Appendix B: Non-Bavesian Theories

Exponential and bayesian conjugate families: Review and extensions

The notion of a conjugate family of distributions plays a very important role in the Bayesian approach to parametric inference. One of the main features of such a family is that it is closed under sampling, but a conjugate family often provides prior distributions which are tractable in various other respects. This paper is concerned with the properties of conjugate families for exponential family models. Special attention is given to the class of natural exponential families having a quadratic variance function, for which the theory is particularly fruitful. Several classes of conjugate families have been considered in the literature and here we describe some of their most interesting feat…

A Bayesian approach to assess data from radionuclide activity analyses in environmental samples

A Bayesian statistical approach is introduced to assess experimental data from the analyses of radionuclide activity concentration in environmental samples (low activities). A theoretical model has been developed that allows the use of known prior information about the value of the measurand (activity), together with the experimental value determined through the measurement. The model has been applied to data of the Inter-laboratory Proficiency Test organised periodically among Spanish environmental radioactivity laboratories that are producing the radiochemical results for the Spanish radioactive monitoring network. A global improvement of laboratories performance is produced when this pri…

An introduction to Bayesian reference analysis: inference on the ratio of multinomial parameters

This paper offers an introduction to Bayesian reference analysis, often described as the more successful method to produce non-subjective, model-based, posterior distributions. The ideas are illustrated in detail with an interesting problem, the ratio of multinomial parameters, for which no model-based Bayesian analysis has been proposed. Signposts are provided to the huge related literature.

Statistical inference and Monte Carlo algorithms

This review article looks at a small part of the picture of the interrelationship between statistical theory and computational algorithms, especially the Gibbs sampler and the Accept-Reject algorithm. We pay particular attention to how the methodologies affect and complement each other.

Simulated Annealing in Bayesian Decision Theory

Since the seminal paper by Kirkpatrick, Gelatt and Vechhi (1983), a number of papers in the scientific literature refer to simulated annealing as a powerful random optimization method which promises to deliver, within reasonable computing times, optimal or nearly optimal solutions to complex decision problems hitherto forbidding. The algorithm, which uses the physical process of annealing as a metaphor, is special in that, at each iteration, one may move with positive probability to solutions with higher values of the function to minimize, rather than directly jumping to the point with the smallest value within the neighborhood, thus drastically reducing the chances of getting trapped in lo…

Probing Public Opinion: the State of Valencia Experience

This paper summarizes the procedures which have been set up during the last years at the Government of the State of Valencia, Spain, to systematically probe its public opinion as an important input into its decision processes.

Approximations in Statistics from a Decision-Theoretical Viewpoint

The approximation of the probability density p(.) of a random vector x∊X by another (possibly more convenient) probability density q(.) which belongs to a certain class Q is analyzed as a decision problem where the action space is the class Qof available approximations, the relevant uncertain event is the actual value of the vector x and the utility function is a proper scoring rule. The logarithmic divergence is shown to play a rather special role within this approach. The argument lies entirely within a Bayesian framework.

Reference Posterior Distributions for Bayesian Inference

Comparing normal means: new methods for an old problem

Comparing the means of two normal populations is an old problem in mathematical statistics, but there is still no consensus about its most appropriate solution. In this paper we treat the problem of comparing two normal means as a Bayesian decision problem with only two alternatives: either to accept the hypothesis that the two means are equal, or to conclude that the observed data are, under the assumed model, incompatible with that hypothesis. The combined use of an information-theory based loss function, the intrinsic discrepancy (Bernardo and Rueda 2002}, and an objective prior function, the reference prior \citep{Bernardo 1979; Berger and Bernardo 1992), produces a new solution to this…

Reference Priors in a Variance Components Problem

The ordered group reference prior algorithm of Berger and Bernardo (1989b) is applied to the balanced variance components problem. Besides the intrinsic interest of developing good noninformative priors for the variance components problem, a number of theoretically interesting issues arise in application of the proposed procedure. The algorithm is described (for completeness) in an important special case, with a detailed heuristic motivation.

A Decision Analysis Approach to Multiple-Choice Examinations

We present a decision analysis approach to the problems faced by people subject to multiple-choice examinations, as often encountered in their education, in looking for a job, or in getting a driving permit.

Bayesian Estimation of Political Transition Matrices

A decision framework is used to propose a procedure designed to estimate the reallocation of the vote of each individual party between two consecutive political elections, given the results of the elections, the information provided by a sample survey, and some assumptions on the hierarchical structure of the population.

The foundations of decision theory: An intuitive, operational approach with mathematical extensions

A new axiomatic basis for the foundations of decision theory is introduced and its mathematical development outlined. The system combines direct intuitive operational appeal with considerable structural flexibility in the resulting mathematical framework.

Overall Objective Priors

In multi-parameter models, reference priors typically depend on the parameter or quantity of interest, and it is well known that this is necessary to produce objective posterior distributions with optimal properties. There are, however, many situations where one is simultaneously interested in all the parameters of the model or, more realistically, in functions of them that include aspects such as prediction, and it would then be useful to have a single objective prior that could safely be used to produce reasonable posterior inferences for all the quantities of interest. In this paper, we consider three methods for selecting a single objective prior and study, in a variety of problems incl…

Es usted… ¿Ortodoxo o Bayesiano?

Appendix A: Summary of Basic Formulae