Cristiano Villa | University of Kent (original) (raw)

Papers by Cristiano Villa

This paper is concerned with the well known Jeffreys-Lindley paradox. In a Bayesian set up, the s... more This paper is concerned with the well known Jeffreys-Lindley paradox. In a Bayesian set up, the so-called paradox arises when a point null hypothesis is tested and an objective prior is sought for the alternative hypothesis. In particular, the posterior for the null hypothesis tends to one when the uncertainty, i.e. the variance, for the parameter value goes to infinity. We argue that the appropriate way to deal with the paradox is to use simple mathematics, and that any philosophical argument is to be regarded as irrelevant.

We present a novel approach to constructing objective prior distributions for discrete parameter ... more We present a novel approach to constructing objective prior distributions for discrete parameter spaces. These type of parameter spaces are particularly problematic, as it appears that common objective procedures to design prior distributions are problem specific. We propose an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. We systematically apply this criterion to a series of discrete scenarios, previously considered in the literature, and compare the priors. The proposed approach applies to any discrete parameter space, making it appealing as it does not involve different concepts according to the model. This article has supplementary material online.

In this paper, we construct an objective prior for the degrees of freedom of a t distribution, wh... more In this paper, we construct an objective prior for the degrees of freedom of a t distribution, when the parameter is taken to be discrete. This parameter is typically problematic to estimate and a problem in objective Bayesian inference since improper priors lead to improper posteriors, whilst proper priors may dominate the data likelihood. We find an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. Truncating the prior on the degrees of freedom is necessary, as the t distribution, above a certain number of degrees of freedom, becomes the normal distribution. The defined prior is tested in simulation scenarios, including linear regression with t-distributed errors, and on real data: the daily returns of the closing Dow Jones index over a period of 98 days.

We discuss the problem of selecting among alternative parametric models within the Bayesian frame... more We discuss the problem of selecting among alternative parametric models within the Bayesian framework. For model selection problems which involve non-nested models, the common objective choice of a prior on the model space is the uniform distribution. The same applies to situations where the models are nested. It is our contention that assigning equal prior probability to each model is over simplistic. Consequently, we introduce a novel approach to objectively determine model prior probabilities conditionally on the choice of priors for the parameters of the models. The idea is based on the notion of the worth of having each model within the selection process. At the heart of the procedure is the measure of this worth using the Kullback-Leibler divergence between densities from different models.

This paper is concerned with the well known Jeffreys-Lindley paradox. In a Bayesian set up, the s... more This paper is concerned with the well known Jeffreys-Lindley paradox. In a Bayesian set up, the so-called paradox arises when a point null hypothesis is tested and an objective prior is sought for the alternative hypothesis. In particular, the posterior for the null hypothesis tends to one when the uncertainty, i.e. the variance, for the parameter value goes to infinity. We argue that the appropriate way to deal with the paradox is to use simple mathematics, and that any philosophical argument is to be regarded as irrelevant.

We present a novel approach to constructing objective prior distributions for discrete parameter ... more We present a novel approach to constructing objective prior distributions for discrete parameter spaces. These type of parameter spaces are particularly problematic, as it appears that common objective procedures to design prior distributions are problem specific. We propose an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. We systematically apply this criterion to a series of discrete scenarios, previously considered in the literature, and compare the priors. The proposed approach applies to any discrete parameter space, making it appealing as it does not involve different concepts according to the model. This article has supplementary material online.

In this paper, we construct an objective prior for the degrees of freedom of a t distribution, wh... more In this paper, we construct an objective prior for the degrees of freedom of a t distribution, when the parameter is taken to be discrete. This parameter is typically problematic to estimate and a problem in objective Bayesian inference since improper priors lead to improper posteriors, whilst proper priors may dominate the data likelihood. We find an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. Truncating the prior on the degrees of freedom is necessary, as the t distribution, above a certain number of degrees of freedom, becomes the normal distribution. The defined prior is tested in simulation scenarios, including linear regression with t-distributed errors, and on real data: the daily returns of the closing Dow Jones index over a period of 98 days.

We discuss the problem of selecting among alternative parametric models within the Bayesian frame... more We discuss the problem of selecting among alternative parametric models within the Bayesian framework. For model selection problems which involve non-nested models, the common objective choice of a prior on the model space is the uniform distribution. The same applies to situations where the models are nested. It is our contention that assigning equal prior probability to each model is over simplistic. Consequently, we introduce a novel approach to objectively determine model prior probabilities conditionally on the choice of priors for the parameters of the models. The idea is based on the notion of the worth of having each model within the selection process. At the heart of the procedure is the measure of this worth using the Kullback-Leibler divergence between densities from different models.