Conditional probability

Abstract: Consider a group of individuals who must act together as a team or committee, and suppose that each individual in the group has his own subjective probability distribution for the unknown value of some parameter. A model is presented which describes how the group might reach agreement on a common subjective probability distribution for the parameter by pooling their individual opinions. The process leading to the consensus is explicitly described and the common distribution that is reached is explicitly determined. The model can also be applied to problems of reaching a consensus when the opinion of each member of the group is represented simply as a point estimate of the parameter rather than as a probability distribution.

Abstract: The diagnosis of coronary-artery disease has become increasingly complex. Many different results, obtained from tests with substantial imperfections, must be integrated into a diagnostic conclusion about the probability of disease in a given patient. To approach this problem in a practical manner, we reviewed the literature to estimate the pretest likelihood of disease (defined by age, sex and symptoms) and the sensitivity and specificity of four diagnostic tests: stress electrocardiography, cardiokymography, thallium scintigraphy and cardiac fluoroscopy. With this information, test results can be analyzed by use of Bayes' theorem of conditional probability. This approach has several advantages. It pools the diagnostic experience of many physicians and integrates fundamental pretest clinical descriptors with many varying test results to summarize reproducibly and meaningfully the probability of angiographic coronary-artery disease. This approach also aids, but does not replace, the physician's ju...

Abstract: Many econometric testing problems involve nuisance parameters which are not identified under the null hypotheses. This paper studies the asymptotic distribution theory for such tests. The asymptotic distributions of standard test statistics are described as functionals of chi-square processes. In general, the distributions depend upon a large number of unknown parameters. We show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters, and we show that this transformation can be easily approximated via simulation. The theory is applied to threshold models, with special attention given to the so-called self-exciting threshold autoregressive model. Monte Carlo methods are used to assess the finite sample distributions. The tests are applied to U.S. GNP growth rates, and we find that Potter's (1995) threshold effect in this series can be possibly explained by sampling variation.