Envelopes of conditional probabilities extending a strategy and a prior probability

Any strategy and prior probability together are a coherent conditional probability that can be extended, generally not in a unique way, to a full conditional probability. The corresponding class of extensions is studied and a closed form expression for its envelopes is provided. Then a topological characterization of the subclasses of extensions satisfying the further properties of full disintegrability and full strong conglomerability is given and their envelopes are studied.Comment: 2

Optimal properties of some Bayesian inferences

Relative surprise regions are shown to minimize, among Bayesian credible regions, the prior probability of covering a false value from the prior. Such regions are also shown to be unbiased in the sense that the prior probability of covering a false value is bounded above by the prior probability of covering the true value. Relative surprise regions are shown to maximize both the Bayes factor in favor of the region containing the true value and the relative belief ratio, among all credible regions with the same posterior content. Relative surprise regions emerge naturally when we consider equivalence classes of credible regions generated via reparameterizations.Comment: Published in at http://dx.doi.org/10.1214/07-EJS126 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Appropriate Methodology of Statistical Tests According to Prior Probability and Required Objectivity

In contrast to its common definition and calculation, interpretation of p-values diverges among statisticians. Since p-value is the basis of various methodologies, this divergence has led to a variety of test methodologies and evaluations of test results. This chaotic situation has complicated the application of tests and decision processes. Here, the origin of the divergence is found in the prior probability of the test. Effects of difference in Pr(H0 = true) on the character of p-values are investigated by comparing real microarray data and its artificial imitations as subjects of Student's t-tests. Also, the importance of the prior probability is discussed in terms of the applicability of Bayesian approaches. Suitable methodology is found in accordance with the prior probability and purpose of the test.Comment: 16 pages, 3 figures, and 1 tabl

Effects on orientation perception of manipulating the spatio–temporal prior probability of stimuli

Spatial and temporal regularities commonly exist in natural visual scenes. The knowledge of the probability structure of these regularities is likely to be informative for an efficient visual system. Here we explored how manipulating the spatio–temporal prior probability of stimuli affects human orientation perception. Stimulus sequences comprised four collinear bars (predictors) which appeared successively towards the foveal region, followed by a target bar with the same or different orientation. Subjects' orientation perception of the foveal target was biased towards the orientation of the predictors when presented in a highly ordered and predictable sequence. The discrimination thresholds were significantly elevated in proportion to increasing prior probabilities of the predictors. Breaking this sequence, by randomising presentation order or presentation duration, decreased the thresholds. These psychophysical observations are consistent with a Bayesian model, suggesting that a predictable spatio–temporal stimulus structure and an increased probability of collinear trials are associated with the increasing prior expectation of collinear events. Our results suggest that statistical spatio–temporal stimulus regularities are effectively integrated by human visual cortex over a range of spatial and temporal positions, thereby systematically affecting perception

Bayesian model-independent evaluation of expansion rates of the universe

Marginal likelihoods for the cosmic expansion rates are evaluated using the `Constitution' data of 397 supernovas, thereby updating the results in some previous works. Even when beginning with a very strong prior probability that favors an accelerated expansion, we obtain a marginal likelihood for the deceleration parameter $q_0$ peaked around zero in the spatially flat case. It is also found that the new data significantly constrains the cosmographic expansion rates, when compared to the previous analyses. These results may strongly depend on the Gaussian prior probability distribution chosen for the Hubble parameter represented by $h$, with $h=0.68\pm 0.06$. This and similar priors for other expansion rates were deduced from previous data. Here again we perform the Bayesian model-independent analysis in which the scale factor is expanded into a Taylor series in time about the present epoch. Unlike such Taylor expansions in terms of redshift, this approach has no convergence problem.Comment: To appear in Astrophysics and Space Scienc