Ramsey’s … probability
F.P. Ramsey Truth and Probability, 1926.
The conclusions we shall come to as to the meaning of probability in logic must not … be taken as prejudging its meaning in physics.”
The frequency theory
The frequency approach may be valid for science (and presumably economics as science-like), but Ramsey’s subject is logic.
Ramsey finds Keynes’ theory incomprehensible. He seems to view Keynes’ notion as – in practice – involving total orders that correspond to (numeric) degrees of belief, and hence ignores Knightian uncertainty.
“They may, of course, say that it is incomparable with any numerical relation, but a relation about which so little can be truly said will be of little scientific use and it will be hard to convince a sceptic of its existence. Besides this view is really rather paradoxical; for any believer in induction must admit that … .”
Of course, Keynes was not a believer in any simplistic, unconditional, ‘law’ of induction (“Although nature has her habits, due to the recurrence of causes, they are general, not invariable.”)
Degrees of belief
“I do not see how [like Keynes] we can sharply divide beliefs into those which have a position in the numerical scale and those which have not. But I think beliefs do differ in measurability in the following two ways. First, some beliefs can be measured more accurately than others; and, secondly, the measurement of beliefs is almost certainly an ambiguous process leading to a variable answer depending on how exactly the measurement is conducted. The degree of a belief is in this respect like the time interval between two events; before Einstein it was supposed that all the ordinary ways of measuring a time interval would lead to the same result if properly performed. Einstein showed that this was not the case; and time interval can no longer be regarded as an exact notion, but must be discarded in all precise investigations. Nevertheless, time interval and the Newtonian system are sufficiently accurate for many purposes and easier to apply.
Let us then consider what is implied in the measurement of beliefs … . Of course this cannot be accomplished without introducing a certain amount of hypothesis or fiction.”
Thus Ramsey is not proposing that a numeric probability precisely indicates uncertainty. If we have a coin of known provenance and another provided by a trickster, the probability of heads may be the same for both of them, but there may be additional ‘ambiguity’ for one. Thus true Knightian uncertainty is not precluded, it is simply regarded as not relevant ‘for many purposes’.
“The old-established way of measuring a person’s belief is to propose a bet, and see what are the lowest odds which he will accept. This method I regard as fundamentally sound; but it suffers from being insufficiently general, and from being necessarily inexact. It is inexact partly because of the diminishing marginal utility of money, partly because the person may have a special eagerness or reluctance to bet, because he either enjoys or dislikes excitement or for any other reason, e.g. to make a book.
I suggest that we introduce as a law of psychology that his behaviour is governed by what is called the mathematical expectation; that is to say that, if p is a proposition about which he is doubtful, any goods or bads for whose realization p is in his view a necessary and sufficient condition enter into his calculations multiplied by the same fraction, which is called the ‘degree of his belief in p’. We thus define degree of belief in a way which presupposes the use of the mathematical expectation.”
Thus Ramsey views the propensity to bet as being essentially correct as an indication of probability, but with some technical problems that he seeks to overcome. He later regards this aspect as unsucessful (1929, below). But Ramsey’s probabilities are defined in terms of mathematical expectation, so it would be improper to try to use Ramsey to define mathematical or scientific uncertainty, or to use it to discount true uncertainty.
Ramsey also denies the principle of indifference:
“To be able to turn the Principle of Indifference out of formal logic is a great advantage; for it is fairly clearly impossible to lay down purely logical conditions for its validity … .”
Thus Ramsey’s claims are not as strong as contemporary Bayesians.
The logic of truth
“[T]he most generally accepted parts of logic, namely, formal logic, mathematics and the calculus of probabilities, are all concerned simply to ensure that our beliefs are not self-contradictory.”
Ramsey’s work can be seen as an attempt to formalise intuitions about probability rather, as Keynes does, to test them.
“We are all convinced by inductive arguments, and our conviction is reasonable because the world is so constituted that inductive arguments lead on the whole to true opinions. We are not, therefore, able to help trusting induction, nor if we could help it do we see any reason why we should, because we believe it to be a reliable process. …
This is a kind of pragmatism: we judge mental habits by whether they work, i.e. whether the opinions they lead to are for the most part true, or more often true than those which alternative habits would lead to.”
Thus induction would seem to be, at best, scientific and possibly merely pragmatic.
Further Considerations, 1928
“It is, however, obvious that we are not armed with systems giving us a degree of belief in every possible proposition for any basis of factual knowledge. Our systems only cover part of the field; and where we have no system we say we do not know the chances.
The phenomena for which we have systematic chances are games of chance, births, deaths, and all sorts of correlation.”
Thus Ramsey agrees with Bayes and Keynes on simple probabilities. They only differ on possible extensions. Ramsey also mentions the problem of context.
“… [S]tatistical causal analysis presupposes a fundamental system within which it moves and which it leaves unchanged; this neither is nor appears to be treated like a proposition. What appears to be so treated is a narrower system derived or derivable from the fundamental system by the addition of an empirical premiss, and what is really treated as a proposition and modified or rejected is not the narrower system but the empirical premiss on which it is based.”
Thus analysis based on statistics, such as scientific analysis is always dependent on some ‘system’, ‘context’ or ‘assumption’ that is not verified by the data. Ramsey also notes that uncertainty sometimes does exist:
“… ‘Why are chance events subject to law?’ The fundamental answer to this is that they are not, taking the whole field of chance events no generalizations about them are possible (consider e.g. infectious diseases, dactyls in hexameters, deaths from horse kicks, births of great men).
Note on ‘random’: Keynes gives a substantially correct account of this. [Not all non-deterministic variables are random in the sense required for the theory. Actual samples are never actually random: but the theories that describe them may be random, and may be a good match.]”
Probability and Partial Belief, 1929
Ramsey recognized the circularity of his definitions of probability and utility, and hence their weakness.
“The defect of my paper on probability was that it took partial belief as a psychological phenomenon to be defined and measured by a psychologist. But this sort of psychology goes a very little way and would be quite unacceptable in a developed science. … [How is probability derived?] I want to say in accordance with the law of mathematical expectation; but I cannot do this, for we could only use that rule if we had measured goods and bads. But perhaps in some sort of way we approximate to it, as we are supposed in economics to maximize an unmeasured utility. The question also arises why just this law of mathematical expectation. The answer to this is that if we use probability to measure utility, as explained in my paper, then consistency requires just this law. Of course if utility were measured in any other way, e.g. in money, we should not use mathematical expectation.”
Von Neumann & Morgenstern reached similar conclusions, as below.
Ramsey develops a theory of rational based on psychology and ‘homo economicus’. But he also notes:
- The theory is adequate for games of chance and overall birth and death rates in normal times, but not for infectious diseases, dactyls in hexameters, deaths from horse kicks or births of great men.
- It leaves out ‘Knightian uncertainty’, not because there is no such thing but because he supposes that people do not take account of it in normal decision-making.
- The principle of indifference is inadequate to resolve any uncertainty.
- The wider applicability or otherwise of his theory of probability is linked to concepts of rationality and of homo economicus.