are listed. At the end, some alternative approaches are briefly discussed.

Consistency of Statistical Estimators: the Epigraphical View

Epi-convergence as appropriate setting for convergence of optimization problems and epi-convergence as characterization of weak convergence of probability measures are jointly considered to analyze the asymptotic behaviour of statistical functionals. The twofold role is key in deriving consistency for a wide class of statistical estimators which includes most of the cases of interest.

Bayesian robustness on constrained density band classes

SummaryAn interesting class in Bayesian robustness is a ‘band’ of priors: its flexibility allows for different tail behaviours while excluding point masses. In this paper, we consider density band classes of priors with additional constraints modelling different available prior information: quantiles, moments, constraints derived from the probability of observables or from the dependence structure in a multidimensional setting. The proposed techniques allow us to obtain the range of quantities of interest that are not linear or ratio linear functionals. Numerical examples are provided.

Stability of Bayes Decisions and Applications

This article reviews the recent developments on the stability of Bayes decision problems. Following a brief account of the earlier works of Kadane and Chuang, who gave the initial formulation of the problem, the article focuses on a major contribution due to Salinetti and the subsequent developments. The discussion also includes applications of stability to the local and global robustness issues relating to the prior distribution and the loss function of a Bayes decision problem.

A note on the geometry of Bayesian global and local robustness

In this paper a geometric interpretation of the main quantities of interest in Bayesian robustness is presented. It helps in visualizing the relationships among global robustness, local sensitivity measures based on functional derivatives and the so-called linearization technique. An immediate geometric representation of general tools, already available in the literature on finite-dimensional fractional optimization, suggests an efficient algorithm to get the range of ratio linear functionals. The geometric understanding is also used to obtain the range of a local sensitivity measure and hence the probability measures in a class that are locally most (least) robust. Some inequalities are derived connecting the global range of the quantity of interest with local sensitivity measures based on the Gateaux derivative, leading to further considerations about the calibration of the local sensitivity.