To provide a long-term global basis of precipitation information, satellite passive microwave observations have been widely used in the past decades to estimate surface rain rate. Among various approaches, some algorithms have invoked Bayes theorem to retrieve a single instantaneous rain rate and vertical hydrometeor profiles (Evans et al., 1995; Olson et al., 1996; Kummerow et al., 1996; Bauer...

2 Lecture 2: Evaluation of Statistical Procedures I 2 2.1 How to compare δ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Comparing risk function I: Bayes risk . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Bayes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Bayes risk revisited . . . . . . . . . . . . . . . . . . . . . . . . . ...

Improper priors are used frequently, but often formally and without reference to a sound theoretical basis. A consequence is the occurrence of seemingly paradoxical results. The most famous example is perhaps given by the marginalization paradoxes presented by Stone and Dawid (1972). It is demonstrated here that the seemingly paradoxical results are removed by a more careful formulation of the ...

The classical Bayesian posterior arises naturally as the unique solution of several different optimization problems, without the necessity of interpreting data as conditional probabilities and then using Bayes' Theorem. For example, the classical Bayesian posterior is the unique posterior that minimizes the loss of Shannon information in combining the prior and the likelihood distributions. The...

We give an upper bound for the posterior probability of a measurable set A when the prior lies in a class of probability measures 9. The bound is a rational function of two Choquet integrals. If g; is weakly compact and is closed with respect to majorization, then the bound is sharp if and only if the upper prior probability is 2-alternating. The result is used to compute-bounds for several set...