Comparison Between Bayesian and Frequentist Tail Probability Estimates

In this paper, we investigate the reasons that the Bayesian estimator of the tail probability is always higher than the frequentist estimator. Sufficient conditions for this phenomenon are established both by using Jensen's Inequality and by looking at Taylor series approximations, both of which point to the convexity of the distribution function

Pruebas Bayesianas Secuenciales y la Hipótesis de independencia o naive Bayes

Quite frequently diagnosis is not final with one medical test but only after a sequence of tests are applied. How the information given by one test is going to be combined with the information conveyed by a second test? Can we “add” up the information of the medical tests assuming conditional independence which is called “naive” or“independent” Bayes? In this article we develop a very simple and basic exact Bayes Factor to check the independent Bayes Model VS the full Bayes Model, without the assumption of conditional independence. Assuming independence Bayes when in fact is not, overstate the accumulation of two positives in favor of the disease and two negatives against. Here we also illustrate, that even in situations of mild evidence against the independence model the difference between the two models may be strikingly different in the presence of conflicting evidence between the medical tests. As a practical advice, when a sequence of tests are applied in combination routinely, a study should be conducted for which the joint results of a set of patients is kept and studied with and without the assumption of independence, and Bayes Factors should be calculated. This work extends and generalizes the work of Pereira & Pericchi (1990) and Mossman & Berger (2001).Con bastante frecuencia el diagnóstico no es definitivo con una prueba médica, pero solo después que se aplique una secuencia de pruebas. ¿Cómo se combinará la información proporcionada por una prueba con la información transmitida por una segunda prueba? ¿Podemos “agregar” la información de las pruebas clínicas suponiendo independencia condicional conocido como “naive” o “independiente” Bayes? En este artículo desarrollamos un simple y básico exacto Factor de Bayes para verificar el Modelo de Bayes independiente vs el Modelo completo de Bayes, sin el supuesto de independencia condicional. Asumiendo “independiente” Bayes cuando de hecho no lo es, exagera la acumulación de dos positivos a favor de la enfermedad, y dos negativos en contra. Aquí también ilustramos, que incluso en situaciones de evidencia leve contra el modelo de independencia, la diferencia entre los dos modelos puedeser notablemente diferente en presencia de evidencia conflictiva entre las pruebas médicas. Como consejo práctico, cuando se aplica una secuencia de pruebas en combinación de forma rutinaria, se debe realizar un estudio para el cual los resultados de un grupo de pacientes se mantengan y estudien con y sin el supuesto de independencia, y los factores de Bayes deben ser calculados. Este trabajo amplía y generaliza el trabajo de Pereira & Pericchi (1990) y Mossman & Berger (2001)

Sequential Bayesian Tests and the Hypothesis of independent or naive Bayes

Quite frequently diagnosis is not final with one medical test but a sequence of tests are applied. How the information given by one test is going to be combined with the information conveyed by a second test? Can we ”add up” the information of the medical tests assuming conditional independence that is the ”independence”or ”naive” Bayes? In this article we develop a very simple and basic exact Bayes Factor to check the independent Bayes Model VS the full Bayes Model, without the assumption of conditional independence. Assuming independence Bayes when in fact is not, overstate the accumulation of two positives in favor of the disease and two negatives against. Here we also illustrate, that even in situations of mild evidence against the independence model the difference between the two models may be strikingly different in the presence of conflicting evidence between the medical tests. As a practical advice, when a sequence of tests are applied in combination routinely, a study should be conducted for which the joint results of a set of patients is kept and studied with and without the assumption of independence Bayes and Bayes Factors should be calculated. This work extends and generalizes the work of Pereira and Pericchi (1990) and Berger and Moosman (2001)