Monday, July 15, 2013

Keynes’s Interval Probabilities

M. Brady (1993: 360) sees Chapters 15–17 and 26 of Keynes’s Treatise on Probability as being poorly understood by economists who interpret Keynes’s theory of probability.

Now I do not pretend at the moment to have anything but a beginner’s grasp of the relevant specialist literature on Keynes’s probability theory, but some thoughts follow.

At any rate, Chapter XV of the Treatise on Probability is a very short one, and the crucial passage seems to be here:
“5. It is evident that the cases in which exact numerical measurement is possible are a very limited class, generally dependent on evidence which warrants a judgment of equiprobability by an application of the Principle of Indifference. The fuller the evidence upon which we rely, the less likely is it to be perfectly symmetrical in its bearing on the various alternatives, and the more likely is it to contain some piece of relevant information favouring one of them. In actual reasoning, therefore, perfectly equal probabilities, and hence exact numerical measures, will occur comparatively seldom.

The sphere of inexact numerical comparison is not, however, quite so limited. Many probabilities, which are incapable of numerical measurement, can be placed nevertheless between numerical limits. And by taking particular non-numerical probabilities as standards a great number of comparisons or approximate measurements become possible. If we can place a probability in an order of magnitude with some standard probability, we can obtain its approximate measure by comparison.

This method is frequently adopted in common discourse. When we ask how probable something is, we often put our question in the form—Is it more or less probable than so and so ?— where ‘so and so’ is some comparable and better known probability. We may thus obtain information in cases where it would be impossible to ascribe any number to the probability in question. Darwin was giving a numerical limit to a non-numerical probability when he said of a conversation with Lyell that he thought it no more likely that he should be right in nearly all points than that he should toss up a penny and get heads twenty times running. Similar cases and others also, where the probability which is taken as the standard of comparison is itself non-numerical and not, as in Darwin's instance, a numerical one, will readily occur to the reader.

A specially important case of approximate comparison is that of ‘practical certainty.’ This differs from logical certainty since its contradictory is not impossible, but we are in practice completely satisfied with any probability which approaches such a limit. The phrase has naturally not been used with complete precision; but in its most useful sense it is essentially non-numerical—we cannot measure practical certainty in terms of logical certainty. We can only explain how great practical certainty is by giving instances. We may say, for instance, that it is measured by the probability of the sun’s rising to-morrow. The type which we shall be most likely to take will be that of a well-verified induction.

6. Most of such comparisons must be based on the principles of Chapter V. It is possible, however, to develop a systematic method of approximation which may be occasionally useful. The theorems given below are chiefly suggested by some work of Boole’s. His theorems were introduced for a different purpose, and he does not seem to have realised this interesting application of them; but analytically his problem is identical with that of approximation. This method of approximation is also substantially the same analytically as that dealt with by Mr. Yule under the heading of Consistence.” (Keynes 1921: 160–161).
From this, we can summarise Keynes’s ideas:
(1) the domain of strict a priori or mathematical probability is highly restricted. I assume that this restriction was taken by Keynes to include what was later called relative frequency probability.

(2) nevertheless, there are many probabilities that are capable of “inexact numerical comparison.” That is, many probabilities cannot have single, objective numerical values (whether a single cardinal number or decimal), but they appear to be measured in an approximate manner by numerical limits or an interval. Certainly, many people do compare the non-inexact probabilities and rank them.

The idea of an interval estimate is described by M. Brady in one of his (numerous) reviews on For convenience I have corrected the spacing errors:
“For Keynes there are only two types of probability estimates, point estimates and interval estimates. Unfortunately, Keynes decided to call interval estimates ‘non-numerical’ probabilities. His reasoning is really quite obvious. A precise estimate of probability used a single numeral for the point estimate. Therefore, an imprecise estimate of probability used two numerals to denote an interval (set). Thus, an interval estimate is not based on a single numeral but two. These types of probabilities are thus ‘non-numerical’ because you are not using a single numeral. In 1922 and 1926, Frank Ramsey reviewed Keynes’s book based on his reading of chapters 1–4 plus 3 pages from Part two and 4 pages from Part five. Keynes’s discussion of non-numerical probabilities takes place in chapters 5, 10, 15 and 17.

