A cautionary note on using the scale prior for the parameter N of a binomial distribution

Statistical analysis of ecological data may require the estimation of the size of a population, or of the number of species with a certain population. This task frequently reduces to estimating the discrete parameter N representing the number of trials in a binomial distribution. In Bayesian methods, there has been a substantial amount of discussion on how to select the prior for N. We propose a prior for N based on an objective measure of the worth that each value of N has in being included in the model space. This prior is compared (through the analysis of the popular snowshoe hare dataset) with the scale prior which, in our opinion, cannot be understood from solid objective considerations

Objective prior for the number of degrees of freedom of a t distribution

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 dom- inate 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

An Objective Bayesian Criterion to Determine Model Prior Probabilities

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 di?erent models

An objective Bayesian approach for discrete scenarios

Objective prior distributions represent a fundamental part of Bayesian infer­ence. Although several approaches for continuous parameter spaces have been developed, Bayesian theory lacks of a general method that allows to obtain priors for the discrete case. In the present work we propose a novel idea, based on losses, to derive objec­tive priors for discrete parameter spaces. We objectively measure the worth of each parameter values, and link it to the prior probability by means of the self­information loss function. The worth is measured by taking into consideration the surroundings of each element of the parameter space. Bayes theorem is then re-interpreted, where prior and posterior beliefs are not expressed as probabilities, but as losses. The approach allows to retain meaning from the beginning to the end of the Bayesian updating process. The prior distribution obtained with the above approach is identified as the Villa-Walker prior. We illustrate the approach by applying it to various scenarios. We derive ob­jective priors for five specific models: a population size model, the Hypergeometric and multivariate Hypergeometric models, the Binomial-Beta model, and the Bi­nomial model. We also derive the Villa-Walker prior for the number of degrees of freedom of a t distribution. An important result in this last case, is that the objective prior has to be truncated. We finally apply the idea to discrete scenarios other that parameter spaces: model selection, and variable selection for linear regression models. We show how an objective model prior can be obtained, by applying our approach, on the basis of the importance that each model has with respect to the other ones. We illustrate various cases: nested and non-nested models, models with discrete and continuous supports, uniparameter and multiparameter models. For the variable selection scenario, the prior includes a loss component due to the complexity of each regression model

On a Class of Objective Prior from Scoring Rules

Objective prior distributions represent an important tool that allows one to have the advantages of using a Bayesian framework even when information about the parameters of a model is not available. The usual objective approaches work off the chosen statistical model and in the majority of cases the resulting prior is improper, which can pose limitations to a practical implementation, even when the complexity of the model is moderate. In this paper we propose to take a novel look at the construction of objective prior distributions, where the connection with a chosen sampling distribution model is removed. We explore the notion of defining objective prior distributions which allow one to have some degree of flexibility, in particular in exhibiting some desirable features, such as being proper, or log-concave, convex etc. The basic tool we use are proper scoring rules and the main result is a class of objective prior distributions that can be employed in scenarios where the usual model based priors fail, such as mixture models and model selection via Bayes factors. In addition, we show that the proposed class of priors is the result of minimising the information it contains, providing solid interpretation to the method