Objectively Speaking, Keynes on Probability16 October 2008
While I was doing some research to-day, I ran across yet another article that classified John Maynard Keynes as a subjectivist when it came to probability theory. I feel moved to explain why this is incorrect.
First, let me explain something about the general issue. There is an outstanding question about just what a probability is. One could take many courses about probability without ever being alerted to the question. The textbook and lecturer might not ever touch on that basic question, or might present a definition of
probability as if it is universally accepted by all Wise People. But Wise People are not in agreement. When it comes to answers to the basic question, the two dominant answers are very different one from another.
One answer is provided by the frequentists, who say that a probability is some sort of frequency of occurrence. They don't agree amongst themselves as to the precise answer, but the gist of their answers is that if a process is repeated m times, where m is satisfactorily large, and results in some particular outcome n of those times, then the probability of that outcome is n/m.
One problem with this notion of probability is that it is only useful in cases where we are concerned with a sufficiently large sample. If one is concerned only with a single instance, then there is actually no logic to get us from a mere probability to a course of action. A single patient won't have average mortality; she will either live or die.
Another answer is provided by the subjectivists, who assert that a probability is a degree of belief, formed subject to certain rationality constraints. These constraints can be largely motivated in terms of avoiding probability assignments under which believers would accept gambles that they are sure to lose. The rationality constraints themselves are ostensibly objective — rules that should hold for everyone; amongst other things, these rules are to constrain the evolution of one's degrees of belief, as new information is introduced. The subjectivism is present in that one supposedly gets to start with any degrees of belief that don't violate these rules.
One immediate consequence of this notion of probability is that probabilities become largely unarguable. There is no real contradiction in Tim claiming that there is an 80% chance of rain and Bob claiming that there is a 20% chance; each is describing his respective belief per se. (The rationality constraints force a convergence of belief at the limit, but that could take forever.)
The subjectivist notion is often defined in terms such as
degree of rational belief or
rational degree of belief; it's best to be wary of such terms. The rationality constraints themselves only preclude certain sorts of irrationality; aspects of the degrees of belief permitted are at best not irrational. And if we are not somehow required to assign some quantity to that belief, then the assignment violates Ockham's Razor.
Now, Keynes's position is that we can make meaningful statements about the plausibility of uncertain outcomes for which frequencies are unknown or otherwise inapplicable. And he certainly wants to impose rationality constraints much like those of the subjectivists. But he sees no requirement that one always assign a quantity to belief. Indeed, he sees no reason to treat the set of possible outcomes as even necessarily totally ordered; that is to say that he holds that, when asked to compare the likelihood of two events, sometimes one can only shrug, rather than making claims that one event is more likely or that the two are equally likely.
Under Keynes's theory, a rational person says no more about the probability of an event than the application of objective rules to the information set yields, and any other rational person with the same information set would reach exactly the same conclusions about probabilities (except, perhaps, where one person halted consideration where the other continued). Keynes rejects the very thing that is subjective in the subjectivist framework.