Back in 1991, when yours truly earned his first PhD with a dissertation on decision making and rationality in social choice theory and game theory, I 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.”

This, of course, was like swearing in church. My mainstream colleagues were — to say the least — not exactly überjoyed.

The decision theoretical approach I was most critical of, was the one building on the then reawakened Bayesian subjectivist (personalistic) 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:

bFrom the point of view of the “logic of consistency”, 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 1/3 to the probability that the sun will rise tomorrow; or, more cheerfully, to take the probability to be 9/10 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 …

Now 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 …

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 … 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 …

Henry E. Kyburg

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

treatprobVariation, 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. 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 modelled by mainstream economists.

How strange that mainstream economists 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 two-dimensional concepts of evidential weight and uncertainty are not possible to squeeze into a single calculable numerical ‘probability.’ In the quest for calculable risk, one puts a blind eye to genuine uncertainty and looks the other way.