A Presumption of Innocence, Not of Even Odds
Friedman, Richard D., Stanford Law Review
Now I know how the Munchkins felt. Here I have been, toiling in the fields of Evidenceland for some years, laboring along with others to show how use of Bayesian probability theory can assist in the analysis and understanding of evidentiary problems.(1) In doing so, we have had to wage continuous battle against the Bayesioskeptics--the wicked witches who deny much value, even heuristic value, for probability theory in evidentiary analysis.(2) Occasionally, I have longed for law-and-economics scholars to help work this field, which should be fertile ground for them.(3)
So imagine my delight when the virtual personification of law and economics himself, Judge Richard Posner, came down from a star to help till the evidentiary soil with his powerful economic tools.(4) And imagine my further delight when his house landed square on the noggins of the Bayesioskeptics. I do not suppose this will be a fatal blow. Like Napoleon's Old Guard, they refuse to surrender--but neither will they die. Still, I appreciate having such an ally.
It may seem that I should be very gracious and roll out the welcome carpet to Evidenceland for Judge Posner. But I confess that my graciousness is tempered by the fact that Judge Posner commits a serious error in the use of Bayesian analysis. He asserts that an unbiased fact-finder should begin consideration of a disputed case with "prior odds of 1 to 1 that the plaintiff or prosecutor has a meritorious case."(5) Indeed, Judge Posner considers this starting point the essence of the definition of an unbiased fact-finder. I believe that this view is wrong in principle as a matter of probability theory. It is indeterminate and also fundamentally at odds with the presumption of innocence. Finally, it leads to bizarre results that expose Bayesian analysis to gratuitous ridicule from the wicked Bayesioskeptics.
I. THEORETICAL UNDERPINNINGS
Under the view of probability most useful for evidentiary analysis, a probability assessment represents an observer's subjective level of confidence in the truth of a given proposition, based upon the information that the observer has at any given time. Probabilities may be stated on a scale from 0 (representing certainty that the proposition is false) to 1 (representing certainty that the proposition is true). For some purposes, it is easier to represent probabilities in terms of odds. The odds that a proposition are tree equal the probability that the proposition is true divided by the probability that the proposition is not true. Accordingly, odds of 0 represent certainty that the proposition is false; infinitely high odds represent certainty that the proposition is true; and the oxymoronic-sounding "even odds" of 1:1--or 50/50--represent an assessment that the proposition is precisely as likely to be true as to be false.
The use of subjective probability theory in evidentiary analysis does not presuppose that fact-finders actually do, or even should, assess probabilities numerically or consciously at all. It only supposes that rational people act consistently with implicit probability assessments. An adult may run into the street to retrieve a bouncing ball if it is absurdly unlikely that an approaching car would hit her when she tried to do so, but she presumably would let the ball bounce away if that possibility seemed entirely plausible; on the other hand, because the stakes are so much different, she probably would run into the street to retrieve her bouncing baby even if the chances of her being run over during the rescue seemed quite high. None of this decision making requires calculation or numerification of odds.
When an observer receives new evidence relevant to the truth of the proposition at issue, she adjusts her probability assessment to take that evidence into account. An important principle indicating how this should be done rationally is Bayes' Theorem, which is so significant that the theory of subjective probability is often referred to, perhaps misleadingly, as Bayesian theory. …