Francisco Javier Girón González-Torre

Consistency of objective Bayes factors for nonnested linear models and increasing model dimension

Casella et al. [2, (2009)] proved that, under very general conditions, for normal linear models the Bayes factor for a wIDe class of prior distributions, including the intrinsic priors, is consistent when the number of parameters does not grow with the sample size n. The special attention paID to the intrinsic priors is due to the fact that they are nonsubjective priors, and thus accessible priors for complex models.The case where the number of parameters of nested models grows as O(nα) for α ≤ 1 was consIDered in Moreno et al. [13, (2010)], in which it was proved that the Bayes factor for intrinsic priors is consistent for the case where both models are of order O(nα) for α < 1, and for α = 1 is consistent except for a small set of alternative models. The small set of models for which consistency does not hold was characterized in terms of a pseudo-distance between models.The goal of the present article is to extend the above results to the case where the linear models are nonnested. As the comparison of nonnested models calls for a method of encompassing, for proving consistency we use encompassing from below in this paper.ResumenEn Casella et al. [2, (2009)] se demostró que, bajo condiciones muy generales, el factor de Bayes para modelos lineales normales y para una amplia clase de distribuciones a priori, que incluía a las a priori intrínsecas, es consistente cuando el número de parámetros no crece cuando lo hace el tamaño muestral n. Se prestó especial atención a las distribuciones a priori intrínsecas debIDo a que son distribuciones a priori no subjetivas y, por consiguiente, se pueden aplicar a modelos complejos.El caso en que el número de parámetros de los modelos anIDados crece del orden de O(nα) para α ≤ 1 se ha consIDerado en Moreno et al. [13, (2010)], en el que se demuestra que el factor de Bayes para distribuciones intrínsecas es consistente para el caso en que ambos modelos son de orden O(nα) para α < 1 y, para el caso α = 1, también es consistente excepto para un conjunto pequeño de modelos alternativos. Este conjunto, para el cual la consistencia no se da, se caracterizó en términos de una pseudo distancia entre modelos.

Complementing the net benefit approach: A new framework for Bayesian cost-effectiveness analysis

OBJECTIVES The aim of cost-effectiveness analysis is to maximize health benefits from a given budget, taking a societal perspective. Consequently, the comparison of alternative treatments or technologies is solely based on their expected effectiveness and cost. However, the expectation, or mean, poses important limitations as it might be a poor summary of the underlying distribution, for instance when the effectiveness is a categorical variable, or when the distributions of either effectiveness or cost present a high degree of asymmetry. Clinical variables often present these characteristics. METHODS In this study, we present a framework for cost-effectiveness analysis based on the whole posterior distribution of effectiveness and cost. RESULTS An application with real data is included to illustrate the analysis. Decision-making measures such as the incremental cost-effectiveness ratio, incremental net-benefit, and cost-effectiveness acceptability curves, can also be defined under the new framework. CONCLUSIONS This framework overcomes limitations of the mean and offers complementary information for the decision maker.

Una familia de distribuciones conjugadas para un proceso ARE (1)

ResumenEn este artículo se estudia, desde una perspectiva bayesiana, un proceso AR(1) con errores exponenciales, ARE(1); para ello se construye una nueva familia de distribuciones conjugadas, denotada por CDG, que permite construir una especie de filtro de Kalman para la estimación recursiva de los parámetros del modelo.AbstractIn this article estimation of autoregressive processes AR(1) with exponential errors, denoted ARE(1), is considered from a Bayesian perspective. For these processes a new family of conjugate distributions, CDG, is shown to exist which allows for recursive estimation in a Kalman filter-like fashion.

Compatibilidad del método de De Groot para llegar a un consenso con la fórmula de Bayes

SummaryDe Groot’s simple method of reaching a consensus when several decision makers have different prior opinions expressed in terms of p.m.’s is shown to be compatible with Bayes rule when sample information is included as an aid to decision making. It is proven that uptdating the priors of the different d.m.’s by means of Bayes theorem and then applying De Groot’s method for reaching a consensus (whenever possible) yields the same as by first reaching a consensus and the applying Bayes theorem to the unanimons p.m.The key point of the proof is that the transition matrix is also updated in every iteration when sample information is considered by means of the predictive distribution.ResumenEn este artículo se prueba que el sencillo método propuesto por De Groot para llegar a un consenso cuando los varios decisores tienen opiniones diferentes expreadas en términos de distribuciones de probabilidad es compatible con la regla de Bayes cuando se tiene en cuenta la información muestral. Se demuestra que si se calcula primero las distribuciones a posteriori y después se aplica el método de De Groot para alcanzar un consenso (cuando esto sea posible) es lo mismo que realizar primero el consenso y después aplicar el teorema de Bayes a esta distribución consensuada.La clave de la demostración es que tambien la matriz de transición, en cada etapa se transforma, en presencia de la información muestral, mediante la utilización de la distribución predictiva.