Keynes then applies his new approach to induction and analogy in chapters 20 and 22, using his concept of ‘finite probability’ which applies to both precise numerical probabilities and imprecise non-numerical probabilities. …. Keynes then showed that interval estimates, because they overlap, would very likely also, in many cases, be noncomparable and/or nonrankable if a decision maker used such order preserving operators like ‘greater than or equal to’ or ‘less than or equal to.’”
Michael Emmett Brady, May 16, 2005
So some rough interval probabilities are comparable and some are not.

Strangely, the notion of interval probability was allegedly taken up by subjectivist probability theorists such as Koopman, Dempster and Shafer, but not rarely used by later advocates of the logical theory of probability (Courgeau 2012: 87).

(3) people are capable of “practical certainty,” in the sense of a highly probable inductive inference or perhaps by direct observation. For Keynes, this type of probability is a degree of rational belief in the truth of inferences from inductive arguments, or “well-verified” inductions, which, of course, have a truth value that is only ever probable, not absolutely certain.

(4) Keynes then (later in the chapter) uses calculus adopted from Boole to analyse such probabilities, although that will not concern me here.
So for Keynes probability is a relation consisting in a degree of rational belief in the truth of propositions derived from inductive arguments, and many do not take objective numerical values. An additional concept (though not discussed by Keynes in this passage) was the weight of evidence, a measure of the amount of evidence relevant to the proposition involved (but, again, I will leave this for another discussion).

I propose some examples below of what (I assume) Keynes was thinking of in his comments above. For example, what are propositions that have “practical certainty”?

Take this proposition:
Proposition (1): Napoleon was defeated at the battle of Waterloo.
This is what people normally think of when they think of “historical facts,” but this is a synthetic proposition, and its truth is a posteriori.

How do we know it is true? You and I did not live at the time Waterloo was fought, so we have no direct access to the event. Nor is there anyone alive today who was present when the battle was fought.

So how? Of course we have a vast amount of empirical evidence in the form of historical records: newspapers, journals, government records, autobiographies, reported statements of eyewitnesses that all report that Napoleon was defeated at Waterloo. Eyewitnesses continued to live until the late 19th century. Important political events followed, such as his banishment to Saint Helena and the restoration of King Louis XVIII, which are also attested with a vast amount of evidence and are incompatible with the idea that Napoleon won at Waterloo. Using all this evidence, an inductive argument can be constructed to the effect that it is extremely probable or virtually certain that Napoleon was defeated at Waterloo. That inference is, however, an inductive inference, and induction only ever yields, at best, extremely probable conclusions, not absolute certainty. Although it seems virtually impossible – indeed insane – that any evidence would arise or become known to refute the proposition, nevertheless there must be some very tiny chance that it might be false.

This, I suspect, is what Keynes was thinking of by the idea of “practical certainty” in certain propositions.

It makes no sense to say that when asserting the truth of this proposition people do not think it has a high degree of probability.

But does it make any sense to give this a strict and exact numerical measure of probability? Probably not.

But can we use a rough, inexact numerical measure in terms of an interval, and say this proposition is probable at something between .9999999 to .9999999999? (note: I have no idea whether this is a remotely realistic or defensible probability that might be given by, say, a modern Bayesian).

While we know generally what we mean by “practical certainty,” such actual numbers in the interval seem ultimately subjective, even though the manner in which we arrived at the inductive inference itself is not subjective but requires a valid and sound inductive argument, with available and verifiable evidence.

Furthermore, can we compare the probability of this statement with that of another:
Proposition (2): the sun rose and set yesterday (understood scientifically as the earth rotated on its axis and we had the perception of the sun rising and setting yesterday).
Many might say that this is a proposition whose probability is much higher than the historical fact about Napoleon since it has a vast amount of not just empirical evidence, but direct personal experience (you saw it) and a body of verified scientific theory in physics to support it.

Does it have an exact numerical probability measure? Or like the historical proposition can we only give an inexact numerical measure in terms of an interval, and say that Proposition (2)’s truth is probable at something between .9999999999999 to .9999999999999999999? (again note: I have no idea whether this is a remotely realistic or defensible probability that might be given by, say, a modern Bayesian!).

We might even rank them and say that Proposition (2) is more probable than Proposition (1), but the ranking is ultimately intuitive but not based on exact numerical measures.

