As an example, suppose that P(Match | Guilty) = 1 (or nearly so), P(Match | not Guilty) = 1/3000000, and N = 2000000. Then

P(Match)
= P(Guilty) P(Match | Guilty) + P(not Guilty) P(Match | not Guilty)
= 1/2000000 + 1/3000000 (approximately)
= 5/6000000

P(Guilty|Match)
= P(Guilty) P(Match|Guilty) / P(Match)
= 1/2000000 * 1 / (5/6000000) (approximately)
= 3/5
= 60%

In this case, the DNA evidence is good reason to suspect the person, but it doesn’t conclusively prove his guilt.

Science lies. All science is opinion based upon some set of “facts” agreed to by those working in a specific area.

The symmetry for an uninformative prior is between the different people who could have done it, so P(guilt) is going to be 1/population.

This is even better, but you should be careful in that I doubt a guy in Maine raped a woman in Hawaii. If we had good data on the ratio of rapists in the male population we could use that as well, or as you say, include it in X when we do our conditioning. In any event, using different priors is probably a good idea as it gives you an idea of how sensitive the posterior is to the specification of the prior.

P(guilt | DNA match, X) = P(DNA match | guilt, X) * P(Guilt | X) / P(DNA match | X).

It can be really helpful to include the X, as it helps solidify what you’re conditioning against, so that it doesn’t change in different expressions.

Okay, seriously though yes, you absolutely need to do this via Bayes theorem. That is you want to know the following probability:

Prob(Guilt|DNA Match).

This is not necessarily the same as Prob(DNA Match|Guilt).

To find the above though you would use the following equation:

Prob(Guilt|DNA Match) = Prob(DNA Match|Guilt)Prob(Guilt)/Prob(DNA Match).

The probability of DNA Match is the 1 in 3 million probability. The higher that number, all else held constant, the lower the Probability of Guilt given a DNA match.

Think of it this way, are the two probabilities equal:

Prob(Royal Flush|Winning Hand) vs. Prob(Winning Hand|Royal Flush).

Given that I have a royal flush, I’m pretty much going to win. However, one can have a winning hand and not have a royal flush (the first probability). Clearly the first probability is going to be smaller than the second probability.

Oh and the Prob(Guilt), that is the prior probability of guilt. This is tricky to handle, IMO. Depending on your flavor of Bayesian analysis it can be a subjective probability. Also, there is the “innocent until proven guilty” aspect as well. I’d say that a good starting point would be Prob(Guilt) = 0.5. This is often considered a “non-informative” prior. You can also calculate the above probabilities for different priors. Also, you’d eventually want to factor in other evidence as well such as eyewitness testimony, an alibi, etc.

One, “99.99967″ may be the technical accuracy of the test, but it doesn’t include any human sources of error.

Secondly, even if we ignore point number one, a database of all the people in the world (for example) would return a large number of results. The perpetrator would be, with 99.99967% certainty, on the list. But the list would also be longer than one name. So getting a positive result doesn’t mean your DNA was in the underwear, it means you’re one of the handful (or more) who might be the one. But, when the database is so small that you’re the only one who gets a positive flag, you look like the culprit.

Sort of, but sometimes an investigator settles on a suspect too early and starts trying to build a case without considering possible alternatives. This is called “falling in love with your theory of the case” and it’s a well-known trap that investigators are normally taught to avoid.

While all the articles I’ve read say this is rare, I can’t find anything that estimates just how rare or how common it is. It just adds to the issues of errors you’ve already brought up.

I’m not saying cold searches should never be used. Only that police, prosecutors, and the public need to be aware of the statistical probabilities, here, and they need to know that matches are nowhere near a certain indicator of guilt.

I think finding proof that a suspect did it and investigating the crime are the same thing and, also, that you’re creating a distinction without a difference between DNA evidence and any other evidence the cops may have.

Every clue that points towards a perpetrator has some odds of pointing to a random innocent person, only with most of those clues (a witness saw someone that looked like the suspect, there were partial fingerprints at the scene) we don’t know what the odds are. It doesn’t make a difference if the first clue found is DNA evidence or any other evidence that points to a specific person… after that first suspect is found the police are going to try to prove that he did it.

I have concerns about DNA databases, but this just isn’t one of them.