Savage’s Tendencies …
This is a summary of Leonard J Savage, The Foundations of Statistics (John Wiley and Sons, New York, 1954). Lindley said of him:
“Savage, around 1960, is reported to have said to his colleagues: “In 1954 I proved that the only sound methods were Bayesian; yet you continue to use non-Bayesian ideas without pointing out a flaw in either my premises or my proof, why?””
But here Savage is more moderate. He is evidently motivated by economics and believes that he is building on the work of von Neumann and others:
“A certain subjective theory of probability formulated by Ramsey and later and more extensively by de Finetti promises great advantages for statistics. … It gives … a consistent, workable, and unifying analysis for all problems about the interpretation of the theory of probability, a much contested subject. It unifies the treatment of uncertainties, measuring them all by probabilities and emphasizing that they depend not only on patterns of information but on the opinions of individual people.”
Savage supports the controversial likelihood principle, and also the even more contentious notion of prior probabilities.
“[I]t is becoming increasingly accepted that, once an experiment has been done, any analysis or other reaction to the experiment ought to depend on the likelihood-ratio function and on it alone, without any further regard to how the experiment was actually planned or performed. I believe that this doctrine, which contradicts much that was recently most firmly established in statistical theory and practice, is basically correct and that it will soon greatly simplify and strengthen statistics.”
Savage’s first axiom (P1) is that the available acts have a total pre-order. This is questionable.
The Foundations Of Statistics Reconsidered . Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., Vol. 1 (Univ. of Calif. Press, 1961), 575-586.
Again Savage argues for a Bayesian approach, but relies on Ramsey. He gives the following example:
If I offer to pay you $10.00 if any one horse of your choosing wins a given race, your decision tells me operationally which you consider to be the most probable winner.
He refers to Good’s ‘kinds of probability’ favourably. He appears not to be arguing that dogmatic Bayesianism is universally correct, but only that it is the best available method.
Savage’s approach seems reasonable, subject to three observations.
Firstly, he only consider’s decisions that are one-off, such as the once-for all selection of a strategy, not ones that may be the first of a related series or involve learning.
Secondly, von Neumann and Morgenstern showed that not all strategies (acts) were pre-orderable, so that in situations that are not zero-sum or which have more than two competing players, as in prisoners’ dilemma, Savage’s result does not apply.
Thirdly, consideration of Simpson’s paradox shows that Savage’s ‘Sure-Thing’ principle does not hold more widely. For example, if a distribution that has what Jack Good called a ‘type 2’ objective distribution it can be that no subjective ‘type 1’ (simple numeric) distribution is adequate: one might need a type 2 subjective probability.
Savage’s approach is also more suitable to the case where one has a process that is being observed but not influenced, or to short-term analysis of processes that are not being influenced in the short-term.
In Savage’s horse-racing example, above, one can agree with Savage if one can equally well assign subjective probabilities to each horse. However, suppose that in a 3 horse race 2 horses are unknown, while 1 horse has a track record which leads you to assign a slightly greater probability than the unknown horses. It is Savage’s ‘most probable winner’. But if the horses are unknown then they may be much better or worse than our known horse. If they have been selected at random then there is a probability of almost 1/2 that a particular unknown horse is better than the known horse, and hence a probability of almost 3/4 that at least one of the two unknown horses is better than the known horse. Hence if one bets on an unknown horse, selected at random, one has a chance of almost 3/8 of backing the best horse (although it might not win). The probability of the known horse being the best is slightly more than 1/4. Hence you would do well to back an unknown horse, even though you think the known horse better.
If we have more unknown horses, n, then the probability of the known horse winning is about (0.5)n, which becomes vanishingly small compared with 1/n, so that backing the known prospect becomes increasingly poor as a strategy, compared with randomly selecting from an unknown prospect. (The setting comes from Savage, but contradicts his P1 axiom. The insight comes from a former colleague in connection with the management of competitions.)