Bayesianism — a ‘patently absurd’ approach to science

2 Dec, 2014 at 19:49 | Posted in Theory of Science & Methodology | 6 Comments

Back in 1991, when I earned my first Ph.D. — with a dissertation on decision making and rationality in social choice theory and game theory — yours truly concluded that “repeatedly it seems as though mathematical tractability and elegance — rather than realism and relevance — have been the most applied guidelines for the behavioural assumptions being made. On a political and social level it is doubtful if the methodological individualism, ahistoricity and formalism they are advocating are especially valid.”

steer-clear-of-scientologyThis, of course, was like swearing in church. My mainstream neoclassical colleagues were — to say the least — not exactly überjoyed.

The decision theoretical approach I perhaps was most critical of, was the one building on the then reawakened Bayesian subjectivist interpretation of probability.

One of my inspirations when working on the dissertation was Henry E. Kyburg, and I still think his critique is the ultimate take-down of Bayesian hubris (emphasis added):

From the point of view of the “logic of consistency” (which for Ramsey includes the probability calculus), no set of beliefs is more rational than any other, so long as they both satisfy the quantitative relationships expressed by the fundamental laws of probability. Thus I am free to assign the number I/3 to the probability that the sun will rise tomorrow; or, more cheerfully, to take the probability to be 9/1io that I have a rich uncle in Australia who will send me a telegram tomorrow informing me that he has made me his sole heir. Neither Ramsey, nor Savage, nor de Finetti, to name three leading figures in the personalistic movement, can find it in his heart to detect any logical shortcomings in anyone, or to find anyone logically culpable, whose degrees of belief in various propositions satisfy the laws of the probability calculus, however odd those degrees of belief may otherwise be. Reasonableness, in which Ramsey was also much interested, he considered quite another matter. The connection between rationality (in the sense of conformity to the rules of the probability calculus) and reasonableness (in the ordinary inductive sense) is much closer for Savage and de Finetti than it was for Ramsey, but it is still not a strict connection; one can still be wildly unreasonable without sinning against either logic or probability.

kyburgNow this seems patently absurd. It is to suppose that even the most simple statistical inferences have no logical weight where my beliefs are concerned. It is perfectly compatible with these laws that I should have a degree of belief equal to 1/4 that this coin will land heads when next I toss it; and that I should then perform a long series of tosses (say, 1000), of which 3/4 should result in heads; and then that on the 1001st toss, my belief in heads should be unchanged at 1/4. It could increase to correspond to the relative frequency in the observed sample, or it could even, by the agency of some curious maturity-of-odds belief of mine, decrease to 1/8. I think we would all, or almost all, agree that anyone who altered his beliefs in the last-mentioned way should be regarded as irrational. The same is true, though perhaps not so seriously, of anyone who stuck to his beliefs in the face of what we would ordinarily call contrary evidence. It is surely a matter of simple rationality (and not merely a matter of instinct or convention) that we modify our beliefs, in some sense, some of the time, to conform to the observed frequencies of the corresponding events.

There is another argument against both subjestivistic and logical theories that depends on the fact that probabilities are represented by real numbers … The point can be brought out by considering an old fashioned urn containing black and white balls. Suppose that we are in an appropriate state of ignorance, so that, on the logical view, as well as on the subjectivistic view, the probability that the first ball drawn will be black, is a half. Let us also assume that the draws (with replacement) are regarded as exchangeable events, so that the same will be true of the i-th ball drawn. Now suppose that we draw a thousand balls from this urn, and that half of them are black. Relative to this information both the subjectivistic and the logical theories would lead to the assignment of a conditional probability of 1/2 to the statement that a black ball will be drawn on the 1001st draw …

Although it does seem perfectly plausible that our bets concerning black balls and white balls should be offered at the same odds before and after the extensive sample, it surely does not seem plausible to characterize our beliefs in precisely the same way in the two cases …

This is a strong argument, I think, for considering the measure of rational belief to be two dimensional; and some writers on probability have come to the verge of this conclusion. Keynes, for example, considers an undefined quantity he calls “weight” to reflect the distinction between probability-relations reflecting much relevant evidence, and those which reflect little evidence …

Though Savage distinguishes between these probabilities of which he is sure and those of which he is not so sure, there is no way for him to make this distinction within the theory; there is no internal way for him to reflect the distinction between probabilities which are based on many instances and those which are based on only a few instances, or none at all.

Henry E. Kyburg

The reference Kyburg makes to Keynes and his concept of “weight of argument” is extremely interesting.

Almost a hundred years after John Maynard Keynes wrote his seminal A Treatise on Probability (1921), it is still very difficult to find statistics textbooks that seriously try to incorporate his far-reaching and incisive analysis of induction and evidential weight.

treatprobThe standard view in statistics – and the axiomatic probability theory underlying it – is to a large extent based on the rather simplistic idea that “more is better.” But as Keynes argues – “more of the same” is not what is important when making inductive inferences. It’s rather a question of “more but different.”

