G. A. Barnard: 23 Sept 1915-30 July, 2002
Today is George Barnard’s birthday. In honor of this, I have typed in an exchange between Barnard, Savage (and others) on an important issue that we’d never gotten around to discussing explicitly (on likelihood vs probability). Please share your thoughts.
The exchange is from pp 79-84 (of what I call) “The Savage Forum” (Savage, 1962)[i]
BARNARD:…Professor Savage, as I understand him, said earlier that a difference between likelihoods and probabilities was that probabilities would normalize because they integrate to one, whereas likelihoods will not. Now probabilities integrate to one only if all possibilities are taken into account. This requires in its application to the probability of hypotheses that we should be in a position to enumerate all possible hypotheses which might explain a given set of data. Now I think it is just not true that we ever can enumerate all possible hypotheses. … If this is so we ought to allow that in addition to the hypotheses that we really consider we should allow something that we had not thought of yet, and of course as soon as we do this we lose the normalizing factor of the probability, and from that point of view probability has no advantage over likelihood. This is my general point, that I think while I agree with a lot of the technical points, I would prefer that this is talked about in terms of likelihood rather than probability. I should like to ask what Professor Savage thinks about that, whether he thinks that the necessity to enumerate hypotheses exhaustively, is important.
SAVAGE: Surely, as you say, we cannot always enumerate hypotheses so completely as we like to think. The list can, however, always be completed by tacking on a catch-all ‘something else’. In principle, a person will have probabilities given ‘something else’ just as he has probabilities given other hypotheses. In practice, the probability of a specified datum given ‘something else’ is likely to be particularly vague–an unpleasant reality. The probability of ‘something else’ is also meaningful of course, and usually, though perhaps poorly defined, it is definitely very small. Looking at things this way, I do not find probabilities unnormalizable, certainly not altogether unnormalizable.
Whether probability has an advantage over likelihood seems to me like the question whether volts have an advantage over amperes. The meaninglessness of a norm for likelihood is for me a symptom of the great difference between likelihood and probability. Since you question that symptom, I shall mention one or two others. …
On the more general aspect of the enumeration of all possible hypotheses, I certainly agree that the danger of losing serendipity by binding oneself to an over-rigid model is one against which we cannot be too alert. We must not pretend to have enumerated all the hypotheses in some simple and artificial enumeration that actually excludes some of them. The list can however be completed, as I have said, by adding a general ‘something else’ hypothesis, and this will be quite workable, provided you can tell yourself in good faith that ‘something else’ is rather improbable. The ‘something else’ hypothesis does not seem to make it any more meaningful to use likelihood for probability than to use volts for amperes.
Let us consider an example. Off hand, one might think it quite an acceptable scientific question to ask, ‘What is the melting point of californium?’ Such a question is, in effect, a list of alternatives that pretends to be exhaustive. But, even specifying which isotope of californium is referred to and the pressure at which the melting point is wanted, there are alternatives that the question tends to hide. It is possible that californium sublimates without melting or that it behaves like glass. Who dare say what other alternatives might obtain? An attempt to measure the melting point of californium might, if we are serendipitous, lead to more or less evidence that the concept of melting point is not directly applicable to it. Whether this happens or not, Bayes’s theorem will yield a posterior probability distribution for the melting point given that there really is one, based on the corresponding prior conditional probability and on the likelihood of the observed reading of the thermometer as a function of each possible melting point. Neither the prior probability that there is no melting point, nor the likelihood for the observed reading as a function of hypotheses alternative to that of the existence of a melting point enter the calculation. The distinction between likelihood and probability seems clear in this problem, as in any other.
BARNARD: Professor Savage says in effect, ‘add at the bottom of list H1, H2,…”something else”’. But what is the probability that a penny comes up heads given the hypothesis ‘something else’. We do not know. What one requires for this purpose is not just that there should be some hypotheses, but that they should enable you to compute probabilities for the data, and that requires very well defined hypotheses. For the purpose of applications, I do not think it is enough to consider only the conditional posterior distributions mentioned by Professor Savage. Continue reading