Revisiting the foundations of randomness and probability

by
Lars Pålsson Syll
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from Lars Syll

Regarding models as metaphors leads to a radically different view regarding the interpretation of probability. This view has substantial advantages over conventional interpretations …

Probability does not exist in the real world. We must search for her in the Platonic world of ideals. We have shown that the interpretation of probability as a metaphor leads to several substantial changes in interpretations and justifications for conventional frequentist procedures. These changes remove several standard objections which have been made to these procedures. Thus our model seems to offer a good foundation for re-building our understanding of how probability should be interpreted in real world applications. More generally, we have also shown that regarding scientific models as metaphors resolves several puzzles in the philosophy of science.

Asad Zaman

Although yours truly has to confess of not being totally convinced that redefining​ probability as a metaphor is the right way to go forward on these foundational issues, Zaman’s article​ sure raises some very interesting questions on the way the concepts of randomness and probability are used in economics.

Modern mainstream economics relies to a large degree on the notion of probability. To at all be amenable to applied economic analysis, economic observations 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,  mainstream economics actually forces us to (implicitly) interpret events as random variables generated by an underlying probability density function.

This is at odds with reality. Randomness obviously is a fact of the real world (although I’m not sure Zaman agrees but rather puts also randomness in ‘the Platonic world of ideals’). 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.

We simply 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 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 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 modelled with probability measures.

If we agree on this, we also have to admit that much of modern neoclassical economics lacks sound foundations.

When economists and econometricians — often uncritically and without arguments — simply assume that one can apply probability distributions from statistical theory on their own area of research, they are really skating on thin ice.

This importantly also means that if you cannot show that data satisfies all the conditions of the probabilistic nomological machine, then the statistical inferences made in mainstream economics lack sound foundations!​