Ideas in Forensic Mathematics

The problem of claiming identity by DNA

Let

	O_posterior = posterior odds.
	O_prior = prior odds – summarizing all evidence except:
	DNA = likelihood ratio from DNA – presumably very large: DNA>> 1
	ALIBI = likelihood ratio for exculpatory evidence – also >1. (I.e. I arrange the ALIBI likelihood ratio so that the numerator, not the denominator, corresponds to the defense story.)

Then

O_posterior = O_prior * DNA / ALIBI.

Example: A woman has been murdered, her body eventually found in the desert.
	An ex-boyfriend is arrested. Supposing that a juror might assume any given ex-boyfriend is guilty in 5% of such cases,
		O_prior=1/19
	Blood with DNA matching the victim is detected in the trunk of the suspect's car. Ignoring for the sake of the example any possible innocent explanations,
		DNA = 10⁶, let us say.
	Defense witnesses attest that the suspect was not a likely killer. He seems nice. Let's say
		ALIBI = 4.
	Therefore
		O_posterior = (1/19)(10⁶)/4 = 13000 to 1 that he killed her.

From the point of view of any particular juror, there are only two meaningful categories of O_posterior – small enough to leave "reasonable doubt", or big enough to have no significant doubt. Hence there is some minimum threshold value of O_posterior for conviction for each juror, and the largest of these is a value O_threshold sufficient to satisfy the conscience of all the jurors. Then to convict, it is necessary and sufficient that

DNA ≥ ALIBI * O_threshold / O_prior

(1)

If DNA is large enough to satisfy the above, then I do no harm by claiming that DNA is essentially infinite, which is what a claim of "identity" amounts to.

In order to believe that the inequality holds, I need to make some kind of estimates for the factors on the right hand side. Then, I'll need to choose my words carefully in testimony.

As for O_prior, it is reasonable to say that O_prior ≥ 1/N, N=population size. What population: World population? City population? "Suspect" population? I figure that if the suspect was arrested through a dragnet or screening of millions of people, I would have read about it in the newspaper. Therefore I can fairly assume that N is not millions, it is only handfuls at most.

O_prior ≥ 1/handfuls

(2)

ALIBI is a little tricker. However, assuming that it represents non-scientific evidence,

DNA >> ALIBI

(3)

because I know of NO kind of non-scientific evidence that is comparable to good DNA evidence.

Combining (2) and (3), I am comfortable saying "I cannot imagine a circumstance but that

DNA >> ALIBI/O_prior"

(4)

But how much >>? Estimating O_threshold is the trickiest of all. I think O_threshold=1,000,000 is incredibly conservative; even O_threshold=10,000 probably represents a considerably higher standard than exists in practice. Much more than one in 10000 people who are convicted, are innocent. But is there any reason that some particular juror is not allowed to insist on an even higher standard?

So at this point I am stuck. For a sufficiently stubborn and unrealistic juror I can never insist on (1). But perhaps I can make a statement that only summarizes (4): "There is at most an extremely tiny chance – one chance in millions or billions – that the DNA is not from the suspect."

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