P. Dawid, Probability, Causality and the Empirical World: A Bayes-de Finetti-Popper-Borel Synthesis Statistical Science 2004, Vol. 19, No. 1,44-57.
This article expounds a philosophical approach to Probability and Causality: a synthesis of the personalist Bayesian views of de Finetti and Popper’s falsificationist programme. A falsification method for probabilistic or causal theories, based on “Borel criteria,” is described. It is argued that this minimalist approach, free of any distracting metaphysical inputs, provides the essential support required for the conduct and advance of Science.
Theory and Reality
We might interpret … a theoretical probability as a personal degree of belief, or as a propensity, but any such interpretation is irrelevant to our purposes. Rather, we should regard probabilities as entering our scientific theories as instrumental terms, the link between theoretical probabilities and the physical universe being indirect. This approach to interpreting probabilistic models avoids many potential philosophical pitfalls. In particular … it is able to eschew deep but ultimately irrelevant and distracting philosophical inquiry into the “true nature of Probability.”
The last sentence implies that the instrumental view does not go beyond mathematics. In the same way, mathematical Geometry and trigonometry can be regarded as instrumental, irrespective of the actual properties of the physical world. Dawid commends a straightforward Borel criterion:
As a model of the physical universe, [the hypothetical probability distribution] P could be regarded as falsified if, on observation, the corresponding physical property,
“the limiting relative frequency of H is P(H),”
is found to fail.
Note that limiting relative frequencies exist and are unique for all probabilistic models, so the idea that probabilities are appropriate instruments is falsified whenever there are either no limiting relative frequencies or multiple limiting relative frequencies, as when there is genuine innovation.
Probability assignments can be tested empirically, in the large rather than individually, by means of Borel falsifiability criteria such as calibration. Probability forecasts that are empirically valid attain a degree of objectivity: they are asymptotically unique (though typically unknowable). However, even this degree of objectivity remains relative, since the appropriate values depend on the nature and extent of the information available. Causal theories are nothing but ambitious probabilistic theories, positing certain common patterns of behavior across a range of different contexts, and raise no new issues of principle.
Note that one only ever fails to falsify the particular use of the probabilistic instrument in a particular case. One would need a separate argument to support any claim – as some do – that some probabilistic instrument always apply. This would imply that there were always long-run limiting frequencies. But in economics, fore example, this seems doubtful.
A generalisation of the above is to suppose that one is some system or ‘epoch’ such that unique limiting frequencies would exist if the system continued unchanged forever. But this is, perhaps, too metaphysical for the above paper.