arXiv subjects

Pier Giovanni Bissiri

Publications and source records attributed to Pier Giovanni Bissiri.

7 recordsLinked to original sources

arXiv2026

Nonparametric Bayesian inference for the Gini-Simpson index

Many statistical problems concern the analysis of species distributions or, more generally, of discrete labeled quantities. Assessing species diversity constitutes a key step toward understanding population structure, and the Gini-Simpson index is among the most widely adopted diversity measures. In this manuscript, we examine several well-established nonparametric prior models for species frequencies and compare them with a newly proposed distribution within this framework. Specifically, we demonstrate that the conventional symmetric Dirichlet distribution leads to certain undesirable properties in terms of expectation and dispersion. These limitations can be mitigated by adopting an alternative symmetric Dirichlet specification, in which the parameter depends on the number of species. This modified formulation is characterized by analytical tractability and interpretability of its main summaries. Furthermore, when the number of distinct species is infinite, the classical Ferguson Dirichlet process exhibits unsatisfactory behavior compared to the more general Poisson-Dirichlet model. Notably, within this latter model, the posterior mean of the diversity index can be expressed as a convex combination of the optimal classical unbiased estimator and the prior expectation. Theoretical results are further supported by asymptotic analyses and systematically compared with their classical counterparts.

stat.ME↗

arXiv2026

Posterior inference via Hill's prediction model

This paper is concerned with the construction of prior free posterior distributions which rely on the use of one step ahead predictive distribution functions. These are typically more straightforward to motivate than prior distributions. Recent interest has been with Hill's $A_n$ prediction model through what has become known as conformal prediction. This model predicts the next observation to lie with equal probability in the intervals created by the observed data. The prediction model generates complete data sets which can be used to provide posterior inference on any statistic of interest.

stat.ME↗

arXiv2025

A new look at fiducial inference

Since the idea of fiducial inference was put forward by Fisher, researchers have been attempting to place it within a rigorous and well motivated framework. It is fair to say that a general definition has remained elusive. In this paper we start with a representation of Bayesian posterior distributions provided by Doob that relies on martingales. This is explicit in defining how a true parameter value should depend on a random sample and hence an approach to "inverse probability" as originally conceived by Fisher. Taking this as our cue, we introduce a definition of fiducial inference that can be regarded as general.

stat.ME↗

arXiv2019

Relations between Schoenberg Coefficients on Real and Complex Spheres of Different Dimensions

Positive definite functions on spheres have received an increasing interest in many branches of mathematics and statistics. In particular, the Schoenberg sequences in the spectral representation of positive definite functions have been studied by several mathematicians in the last years. This paper provides a set of relations between Schoenberg sequences defined over real as well as complex spheres of different dimensions. We illustrate our findings describing an application to strict positive definiteness.

math.CA↗

arXiv2016

A General Framework for Updating Belief Distributions

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 under the special case of using self information loss. Modern application areas make it is increasingly challenging for Bayesians to attempt to model the true data generating mechanism. Moreover, 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 proposed 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.

math.ST↗

arXiv2011

A definition of conditional probability distribution with non-stochastic information

The current definition of a conditional probability distribution enables one to update probabilities only on the basis of stochastic information. This paper provides a definition for conditional probability distributions with non-stochastic information. The definition is derived as a solution of a decision theoretic problem, where the information is connected to the outcome of interest via a loss function. We shall show that the Kullback-Leibler divergence plays a central role. Some illustrations are presented.

math.PR↗

arXiv2010

On Bayesian learning from Bernoulli observations

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.

math.ST↗