Priors

Recently Dylan Farrow, the adopted daughter of Woody Allen and Mia Farrow, repeated allegations that Allen had sexually molested her when she was 7 years old.  These allegations were first made in 1982 during Allen and Farrow's acrimonious breakup (triggered by Farrow discovering Allen was involved with her adopted daughter Soon-Yi).  The allegations were investigated at the time and no criminal charges were pressed against Allen although the judge in a custody battle between Allen and Farrow seems to have given them some credence. 

The revived charges led to considerable internet discussion with some people quite convinced of Allen's guilt while others were far more skeptical.  I think it is worth noting that this disagreement can reflect different prior beliefs about how common sexual abuse of children is rather than different opinions about the strength of the evidence in Allen's case.

Suppose for example that some people believe that fathers rarely molest their children say .1% of the time while others believe it is fairly common say 10% of the time.  Suppose further that a credible accusation of abuse will be made in 1% of the cases where no abuse occurred and (for simplicity) in 100% of the cases where abuse did occur.  Then what fraction of credible accusations will in fact be true?

If sexual abuse is rare then out of every million men there will be 1000 cases of actual abuse (hence 1000 true accusations) and 9990 false accusations (.01 * 999000).  So only 9.1% of the credible accusations will actually be true.
On the other hand if abuse is common there will be 100000 true accusations and 9000 false accusations.  So in this case 91.7% of the accusations will be true.  Hence in a case like Allen's your prior beliefs can determine whether you see the evidence of guilt as weak or strong.

Of course I made up the above numbers to illustrate my point.  The reader can try different numbers and see how the fraction of false accusations changes.  I will note that assuming (more realistically) that less than 100% of actual abuse is reported leads to an increase in the fraction of false accusations.

This false positive problem also occurs in other contexts.  For example if a medical condition is rare a positive test result may be more likely test error than a true positive even if the test is fairly reliable.