The Dilemma of Probability Theory (Wonkish)

from Lars Syll

Modern neoclassical economics relies to a large degree on the notion of probability.

To at all be amenable to applied economic analysis, economic observations allegedly have to be conceived as random events that are analyzable within a probabilistic framework.

But is it really necessary to model the economic system as a system where randomness can only be analyzed and understood when based on an a priori notion of probability?

When attempting to convince us of the necessity of founding empirical economic analysis on probability models, neoclassical economics actually forces us to (implicitly) interpret events asrandom variables generated by an underlying probability density function.

This is at odds with reality. Randomness obviously is a fact of the real world. Probability, on the other hand, attaches (if at all) to the world via intellectually constructed models, and a fortiori is only a fact of a probability generating (nomological) machine or a well constructed experimental arrangement or “chance set-up”.

Just as there is no such thing as a “free lunch,” there is no such thing as a “free probability.” To be able at all to talk about probabilities, you have to specify a model. If there is no chance set-up or model that generates the probabilistic outcomes or events – in statistics one refers to any process where you observe or measure as an experiment (rolling a die) and the results obtained as the outcomes or events (number of points rolled with the die, being e. g. 3 or 5) of the experiment – there strictly seen is no event at all.

Probability is a relational element. It always must come with a specification of the model from which it is calculated. And then to be of any empirical scientific value it has to be shown to coincide with (or at least converge to) real data generating processes or structures – something seldom or never done!

And this is the basic problem with economic data. If you have a fair roulette-wheel, you can arguably specify probabilities and probability density distributions. But how do you conceive of the analogous nomological machines for prices, gross domestic product, income distribution etc? Only by a leap of faith. And that does not suffice. You have to come up with some really good arguments if you want to persuade people into believing in the existence of socio-economic structures that generate data with characteristics conceivable as stochastic events portrayed by probabilistic density distributions!

From a realistic point of view we really have to admit that the socio-economic states of nature that we talk of in most social sciences – and certainly in economics – are not amenable to analyze as probabilities, simply because in the real world open systems that social sciences – including economics – analyze, there are no probabilities to be had!

The processes that generate socio-economic data in the real world cannot just be assumed to always be adequately captured by a probability measure. And, so, it cannot really be maintained that it even should be mandatory to treat observations and data – whether cross-section, time series or panel data – as events generated by some probability model. The important activities of most economic agents do not usually include throwing dice or spinning roulette-wheels. Data generating processes – at least outside of nomological machines like dice and roulette-wheels – are not self-evidently best modeled with probability measures.

If we agree on this, we also have to admit that much of modern neoclassical economics lacks a sound justification. I would even go further and argue that there really is no justifiable rationale at all for this belief that all economically relevant data can be adequately captured by a probability measure. In most real world contexts one has to argue and justify one’s case. And that is obviously something seldom or never done by practitioners of neoclassical economics.

As the great British mathematician John Edensor Littlewood says in A Mathematician’s Miscellany:

Mathematics (by which I shall mean pure mathematics) has no grip on the real world ; if probability is to deal with the real world it must contain elements outside mathematics ; the meaning of ‘ probability ‘ must relate to the real world, and there must be one or more ‘primitive’ propositions about the real world, from which we can then proceed deductively (i.e. mathematically). We will suppose (as we may by lumping several primitive propositions together) that there is just one primitive proposition, the ‘probability axiom’, and we will call it A for short. Although it has got to be true, A is by the nature of the case incapable of deductive proof, for the sufficient reason that it is about the real world …

We will begin with the … school which I will call philosophical. This attacks directly the ‘real’ probability problem; what are the axiom A and the meaning of ‘probability’ to be, and how can we justify A? It will be instructive to consider the attempt called the ‘frequency theory’. It is natural to believe that if (with the natural reservations) an act like throwing a die is repeated n times the proportion of 6′s will, with certainty, tend to a limit, p say, as n goes to infinity … If we take this proposition as ‘A’ we can at least settle off-hand the other problem, of the meaning of probability; we define its measure for the event in question to be the number p. But for the rest this A takes us nowhere. Suppose we throw 1000 times and wish to know what to expect. Is 1000 large enough for the convergence to have got under way, and how far? A does not say. We have, then, to add to it something about the rate of convergence. Now an A cannot assert a certainty about a particular number n of throws, such as ‘the proportion of 6′s will certainly be within p +- e for large enough n (the largeness depending on e)’. It can only say ‘the proportion will lie between p +- e with at least such and such probability (depending on e and n*) whenever n>n*’. The vicious circle is apparent. We have not merely failed to justify a workable A; we have failed even to state one which would work if its truth were granted. It is generally agreed that the frequency theory won’t work. But whatever the theory it is clear that the vicious circle is very deep-seated: certainty being impossible, whatever A is made to state can only be in terms of ‘probability ‘.

This importantly also means that if you cannot show that data satisfies all the conditions of the probabilistic nomological machine, then the statistical inferences used – and a fortiorineoclassical economics – lack sound foundations!