Wednesday, April 2, 2014

Ramsey: "Truth and Probability" (1926)

Ramsey; from Wikipedia.
F. P Ramsey is a curious figure. Reading Keynes, Russell, and Whitehead at the age of 19, he was obviously a prodigy and seemed to have attracted attention from the Cambridge community of the 1920s, including Wittgenstein, Moore, and others. He died in 1930 from complications following a stomach operation, aged 26.

This posthumous book is a collection of some of his writings, both published and unpublished. I'm reading it mostly for the unpublished essay "Truth and Probability" from 1926, and Ramsey's own 1928 comments on that paper.

Degrees of Gamblability

In that essay, Ramsey embraces an epistemic interpretation of probability although he realizes that there might not be a single person with coherent beliefs in the sense of probability theory.

In his own commentary on the paper, the choice of this philosophy is compressed to a quick dismissal of any "crude frequency theory" and the conclusion:
(2) Hence chances must be defined by degrees of belief; but they do not correspond to anyone's actual degrees of belief; the chances of 1,000 heads, and of 999 heads followed by a tail, are equal, but everyone expects the former more than the latter.
(3) Chances are degrees of belief within a certain system of beliefs and degrees of belief; not those of any actual person, but in a simplified system to which those of actual people, especially the speaker, in part approximate. (p. 206)
In part 3 of the essay (pp. 166–184), he goes on to suggest that degrees of belief might be elicited through observed gambling behavior, and he states one direction of a Dutch book theorem (the easy one). He even calls the betting system a "book" (p. 182).

"Impossible to Say"

Ramsey also denies that there can be a question of rationality in initial assumptions, that is, in the prior distributions assumed in the absence of data.

He imagines reconstructing his own initial beliefs on the basis of his present beliefs and all of the data that he has ever observed (although, I should add, Bayesian updates can't be uniquely reversed like that). Assuming the update mechanism is sound, we then have:
My present degrees of belief can then be considered logically justified if the corresponding initial degrees of belief are justified. But to ask what initial degrees of belief are justified, or in Mr Keynes' system what are the absolutely a priori probabilities, seems to me a meaningless question; and even if it had a meaning I do not see how it could be answered.
If we actually applied this process to a human being, found out, that is to say, on what a priori probabilities his present opinions could be based, we should obviously find them to be ones determined by natural selection, with a general tendency to give a higher probability to the simpler alternatives. But, as I say, I cannot see what could be meant by asking whether these degrees of belief were logically justified. (p. 192–93)

Unconditional distributions cannot be uniquely reconstructed
from the conditional distributions and the conditions.
In the commentary, he makes a related point about independence and relevance assumptions:
E.g., the death-rate for men of 60 is 1/10, but all the 20 red-haired 60-year-old men I've known have lived till 70. What should I expect of a new red-haired man of 60? I can but put the evidence before me, and let it act on my mind. There is a conflict of two 'usually's' which must work itself out in my mind; one is not the really reasonable, the other the really unreasonable. (p. 202)
He concludes that "to introduce the idea of 'reasonable' is really a mistake" (p. 203). So to the question "What ought we to take as relevant?" the answer is that "it is impossible to say" (p. 203).

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