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The Bayesian analog of the p-value is the complement of the posterior probability of the highest density region that is tangent to the null hypothesis. See, for example, Periera and Stern, http://scholar.google.com/citations?view_op=view_citation&hl=en&user=7spXyx8AAAAJ&citation_for_view=7spXyx8AAAAJ:u5HHmVD_uO8C, although the idea goes back to Box and Tiao, Bayesian Inference in Statistical Analysis (1973), if not earlier. In some cases with non-informative priors this definition of "Bayesian p value" (BPV) is numerically the same as the Frequentist p-value (F-tests, for example). In the one-dimensional case with a symmetric, unimodal posterior distribution the BPV is indeed double the "smaller of the one-tailed p-values. However, for asymmetric posterior densities, assuming the smaller tail area is to the left of the null hypothesis, the BPV is the smaller tail area PLUS the tail area to the right of the parameter value with the same posterior density as the null value. In other words, it is the area under the density curve over parameter values with posterior density equal to or less than the posterior density at the null hypothesis. To compute the exact BPV in BUGS you'd need a closed-form expression for the marginal posterior density of theta up to a constant, call this L(theta), then the BPV would be the mean of step(L(theta0)-L(theta)), where theta0 is the null-hypothesis value of theta. This works in any number of dimensions. Madruga, Esteves, and Wechsler discuss whether using the BPV as a test is a Bayesian decision rule. http://www.ime.usp.br/~jstern/miscellanea/citacoes/swtest1.pdf George Woodworth From: (The BUGS software mailing list) [mailto:[log in to unmask]] On Behalf Of Ryan Black Sent: Thursday, May 29, 2014 15:14 To: [log in to unmask] Subject: Bayesian p-value Hi all, I'm about to ask about something that is a bit controversial (or at least seems that way when I read the literature and think about Bayes Theory). I came across a website by a Bayesian analyst who indicated that one can obtain "one-tailed" p-values using the step function in WINBUGS where the mean statistic of the posterior distribution is the p-value. Furthermore, the analyst indicated that if you multiply the smaller of the one-tailed p-values by 2, that would produce what is essentially a 2-sided p-value. For example, suppose H0: slope=0. Right-Tailed p-value = step(slope - 0) Left-Tailed p-value = 1 - step(slope - 0) Let's assume for this example that the mean slope ends up being (+) 0.5. Further, suppose the right tailed p-value is 0.99xx and the left tailed p-value is .01xx, applying the recommendation to obtain the 2-sided p-value, the left-tailed p-value would be multiplied by 2. Question: How exactly would this two-sided p-value be interpreted? Is it similar to that of a Frequentist interpretation? If not, how would it differ? Thanks! Ryan ------------------------------------------------------------------- This list is for discussion of modelling issues and the BUGS software. For help with crashes and error messages, first mail [log in to unmask]<mailto:[log in to unmask]> To mail the BUGS list, mail to [log in to unmask]<mailto:[log in to unmask]> Before mailing, please check the archive at www.jiscmail.ac.uk/lists/bugs.html<http://www.jiscmail.ac.uk/lists/bugs.html> Please do not mail attachments to the list. To leave the BUGS list, send LEAVE BUGS to [log in to unmask]<mailto:[log in to unmask]> If this fails, mail [log in to unmask]<mailto:[log in to unmask]>, NOT the whole list ------------------------------------------------------------------- This list is for discussion of modelling issues and the BUGS software. For help with crashes and error messages, first mail [log in to unmask] To mail the BUGS list, mail to [log in to unmask] Before mailing, please check the archive at www.jiscmail.ac.uk/lists/bugs.html Please do not mail attachments to the list. To leave the BUGS list, send LEAVE BUGS to [log in to unmask] If this fails, mail [log in to unmask], NOT the whole list