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.Comment: This is the pre-peer reviewed version of the article "A General Framework for Updating Belief Distributions", which has been accepted for publication in the Journal of Statistical Society - Series B. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archivin

Multivariate isotropic random fields on spheres: Nonparametric Bayesian modeling and Lp fast approximations

We study multivariate Gaussian random fields defined over d-dimensional spheres. First, we provide a nonparametric Bayesian framework for modeling and inference on matrix-valued covariance functions. We determine the support (under the topology of uniform convergence) of the proposed random matrices, which cover the whole class of matrix-valued geodesically isotropic covariance functions on spheres. We provide a thorough inspection of the properties of the proposed model in terms of (a) first moments, (b) posterior distributions, and (c) Lipschitz continuities. We then provide an approximation method for multivariate fields on the sphere for which measures of L^p accuracy are established. Our findings are supported through simulation studies that show the rate of convergence when truncating a spectral expansion of a multivariate random field at a finite order. To illustrate the modeling framework developed in this paper, we consider a bivariate spatial data set of two 2019 NCEP/NCAR FluxReanalyses

Random Partition model and finitary Bayesian statistical inference

Exchangeability of observations corresponds to a condition shared by the vast majority of applications of the Bayesian paradigm. By de Finetti's rep- resentation theorem, if exchangeable observations form an infinite sequence of random variables, then they are conditionally independent and identi- cally distributed given some random parameter, which is the main object of statistical inference. Such parameter is a limiting mathematical entity and therefore hypotheses related to it might be not verifiable. For this reason, statistical analysis should be directed toward the prevision of the empirical distribution of N observations. In view of these considerations, specific forms of (finitary) exchangeable laws based on sequences of nested partitions have been introduced and studied in Bassetti and Bissiri (2007). In this paper, we intend to carry on this line of research studying another class of exchangeable laws, which rests on the concept of exchangeable random partition. These distributions are related to species sampling sequences, but allow negative correlation between observations. Marginal and predictive distributions are calculated together with the posterior distribution of the empirical process, and finally, it is shown how the predictive mean can be approximated by importance sampling

Finitary Bayesian statistical inference through partitions tree distributions

According to the Bayesian theory, observations are usually considered to be part of an infinite sequence of random elements that are conditionally inde pendent and identically distributed, given an unknown parameter. Such a parameter, which is the object of inference, depends on the entire sequence. Consequently, the unknown parameter cannot generally be observed, and any hypothesis about its realizations might be devoid of any empirical meaning. Therefore it becomes natural to focus attention on finite sequences of obser vations. The present paper introduces specific laws for finite exchangeable sequences and analyses some of their most relevant statistical properties. These laws, assessed through sequences of nested partitions, are strongly reminiscent of Polya-tree distributions and allow forms of conjugate analy sis. As a matter of fact, this family of distributions, called partitions tree distributions, contains the exchangeable laws directed by the more familiar Polya-tree processes. Moreover, the paper gives an example of partitions tree distribution connected with the hypergeometric urn scheme, where negative correlation between past and future observations is allowed

Finitary Bayesian statistical inference through partitions tree distributions

In Bayesian theory, observations are usually assumed to be part of an infinite sequence of random elements that are conditionally independent and identically distributed given some unknown parameter. Such a parameter, which is the object of inference, depends on the entire sequence. Consequently, it cannot generally be observed, and any hypothesis about the realizations of the unknown parameter might be devoid of any empirical meaning. This situation leads to ponder the advisability of directing statistical analysis toward the prevision of the empirical distribution of N observations or more generally toward functionals of such a distribution. According to this stance, it becomes natural to focus attentation on finite sequences of observations. In the present paper, specific laws for finite exchangeable sequences are given, and some of their most relevant statistical properties are studied. These laws, assessed through sequences of nested partitions, are strongly reminiscent of P?lya-tree distributions and allow forms of conjugate analysis. In fact, the class of these distributions, called partitions tree distributions contains the exchangeable laws directed by the more familiar P?lya-tree processes. Moreover, it encompasses an example of distribution connected with the hypergeometric urn scheme. It is worth stressing that, in this latter case, negative correlation between past and future observations are allowed

Random Partition model and finitary Bayesian statistical inference

Exchangeability of observations corresponds to a condition shared by the vast majority of applications of the Bayesian paradigm. By de Finetti's rep- resentation theorem, if exchangeable observations form an infinite sequence of random variables, then they are conditionally independent and identi- cally distributed given some random parameter, which is the main object of statistical inference. Such parameter is a limiting mathematical entity and therefore hypotheses related to it might be not verifiable. For this reason, statistical analysis should be directed toward the prevision of the empirical distribution of N observations. In view of these considerations, specific forms of (finitary) exchangeable laws based on sequences of nested partitions have been introduced and studied in Bassetti and Bissiri (2007). In this paper, we intend to carry on this line of research studying another class of exchangeable laws, which rests on the concept of exchangeable random partition. These distributions are related to species sampling sequences, but allow negative correlation between observations. Marginal and predictive distributions are calculated together with the posterior distribution of the empirical process, and finally, it is shown how the predictive mean can be approximated by importance sampling

Clustering via copula-based dissimilarity measures

A theoretical framework for clustering data is presented according to the dissimilarity behaviour as measured via a suitable copula-based coefficient and study its main properties. The coefficients are defined in terms of copulas, which may or may not be Gaussian. Applications to real data are used to illustrate the usefulness and importance of our proposal