Deciding, Estimating, Computing, Checking. How are Bayesian posteriors used, computed and validated?. Fundamentalist Bayes: The posterior is ALL knowledge you have about the state.
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How are Bayesian posteriorsused, computed and validated?
Is sample from givendistribution?
Test statistic d is maxdeviation of empiricalcumulative distributionfrom theoretical.
If d*sqrt(n) > 2.5,
Sample is (probably)not from target distr
Graphical posterior predictivemodel checking takes first place inauthoritative book.Left column is 0-1 coding of logistic regression of sixsubjects response (row) to stimulus(column). Replicationsusing posterior and likelihooddistribution in right six columns. There is clear micro-structure in left column not present in the right ones. Thus,the fitting appears to have beendone with inappropriate(invalid)model.
Cumulative counts of real coal-mining disasters (lower red)Comparing with 100 scenarios of same number of simulateddisasters occuring randomly: The real data cannot reasonablybe produced by a constant-intensity process.
R A Fisher (1890--1962).
In his paper Inverse Probability, he rejected Bayesian Analysis on grounds of its dependency on priors and scaling.
He launched an alternative concept, 'fiducial analysis'. Although this concept was not developed after Fishers time, the standard definition of confidence intervals has a similar flavor. The fiducial argument was apparently the starting point for Dempster in developing evidence theory.