Probabilistic reasoning and its limits (wonkish)

13 Apr, 2013 at 17:24 | Posted in Statistics & Econometrics | Comments Off on Probabilistic reasoning and its limits (wonkish)

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

The 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.

John Kay raises  similar concerns about simplistic probabilistic reasoning in a recent article on his blog: 

Probabilistic reasoning has become the dominant method of structured thinking about problems involving risk and uncertainty – to such an extent that people who do not think this way are derided as incompetent and irrational. Yet this probabilistic approach, a recent intellectual development, was heavily implicated in the 2008 financial crisis. Legal systems have evolved over hundreds if not thousands of years, applying different modes of reasoning.

It is possible – common, even – to believe something is true without being confident in that belief. Or to be sure that, say, a housing bubble will burst without being able to attach a high probability to any specific event, such as “house prices will fall 20 per cent in the next year”. A court is concerned to establish the degree of confidence in a narrative, not to measure a probability in a model.

Such narrative reasoning is the most effective means humans have developed of handling complex and ill-defined problems. A court can rarely establish a complete account of the probabilities of the events on which it is required to adjudicate. Similarly, an individual cannot know how career and relationships will evolve. A business must be steered into a future of unknown and unknowable dimensions.

So while probabilistic thinking is indispensable when dealing with recurrent events or histories that repeat themselves, it often fails when we try to apply it to idiosyncratic events and open-ended problems. We cope with these situations by telling stories, and we base decisions on their persuasiveness. Not because we are stupid, but because experience has told us it is the best way to cope. That is why novels sell better than statistics texts.

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 economists and other social scientists 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.