HCOL 196, March 21, 2011

 

Today’s discussion was about how to compute posterior probabilities of guilt. The states of nature are G, I, and the decisions are C, A. We need a prior on the states of nature. We cannot use the fact that the accused has been indicted, since that would already use evidence that will be used at trial, and it is wrong to use the same piece of evidence twice (and in fact, it is impossible to do so if the rules of probability are used correctly. I’ll say something about this on Wednesday.) One really should use a prior that expresses ignorance about innocence or guilt, and that prior would basically say, what is the probability of guilt, given that the accused was just randomly picked up?

Thought about this way, if N is the number of people in the area, the probability of picking someone out at random is 1/N, and that should be the prior on guilt. If we work with odds (ratio of P(G)/P(I)) then the result, for the number Ncc of people in Chittenden County, is as on the top line of the whiteboard shot below. (The lower lines were things we put in later).

To consider what would happen if we now have some evidence presented at trial, we imagine that evidence has been presented that shows that the accused had motive for the crime. The likelihood is shown below, where NMotive is the number of people in Chittenden County that have motive (this may not appear in the trial, we may have to estimate it using our knowledge of things in general). A guilty person is certain to have motive, but the probability that an innocent person has motive depends on the number of people that have motive.

When this is used to update the odds ratio, the results are shown in the second and third lines in the whiteboard shot above.

We then talked about DNA (or Fingerprint) matches. After some discussion, we found that the expert witness is going to give us a math probability, that is, the probability that there would be a match, given that the accused is just a random person. (Again, the probability of a match is 1 if the accused is guilty.) But this should not be confused with the probability of innocence, given a match. That requires us to use Bayesian reasoning. Confusing P(match|innocent) with P(innocent|match) is a mistake known as the Prosecutor’s Fallacy.

I pointed out that you can only add probabilities over mutually exclusive cases when the variable lies to the left of the conditioning bar |

See the second and third blue lines below.

The bit below is just reminding us that we need to get probabilities of guilt or innocence given a match, not the other way around.

And the calculation from prior through likelihood to posterior if the only evidence we look at is the match evidence, is shown below. Notice that our prior, which involves the number of people in Chittenden County, makes the evidence P(match|innocent) still yield a posterior (with these example numbers) that isn’t enough to convict.

You convert an odds ratio, letter O (terrible notation, it looks like a zero) into a probability using the following math:

I noted that on Wednesday we will discuss the O. J. Simpson case and I urged you to check that chapter in “Calculated Risks”.

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