Talk:Prosecutor's fallacy

From the original article on the so called Prosecutor's fallacy:

Consider for instance the case of Sally Clark, who was accused in 1998 of having killed her first child at 11 weeks of age, then conceived another child and killed it at 8 weeks of age. The prosecution had an expert witness testify that the probability of two children dying from sudden infant death syndrome is about 1 in 73 million. To provide proper context for this number, the probability of a mother killing one child, conceiving another and killing that one too, should have been estimated and compared to the 1 in 73 million figure, but it wasn't. Ms. Clark was convicted in 1999, resulting in a press release by the Royal Statistical Society which pointed out the mistake (http://www.rss.org.uk/archive/reports/sclark).

The reason the case against Clark was flawed has nothing to do with the Prosecutor's fallacy, or the misapplication of statistics; though I agree that the way in which statistics were used here was simply wrong.

The case against Clark was flawed because it rests on a false dichotomy: Either Sally Clark's children both died from sudden infant death syndrome or she killed them.

While it's true that SIDS and infanticide are contrary explanations, showing one to be false is insufficient by itself to establish the truth of the alternative.

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I removed:

Of course some consider the Prosecutor's fallacy no fallacy at all, because a logical fallacy, by Wikipedia's own definition, is any way in which an argument fails to be valid or sound. Based as it is upon probability, the Prosecutor's fallacy is not a deductive argument to begin with, and therefore questions of its validity or soundness are themselves variants of the fallacy of stolen concept.

I don't think anyone, not even prosecutors or Bayesians, consider the reasoning behind the Prosecutor's fallacy to be sound. Furthermore, reasoning about probabilities can very well be fallacious, like the Gambler's fallacy for example.

I also cut the remainder of the article significantly, pointing out that some Bayesians may find it justified to use an a priori probability of 1/2, which is what the Prosecutor's fallacy implicitly assumes. AxelBoldt 18:58 Oct 13, 2002 (UTC)

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I don't see how the part of the article that says

One formulation of Bayes' theorem then states:

Odds(G|E) = Odds(G) · P(E|G)/P(E|~G)

is justified. Using Bayes' theorem to find the probablility of guilt gives:

Odds(G|E) = Odds(G) · P(E|G) / P(E)

where due to mutual exclusion:

P(E) = P(E|~G) · P(~G) + P(E|G) · P(G)

The article's formula holds only when you make the additional, unstated assumption that P(G) is small enough to make P(E|~G) a good approximation for P(E).

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Does this happen everywhere int the World or limited to in U.S.? -- Taku

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No, it has occurred in cases in the UK -- Tony Vignaux 09:00 18 May 2003 (UTC)

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I'm not sure that the Clark example illustrates the Prosecutor's Fallacy very well, because I think there's another closely related fallacy. I think I understand the Bayesian analysis, but I think there's a perhaps more obvious error than not computing the prior Odds(G): the calculation of Odds(G|E) doesn't take into account that any family that has 2 children who die of SIDS could find themselves in the same situation as Clark, and there are probably millions of such families.

Suppose 50% of people are guilty and we believe that the probability of two children in the same household dying of SIDS are 1 in 73 million. We still shouldn't be convinced that Clark is guilty. Rather, if there are say 10 million households with two children, and any time two children die there's a police inquiry because it's suspicious when 2 children in the same household die, then the odds of an unfortunate 2-time SIDS mother coming before the court is approximately 1/7.

BTW, the fallacy of the other example given in this article seems more related to the number of people tested than to failing to take into account the a priori probability.

Here's a an example of not taking into account the a priori prob, that has no problem with the number of people tested: suppose I believe that I have magical abilities that allow me to win the lottery, even though the odds of winning with random numbers are 1 in 1 billion. I play and win the lottery. Are you convinced I have magical abilities? I'd say it's certainly worth checking out, but anyone's reasonable estimate of the a priori odds that I have magical powers is very very low. So, it's more likely that I was simply very lucky than that I have magical powers. Note that this example doesn't require any argument about how many other people declare that they have magical powers and play the lottery -- I could be the only one, and my claim would still be dubious.

What are people's thoughts on my reasoning here? If I'm sound, does anyone know which is the prosecutor's fallacy: failing to account for the a priori probability, or failing to account for the number of tests done?

Zashaw 05:21 24 May 2003 (UTC)