A Commentary on Bayesian Games etc.

  The last few blogs dealt with problems related to Bayesian inference. The chief obstacle to such an analysis is getting a good estimate of the conditional probabilities when the reliability of the observers are unknown. We found that using two observers can give an improved estimate of the expected values of the likely unknown counts which are needed to determine the conditional probabilities.

The formulas for the hidden probabilities require counts of the number of events that the observers agree on. Rutherford gives the formula for computing the probability of two simultaneous events P and Q occurring but his papers don't indicate that he used these formulas to get improved estimates of rates based on the observations of two observers. Fuller's proofreading problem doesn't indicate a method of solution either. I have a copy of Fuller book (3rd Ed. see p. 170) and derived the formulas for the solution to the problem on my own. The proofreading problem and formulas can be found in Ross, Introduction to Probability and Statistics for Engineers and Scientists, p. 234f.

A related problem in Bayesian inference would be the determination of the likely number of false positives and false negatives for the observers using a particular method for assessments. If the conditional probabilities are known one can do so. A problem with statistical analysis is that rare events tend to be excluded from observations. Another problem is extending restricted studies to more general cases where the rates are likely not to be the same.