I am not a great follower of the sport of cycling. Apparently it is bad for the health of male genitals, and I can see why. But recently a successful cyclist, Alberto Contador, has tested positive for a minute trace of a substance that is banned. His explanation is that it must have been in meat he innocently ate. Farmers do use that drug to improve the muscle mass of their cattle. It seems to be thought that the controversy will be resolved by a second test of Mr Contador's body fluid samples.
Nonsense. The occurrence of the drug in people who eat meat from the same source as Mr Contador's meal must be examined. The probability of getting Mr Contador's test result, on the assumption that he is guilty of deliberately taking the drug, must be compared with the probability of getting his test result on the assumption that he is innocent. This latter is the proportion of people who have a similar test result who got that result innocently from eating.
Scientific reasoning is comparable to legal reasoning. In science the method of investigation involves attempting to disprove a null hypothesis. For example, if the null hypothesis was “this drug test result could not have been caused by food”, scientific inquiry would involve looking for an instance where the relevant sort of food consumption caused the same test result. Falsification of the null hypothesis was the criterion for scientific advance recognised in the scientific community and famously described by Karl Popper.
In legal reasoning applicable to criminal trials, the prosecution’s hypothesis is the null hypothesis (“this drug test result could not have been caused by food”). But it is not for the defence to disprove the null hypothesis. Of course, the defence could seek to do so, and would win if it did produce evidence that the null hypothesis was false. But generally it is for the prosecution to prove that there is nothing to falsify the null hypothesis. “Progress” in this legal context occurs where there can be no disproof of the null hypothesis, whereas in science progress is disproof of the null hypothesis. Obviously, whereas disproof of the null hypothesis occurs by a specific event, the prosecution’s task of showing there is no disproof of its hypothesis can only be a matter of likelihood.
A disadvantage of the scientific method is that disproof may be a long time coming, and this will slow down progress. Disproof has, in recent times, been complemented by another technique: asking what is the most likely hypothesis behind given observations. Given the drug test result, what is the most likely explanation? Law is similar: given the evidence, is the defendant’s guilt the most likely (to the necessary high standard) explanation? On this approach, conditional probabilities come into play. Hypotheses are compared as explanations for the observations or for the evidence. Bayes’ Theorem is a means of assessing the likelihood of an hypothesis as an explanation for an observed fact.
Mistakes in logic can be identified using Bayes’ Theorem, and it is not necessary for this that actual probabilities are known. A common error in logic is to say that the probability of A, given B, is the same as the probability of B, given A. Using the example of a (any) cyclist, the error would be in saying that the probability of this test result, given that the drug was taken deliberately, is the same as the probability that the drug was taken deliberately, given this test result. Another error of logic is to suppose that the likelihood of the cyclist having cheated can be derived directly from the likelihood of the drug having been in his food. This error is that of ignoring the other probabilities of the cyclist having cheated, taking into account all the relevant facts. A Bayesian approach avoids both these sorts of errors.
