Pier Giovanni Bissiri

A general framework for updating belief distributions

Summary We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data‐generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.

Multilevel Functional Principal Component Analysis of Façade Sound Insulation Data

This work analyzes data from an experimental study on facade sound insulation, consisting of independent repeated measurements executed by different laboratories on the same residential building. Mathematically, data can be seen as functions describing an acoustic parameter varying with frequency. The aim of this study is twofold. On one hand, considering the laboratory as the grouping variable, it is important to assess the within-group and between-group variability in the measurements. On the other hand, in building acoustics, it is known that sound insulation is more variable at low frequencies (from 50 to 100Hz), compared with higher frequencies (up to 5000Hz), and therefore, a multilevel functional model is employed to decompose the functional variance both at the measurement level and at the group level. This decomposition also allows for the ranking of the laboratories on the basis of measurement variability and performance at low frequencies (relative high variability) and over the whole spectrum. The former ranking is obtained via the principal component scores and the latter via an original Bayesian extension of the functional depth. Copyright

On Bayesian Learning from Bernoulli Observations

Abstract We provide a reason for Bayesian updating, in the Bernoulli case, even when it is assumed that observations are independent and identically distributed with a fixed but unknown parameter θ 0 . The motivation relies on the use of loss functions and asymptotics. Such a justification is important due to the recent interest and focus on Bayesian consistency which indeed assumes that the observations are independent and identically distributed rather than being conditionally independent with joint distribution depending on the choice of prior.

A Definition of Conditional Probability with Non-Stochastic Information

The current definition of a conditional probability enables one to update probabilities only on the basis of stochastic information. This paper provides a definition for conditional probability with non-stochastic information. The definition is derived by a set of axioms, where the information is connected to the outcome of interest via a loss function. An illustration is presented.

On the topological support of species sampling priors

Abstract: In Bayesian nonparametric statistics, it is crucial that the support of the prior is very large. Here, we consider species sampling priors. Such priors are widely used within mixture models and it has been shown in the literature that a large support for the mixing prior is essential to ensure the consistency of the posterior. In this paper, simple conditions are given that are necessary and sufficient for the support of a species sampling prior to be full. In particular, for proper species sampling priors, the condition is that the maximum size of the atoms of the corresponding process is small with positive probability. We apply this result to show that the main classes of species sampling priors known in literature have full support under mild conditions. Moreover, we find priors with a very simple construction still having full support.