THE REVEREND THOMAS BAYES

lived in England (1702-61) and wrote on theology and philosophy; but he was also an outstanding mathematician and is mainly remembered for a theorem contained in his "Essay Towards Solving a Problem in the Doctrine of Chances" (1763), today known as "Bayes' Theorem."

Probabilities are normally forward looking: from several causes we calculate the probability of an effect that may occur in the future. Bayes showed how to reverse the process and calculate probabilities a posteriori, or probabilities looking into the past: after an effect has already occurred, one can calculate the probabilities of the individual causes that must have been responsible for it. The most probable of these causes can then be identified, and the relative likelihoods of the individual causes can be estimated.

The reason why the Rev. Bayes posthumously visits our newsletter is Senate Bill S-1483, "The Atomic Fallout Compensation Act," sponsored by Sen. Orrin Hatch (R-Utah). The law (now still in committee, but presumably to be taken up by the next Congress) is intended to compensate victims of radioactive fallout from the Nevada bomb tests in the 1950's. Although Bayes' Theorem is not mentioned in the draft, the idea is to compensate victims by "attributable risk," for of thousands of cancers only some are caused by radiation, and of these, only some are caused by fallout from the bomb tests. Since the probabilities of cancer incidence from radiation, and from radiation under various conditions, are known from statistical data, they can be reversed by Bayes' Theorem to yield the probability that a cancer, once it appears, was caused by the fallout.

Hatch's bill would tabulate the risks and give federal courts the likelihoods that a given type of cancer was due to fallout radiation, and it would compensate the probable victims in proportion. For example, a victim with 10% probability (that the cancer is of bomb-test origin) would share one award with 9 others who have the same 10% probability. Thus the treasury would pay the total due to a single (indisputable) case, but it would be shared among all who have the same (probabilistic) claim to it. And that is surely an optimization of justice under difficult circumstances.

There are two interesting points about this. First, Sen. Hatch was moved to use this method after hearing the testimony by Dr Victor Bond of the Brookhaven National Lab, who suggested that scientific considerations be used to find an equitable solution. Before that, a bill by Sen. Kennedy used this emotional issue to make political hay.

The bill proposes to compensate only victims with an attributable risk probability of more than 10%. If the compensation is shared among too many people with small probabilities, it loses its compensatory value. This is reminiscent of the pauper who reportedly approached J.P. Morgan and appealed to Morgan's conscience to share his wealth with the rest of mankind. Morgan is said to have pulled out a dime and to have asked "You want your share now?"