E.T. Jaynes in his posthumous Probability Theory: The Logic of Science (on Bayesian statistics) includes a chapter 5 on “Queer Uses For Probability Theory”, discussing such topics as ESP; miracles; heuristics & biases; how visual perception is theory-laden; philosophy of science with regard to Newtonian mechanics and the famed discovery of Neptune; horse-racing & weather forecasting; and finally—section 5.8, “Bayesian jurisprudence”. Jaynes’s analysis is somewhat similar in spirit to my above analysis, although mine is not explicitly Bayesian except perhaps in the discussion of gender as eliminating one necessary bit.
The following is an excerpt; see also “Bayesian Justice”.
It is interesting to apply probability theory in various situations in which we can’t always reduce it to numbers very well, but still it shows automatically what kind of information would be relevant to help us do plausible reasoning. Suppose someone in New York City has committed a murder, and you don’t know at first who it is, but you know that there are 10 million people in New York City. On the basis of no knowledge but this, e(Guilty|X) = −70 db is the plausibility that any particular person is the guilty one.
How much positive evidence for guilt is necessary before we decide that some man should be put away? Perhaps +40 db, although your reaction may be that this is not safe enough, and the number ought to be higher. If we raise this number we give increased protection to the innocent, but at the cost of making it more difficult to convict the guilty; and at some point the interests of society as a whole cannot be ignored.
For example, if 1,000 guilty men are set free, we know from only too much experience that 200 or 300 of them will proceed immediately to inflict still more crimes upon society, and their escaping justice will encourage 100 more to take up crime. So it is clear that the damage to society as a whole caused by allowing 1,000 guilty men to go free, is far greater than that caused by falsely convicting one innocent man.
If you have an emotional reaction against this statement, I ask you to think: if you were a judge, would you rather face one man whom you had convicted falsely; or 100 victims of crimes that you could have prevented? Setting the threshold at +40 db will mean, crudely, that on the average not more than one conviction in 10,000 will be in error; a judge who required juries to follow this rule would probably not make one false conviction in a working lifetime on the bench.
In any event, if we took +40 db starting out from −70 db, this means that in order to ensure a conviction you would have to produce about 110 db of evidence for the guilt of this particular person. Suppose now we learn that this person had a motive. What does that do to the plausibility for his guilt? Probability theory says
e(Guilty|Motive)=e(Guilty|X)+10log10P(Motive|Guilty)P(Motive|Not Guilty) (5-38)
≃−70−10log10P(Motive|Not Guilty)
since P(Motive|Guilty)≃1, i.e. we consider it quite unlikely that the crime had no motive at all. Thus, the [importance] of learning that the person had a motive depends almost entirely on the probability P(Motive|Not Guilty) that an innocent person would also have a motive.
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