Matters are complicated by future events. What about:
Proposition (3): it will not rain tomorrow.
Of course, one can justify one’s proposition inductively by appealing to current weather data and say there is no sign of any phenomena that could give rise to precipitation tomorrow, and note that we are in a season which has low rainfall, and so on. But what sort of probability can this statement have? Its probability seems much vaguer than (1) or (2). Perhaps we can only estimate its probability vaguely with an interval of 0.6–0.9, but such estimates are now becoming variable, imprecise and surely subjective.

Another example could be:
Proposition (4): The interest rate will be 4.3% on January 10, 2033.
At this point, anyone proposing a probability for this proposition, even an interval one, is surely engaged in highly dubious and extremely subjective estimate. The evidence upon which one could base one’s inductive reasoning would be flimsy or non-existent; in contrast, the evidence for which one could base one’s subjective interval probability score for Propositions (1) or (2) above is not.

There is clearly a complex continuum of inductive propositions ranging from those nearly certain (on the basis of good evidence) to those extremely doubtful, and anything in between, which might have vague or imprecise probability intervals, and those where even giving probability intervals is a complete waste of time.

By 1937, Keynes was referring to the type of future events as in Proposition (4) as characteristic of fundamental uncertainty:
“By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty; nor is the prospect of a Victory bond being drawn. Or, again, the expectation of life is only slightly uncertain. Even the weather is only moderately uncertain. The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention, or the position of private wealthowners in the social system in 1970. About these matters there is no scientific basis on which to form any calculable probability whatever” (Keynes 1937: 213–214).
So uncertainty means not only that no objective probability score can be given, but also pertains especially to future situations about which inductive reasoning or statistic inference are highly limited or useless.

It seems difficult to conceive that Keynes (at least by 1937) would have regarded Proposition (4) as anything but fundamentally uncertain.

Thus interval probability estimates must at some point transform into essentially worthless subjective probabilities as one faces situations (especially in the future) where evidence is lacking and there is no serious basis even for defensible inductive reasoning.

But I assume that most of what I have said above seems consistent with Keynes’s theory of probability: inferences from induction do not have single objective probability scores, and often only estimates and more often only vague estimates are possible that, if they can be represented at all, look like intervals. Nevertheless, our inductive reasoning involves a degree of objectivity, such as the necessity of having valid and sound arguments, and weight of evidence can increase the worth of an interval.

Two final questions I have not yet resolved:
(1) did Keynes himself think that all or most interval probability estimates were subjective?

(2) Did Keynes later change his mind on his fundamental views of probability after the comments of critics? Some scholars contend that he did: Bateman (1987), Davis (2003), and Gillies (2003); and others that he did not: Gerrard (2003) and O’Donnel (2003).
Bateman, Bradley W. 1987. “Keynes’s Changing Conception of Probability,” Economics and Philosophy 3: 97–119.

Brady, Michael Emmett, 1993. “J. M. Keynes’s Theoretical Approach to Decision-Making under Conditions of Risk and Uncertainty,” The British Journal for the Philosophy of Science 44.2: 357–376.

Courgeau, Daniel. 2012. Probability and Social Science: Methodological Relationships between the Two Approaches. Springer, Dordrecht and New York.

Davis, John B. 2003. “The Relationship between Keynes’s Early and Later Philosophical Thinking,” in Jochen Runde and Sohei Mizuhara (eds.), The Philosophy of Keynes’ Economics: Probability, Uncertainty and Convention. Routledge, London and New York. 100–110.

Gerrard, B. 2003. “Keynesian Uncertainty: What Do We Know?,” in Jochen Runde and Sohei Mizuhara (eds.), The Philosophy of Keynes’ Economics: Probability, Uncertainty and Convention. Routledge, London and New York. 239–251.

Gillies, Donald. 2003. “Probability and Uncertainty in Keynes’s Economics,” in Jochen Runde and Sohei Mizuhara (eds.), The Philosophy of Keynes’ Economics: Probability, Uncertainty and Convention. Routledge, London and New York. 111–129.

Keynes, John Maynard. 1921. A Treatise on Probability. Macmillan, London.

O’Donnel, R. 2003. “The Thick and the Think of Controversy,” in Jochen Runde and Sohei Mizuhara (eds.), The Philosophy of Keynes’ Economics: Probability, Uncertainty and Convention. Routledge, London and New York. 85–99.

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