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.
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.(6) A simple statement of Bayes' Theorem uses three terms. One is the prior odds of a proposition--that is, the odds as assessed before receipt of the new evidence. The second is the posterior odds of the proposition--that is, the odds that the proposition is true as assessed after receipt of the new evidence. And the third is the likelihood ratio. Simply defined, the likelihood ratio of a given body of evidence with respect to a given proposition is the ratio of the probability that the evidence would arise given that the proposition is true to the probability that the evidence would arise given that the proposition is false.(7)
Bayes' Theorem posits that the posterior odds of the proposition equal the prior odds times the likelihood ratio. In simple notation,
(1) [O.sub.pos] = [O.sub.pr] x L.
Thus, all other things being equal, the posterior odds will be higher: (a) the higher the prior odds; (b) the higher the probability that the evidence would arise given the troth of the proposition; and (c) the lower the probability that the evidence would arise given that the proposition is false. A likelihood ratio greater than 1 means that the proposition appears more probable in light of the new evidence; a likelihood ratio less than 1 means that the new evidence makes the proposition appear less probable; and a likelihood ratio of precisely 1 means that the new evidence leaves the probability unchanged.
Now, one problem in application of the theory is determination of the prior odds of a proposition in the face of virtual ignorance of information bearing on the troth of the proposition. I speak of virtual rather than of total ignorance, because if the observer understands the proposition at all then she must have at least some minimal level of information about the world and about the terms in which the proposition is articulated. Thus, she can make a very tentative assessment of how probable she believes the proposition to be.(8) That assessment is, of course, subject to being altered radically by the receipt of new evidence.(9)
It seems plain that the prior odds that an observer attaches to a proposition--even in the face of virtual ignorance--may be very low, very high, or somewhere in the middle, depending on the nature of the proposition and the circumstances in which it is posed. A fact-finder would certainly be rational in assigning low prior odds to the proposition, "Yesterday it literally rained cats and dogs," and high prior odds to the proposition, "All mountains that exist today will exist tomorrow."
Even in the face of virtual ignorance, there is no reason for an observer to adopt a general rule of setting prior odds of 1:1 for a proposition. Such a rule would be arbitrary and wrongheaded, putting out of mind all useful information with which the observer addresses a new problem. It would assume that, simply because a proposition has been articulated, the proposition is exactly as likely to be true as false. There is no justification for such a conclusion.(10)
I do not mean to deny that there are some propositions for which an observer may quite reasonably assess prior odds of 1:1, or even that there are some for which prior odds much different from 1:1 would not be reasonable. Suppose the observer is shown a disk that is perfectly symmetrical, except that one side is blue and the other side is red, and asked to assess the odds that the disk, if flipped from a vertical position, will land red side up. She may very well assess these odds as even, for she may have no basis at all for believing that one side is more likely to land up than the other. But, as the rather contrived nature of this hypothetical suggests, this is a relatively narrow type of situation.(11)
Whatever appeal prior even odds may retain disappears altogether when we consider partitions of the possibilities into more than two groupings. Suppose that instead of a disk it is a perfectly fair cubic die that is to be tossed. Would a rational observer assess the odds that Face 1 will land up as 1:1? Of course not; that would mean that it is equally probable that the die will land on Face 1 as that it will not land on Face 1--and that, of course, is inconsistent with the premise that the die, with six faces, is fair. As the range of possibilities is partitioned into more and more smaller and smaller pieces--eventually becoming a continuous distribution--the point becomes stronger still. Suppose we select one point on our disk and then roll the disk, east to west. Eventually it will drop. The probability that the selected point will then be the easternmost point on the disk is infinitesimally small.
One might respond that the assumption of prior even odds really is only a specialized application for the two-possibility case--blue side or red side up, for example--of a more general "principle of indifference," that all possibilities should be assumed to have equal prior odds. But for at least two reasons this response does not work. First, it makes the prior odds of a given hypothetical proposition depend on how the competing hypotheses are defined. Suppose there are balls in an urn, some red, some powder blue, and the rest royal blue, but we do not know how many there are of each. What are the prior odds that the first ball selected at random will be red? If we define the competing possibilities as red versus blue, then this indifference principle would prescribe that the prior odds that the ball is red are 1:1--but if the possibilities are defined as red, powder blue, and royal blue, then it would prescribe prior odds of 1:2.(12) 11 Second, in the poly-possibility case, just as in the two-possibility case, it is not usually sound to assign equal prior odds to each possibility. Suppose you throw a pin up in the air. Where will its tip land? Points directly under the spot where you released the pin are substantially more probable than distant points.
It is not surprising...