## Objectively Speaking, Keynes on Probability

16 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.

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### 7 Responses to Objectively Speaking, Keynes on Probability

• Blue Aurora says:

Although this post was done back in 2008...

I would have to say that you are correct to say that J.M. Keynes is not a thinker of the subjective interpretation of probability. He is a thinker in the logical interpretation of probability. There are points where the logical interpretation of probability and the subjective interpretation will agree on, but the former would disagree with the latter to the extent they take it.

That stated...have you encountered any of these articles before, sir?

http://bjps.oxfordjournals.org/content/44/2/357.extract

http://dx.doi.org/10.1080/02698599408573487

http://www.sciencedirect.com/science/article/pii/S0167487008001104

http://www.torrossa.it/resources/an/2569189

• Daniel says:

No, I've not encountered any of those articles (and I appreciate the links). (Keynes, Boole and the Interval Approach to Probability by Brady and Arthmar provoked some interest in me, until I realized that I'd elsewhere encountered Brady making logically incompetent and factually false arguments about Keynes, about Ramsey, and about me. Possibly his work is less BFC when controlled by a coäuthor or written for a journal.)

My own present concerns are with identifying how far one may go with incomplete preörderings of events by conditional plausibility. An interval approach can proxy an incomplete ordering, but I think that such an approach often amounts to affixing nails to the subject in order that a hammer might be used.

• Blue Aurora says:

Oh, I am well aware that comments on older posts don't get much attention for a reason. That stated - I take it that you were the same Daniel McKiernan to respond to that book review by M.E. Brady on a book by Irving J. Good?

That stated - yes, I did get the same initial impression that he was eccentric and strange due to his poor use of punctuation, but over time, I decided to put that aside.

Would you like to continunue this discussion on e-mail?

• Daniel says:

Your initial paragraph invites confusion, as you are, effectively, replying in a comment to unquoted e.mail.

I'm the fellow who responded to Brady.

The problems of his punctuation don't much concern me. That could indeed be considered a matter of eccentricity, as could his hijacking of an entry on Good to talk about Keynes. Rather, I have a problem with the strength of his understanding of mathematics, and a greater problem with his active casting aside of logic.

Preörderings are logically prior to quantifications, and in the case of plausibility judgments are millennia prior to quantifications. (On the latter point, see The Science of Conjecture: Evidence and Probability before Pascal by James Franklin.) Indeed, if we recognize that quantifications are a special case of preörderings, then we should realize that constructing every preördering in terms of intervals would entail an infinite regress. While Brady may be, for all that I know, quite able with the mathematics of quantification, his criticism of Ramsey shows a failure to understand how the mathematics of ordering underlies the mathematics of quantification.

Brady has recognized that interval structures can be fitted to Keynes's propositions about probability, and thence Brady insists that Keynes therefore was necessarily thinking in terms of intervals. This much is like inferring that, because a translation of Keynes's work into French is possible, Keynes necessarily conceived the work in French, albeït that he presented it in English.

I made these points to Brady (laboring the point on isomorphism more thoroughly); his response was then to insist that I hadn't read anything of Keynes's Treatise on Probability other than its third chapter, and that Ramsey had only read its third and fourth chapters; and Brady claimed that I was being illogical without even attempting to show anything illogical in what I'd written.

As to continuing this discussion in e.mail, well, if I'm treating Brady unfairly here, then it would be best if he were defended here. If I'm not treating Brady unfairly here, yet there's a good reason to attend to his work, then my guess would be that anyone else happening upon this entry should be given that reason.

• Blue Aurora says:

Yes, you are correct that this might cause other readers confusion. I don't think that you are treating him unfairly. But I'm not him, nor canl I truly claim to speak for him. I simply shared the links to these articles (not all of them involve Dr. Michael Emmett Brady, but the most of them are) because I thought you might be interested in it. Your blog clearly indicates that you have an interest in probability theory, decision theory, and economics.

I believe that both you and Dr. Brady would agree that J.M. Keynes is a thinker who belongs to the logical interpretation of probability, and anyone who characterizes J.M. Keynes as a thinker who belongs to the subjective interpretation of probability is incorrect.

The reason why I shared these articles on a public forum like this, is also because I believe that the links to the articles are worth reading from the stand-point of intellectual history. To get more specific - as far as I know, his article (which was co-written with a Brazilian historian of economics named Rogerio Arthmar) on George Boole and J.M. Keynes published in the secondary source literature for the history of economc thought that examines the connection between these two figures in depth. Of course, you can find scholarly research on George Boole's role in the history of mathematics, but as far as I know, the connection between Mr. Boole and Lord Keynes hasn't been discussed particularly extensively among historians of economic thought or the wider economics profession.

IIRC, A Treatise on Probability by J.M. Keynes is cited and mentioned in the secondary source literature (for the history of mathematics) dealing with George Boole's Challenge Problem. David W. Miller and Theodore Hailperin are two such scholars. Although the material we are talking about inevitably involves technicalities that wouldn't be known by the layman, I believe that naming intellectual history is a good reason in itself for any interested person to take a look at any of the links above.

• Daniel says:

Your description of how I see Keynes is accurate.

I welcome the idea of finding parallels and connections between the work of Boole and that of Keynes. I think that your drawing attention to the paper by Brady and Arthmar was quite appropriate. But I add that I lost considerable hope for the paper when I recognized the name of one of the authors; that Brady insists upon the certainty of a particular interpretation of Keynes which is in fact dubious, misinterprets Ramsey, and leapt to a dogmatic and false conclusion about me. The paper with Arthmar may in fact have great merit in spite of Brady's vices.

• Daniel says:

I note that Brady was, one way or another, led to this entry.

Then, acting not merely in the manner of a crack-pot but in that of a stereotypical crack-pot, he grabbed for a passage in the entry that he could, by strained interpretation, present as ostensible proof that I didn't understand the nature of subjectivism and difference between subjectivism and logicism, and used this passage and his interpretation, without directing his readers to its source, in a return comment to the Amazon review.

It has not gone well for him.

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