Variation, not replication, is at the core of induction. Finding that p(x|y) = p(x|y & w) doesn’t make w “irrelevant.” Knowing that the probability is unchanged when w is present gives p(x|y & w) another evidential weight (“weight of argument”). Running 10 replicative experiments do not make you as “sure” of your inductions as when running 10 000 varied experiments – even if the probability values happen to be the same.

According to Keynes we live in a world permeated by unmeasurable uncertainty – not quantifiable stochastic risk – which often forces us to make decisions based on anything but “rational expectations.” Keynes rather thinks that we base our expectations on the confidence or “weight” we put on different events and alternatives. To Keynes expectations are a question of weighing probabilities by “degrees of belief,” beliefs that often have preciously little to do with the kind of stochastic probabilistic calculations made by the rational agents as modeled by “modern” social sciences. And often we “simply do not know.” As Keynes writes in Treatise:

The kind of fundamental assumption about the character of material laws, on which scientists appear commonly to act, seems to me to be [that] the system of the material universe must consist of bodies … such that each of them exercises its own separate, independent, and invariable effect, a change of the total state being compounded of a number of separate changes each of which is solely due to a separate portion of the preceding state … Yet there might well be quite different laws for wholes of different degrees of complexity, and laws of connection between complexes which could not be stated in terms of laws connecting individual parts … If different wholes were subject to different laws qua wholes and not simply on account of and in proportion to the differences of their parts, knowledge of a part could not lead, it would seem, even to presumptive or probable knowledge as to its association with other parts … These considerations do not show us a way by which we can justify induction … /427 No one supposes that a good induction can be arrived at merely by counting cases. The business of strengthening the argument chiefly consists in determining whether the alleged association is stable, when accompanying conditions are varied … /468 In my judgment, the practical usefulness of those modes of inference … on which the boasted knowledge of modern science depends, can only exist … if the universe of phenomena does in fact present those peculiar characteristics of atomism and limited variety which appears more and more clearly as the ultimate result to which material science is tending.

Science according to Keynes should help us penetrate to “the true process of causation lying behind current events” and disclose “the causal forces behind the apparent facts.” Models can never be more than a starting point in that endeavour. He further argued that it was inadmissible to project history on the future. Consequently we cannot presuppose that what has worked before, will continue to do so in the future. That statistical models can get hold of correlations between different “variables” is not enough. If they cannot get at the causal structure that generated the data, they are not really “identified.”

How strange that writers of statistics textbook as a rule do not even touch upon these aspects of scientific methodology that seems to be so fundamental and important for anyone trying to understand how we learn and orient ourselves in an uncertain world. An educated guess on why this is a fact would be that Keynes concepts are not possible to squeeze into a single calculable numerical “probability.” In the quest for quantities one puts a blind eye to qualities and looks the other way – but Keynes ideas keep creeping out from under the statistics carpet.

It’s high time that statistics textbooks give Keynes his due — and to re-read Henry E. Kyburg!

6 Comments

  1. Just found yourself another rss subscriber.

  2. “Science according to Keynes should help us penetrate to “the true process of causation lying behind current events” and disclose “the causal forces behind the apparent facts.” ”

    But this is metaphysics professor. There are infinite theories of causal nature that produce the same correlations. This is the underdetermination problem.

    Please do this as a thought experiment only to avoid breaking something: rorate a rock attached to a rope above your head. Does the rotation give rise to the centripetal force on the rock or the centripetal force gives rise to the rotation?

    Apparently, anyones’s guess is as good. Causality requires a metaphyiscal commitment, it cannot be founded on empirical evidence. Keynes was flat wrong.

    • Keynes has some ‘Notes on Part II’ (ii) ‘On the use of the term Cause’. I would not express myself the way he does, but he does address these metaphysical issues.

      • Everyone seems to address the issues but very few respect them. Causality is unattainable.

        • I’ve only just had DCosmology’s posting by email. I wonder what ’caused’ that delay.

          Anyway, I agree that naïve conceptions of probability and causality are dangerous, and would be tempted to deny that they exist. But it seems to me that something like them does exist. So for me the challenge is to develop meaningful relations that could replace the naïve ones.

          It seems to me that I know how to cause my car to go, subject to the usual caveats. These concern my limited knowledge of cars and the ever-present possibility of innovation. The problem with probabilistic and causal reasoning is, I think, that people overlook the caveats. What we lack is an explanation of ‘the usual caveats’ that allows us to get through the day without falling into danger. Keynes is close, but not an easy read.

  3. I agree with your conclusions, but Kyburg’s notion of what is ‘logical’ and ‘rational’ is quite different from mine, and Keynes’. My own view is that combining common-sense and mathematical notions in the way that Kyburg does could explain many of the apparent problems with Bayesianism. Keynes’ critique is more potent, if sadly less accessible.


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