Re: Koehler: Conditional Probability

From: HARNAD Stevan (harnad@coglit.soton.ac.uk)
Date: Tue Jun 02 1998 - 22:32:11 BST


> From: Dock Jenny <jmd195@soton.ac.uk>
>
> I get everthing except this paragraph:
>
> > Suppose the probability that the DNA will call someone guilty when they
> > are really innocent is very low, and the jury misinterprets this as
> > meaning that the probability that they are really innocent when the DNA
> > calls them guilty is very low; so they vote guilty.
>
> I think I understand what you are trying to say, but have
> you not just the same thing twice in this sentence? If not,
> then I think I am more confused that I thought.

The language is still against us. Let me try it like this:

P(G|I): Pull an INNOCENT person out of a hat. The probability that if
you gave them a DNA test, it would find them guilty, is very low. Try
it over and over, and you find it is very rare.

P(I|G) Now pull a GUILTY DNA test result out of a hat. The probability
that the person it was based on was innocent may not be as low: It
could be, but it need not be, and juries are making a mistake when they
assume it needs to be very low, because P(G|I) is very low.

The reason they need not both be low is that they also depend on (1)
the probability that someone you pull out of a hat is innocent (i.e.,
are the tests done on mostly innocent people or mostly guilty people
and (2) the probability that a DNA test will come out guilty: Maybe
most come out innocent, maybe most come out guilty.

Using Bayes Rule:

P(G|I)= P(I|G) x P(G) / P(I)

Pick any P(G|I) and you can make P(I|G) come out anything you like,
depending on what you make P(G) and P(I).

For example, if there is and has been only one guilty person, ever, in
the history of the universe, and everyone else is innocent, then even a
test that rarely calls someone innocent guilty, on those rare occasions
when it says guilty, has an extremely high probability of being wrong!

Cheers, Stevan



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