Local robustness measures for posterior summaries

This paper deals with measures of local robustness for particular Bayesian quantities, i.e. posterior summaries. We build a framework where any Bayesian quantity can be seen as a posterior functional and its sensitivity to all inputs is checked. First, we use the Gateaux derivatives to measure the impact on posterior summaries of perturbations of prior or sampling models, giving some general expressions. Such quantities capture both a ’data effect’ and a ’model effect’ on the functional. Secondly, we check the sensitivity to one observation in the sample, once a particular combination of prior/sampling models has been chosen. Moreover, we propose a new estimator of the Bayes factor for practical implementation. Finally, illustrative examples on sensitivity analysis are provided and discussed.

Cervical mucus symptom and daily fecundability: First results from a new data base

A prospective study allowed, with the collaboration of Italian centres providing services on natural family planning, to collect data on 2755 menstrual cycles of 193 women. With the information available on daily characteristics of the cervical mucus and intercourse episodes, was constructed a data base. Taking the day of the peak mucus as a conventional marker of ovulation, the base allowed to identify length (12 days) and location of a window of potential fertility, with the highest level of conception probabilities confined to the central 5-6 days. Univariate analysis provided evidence of the impact on fecundability of woman?s age and basic infertile pattern of a cycle. Several approaches of analysis allowed to underline the relationship between daily mucus characteristics and fecundability levels

A robust bayesian approach to portfolio selection

This thesis aims at studying the local robustness properties of Bayesian posterior summaries and deriving a robust procedure to estimate Bayesian Mean-Variance weights in a portfolio selection problem. In the first part, we study the local robustness of Bayesian estimators. In particular, we build a framework wherein any Bayesian quantity can be seen as a posterior functional. In this way it becomes possible to construct different robustness measures. We derive local influence measures for posterior summaries with respect both to prior and sampling distributions and to observations. Then we address the issue of efficient implementation of the derived measures through MCMC algorithms. In the second part, we deal with the problem of robust estimation in a Bayesian context, providing a useful result to generalize univariate robust distributions to the multivariate case. We also propose criteria to assess in which cases a robust model is recommended and how to choose among estimates obtained with different distributions. Finally, we consider in the third part the Mean-Variance portfolio selection problem. We provide evidence that if the data are normally distributed the Bayesian approach works better than the Certainty Equivalence approach, nevertheless this is no longer true when the data contain few outlying observations. Moreover, we compute useful measures of sensitivity of Bayesian weights and we construct and implement a new estimator which is robust with respect to the presence of 'extreme' observations

Cervical mucus symptom and daily fecundability: First results from a new data base

A prospective study allowed, with the collaboration of Italian centres providing services on natural family planning, to collect data on 2755 menstrual cycles of 193 women. With the information available on daily characteristics of the cervical mucus and intercourse episodes, was constructed a data base. Taking the day of the peak mucus as a conventional marker of ovulation, the base allowed to identify length (12 days) and location of a window of potential fertility, with the highest level of conception probabilities confined to the central 5-6 days. Univariate analysis provided evidence of the impact on fecundability of woman?s age and basic infertile pattern of a cycle. Several approaches of analysis allowed to underline the relationship between daily mucus characteristics and fecundability levels