Edit: Note that this question is not about multiple unreliable witnesses to the same incident, but rather multiple incidents with only one witness each. Should the accumulation of separate alleged incidents change our level of doubt about any individual alleged incident?
Here's a question about the informal statistical reasoning that many people do about crime and false accusations. I'd like to know whether the "common sense" response makes mathematical sense.
It's the middle of Prohibition. A prominent politician is accused of drinking copious amounts of alcohol during a party. He says that he has only ever consumed alcohol medicinally after strict advice from his doctor. He says that these are scurrilous, completely false accusations by his political enemies.
Most people say, "It's no more reliable than a rumour; he has lots of political enemies who'd have reason to falsely accuse him; much doubt remains."
Then a number of other people make the same accusation about him; different times, different places, different parties, but always lots of drinking.
After these reports, most people say, "With just one accuser, we were doubtful; now, with many additional accusations, there's little doubt that the first accusation was true."
Is this common informal statistical reasoning sound?
If you're approaching the question with Bayesian reasoning, keep in mind that the question is not "do the additional accusations change the likelihood that the first accusation is false?", but rather, "should the additional accusations cause a large change our level of doubt about the first accusation?"
Now let's add a series of prior assumptions. Do any of them change the soundness of the informal statistical reasoning?
- The amount of drinking is highly skewed, with a few people drinking the majority of alcohol. (This assumption is still true today, as it happens, though the skew is probably less extreme than it was during the height of Prohibition.)
Would that assumption make any difference to the way that our doubt should change as new accusations emerge?
- The vast majority of people - over 99%, let's say - are honest, and would never make a false accusation like this. (Note that most of our informal reasoners are wrong about this; they assume much more dishonesty than actually exists.)
Would that make any difference?
- The distribution of accusers is also skewed: Most people who could make true accusations keep their mouth shut, since admitting they were at a party would damage their own reputation; most people who do make accusation make true accusations; a very small number of people make repeated false accusations. However, we have no evidence to determine which group the original accuser falls into.
Would that make a difference? If it would, what effect would the amount of skew have?
If these factors make a difference, which one makes the biggest difference?
I eagerly await input from the finest minds of Stack Exchange. Again, remember that the question is whether our level of doubt about the initial accusation should change a little, a lot, or not at all after more accusations surface.