Are there important philosophical interpretations of probability? What are the major "schools" or frameworks? What is their relation to formal systems of probability (for instance - the orthodox axiomatic system of Kolmogorov)? And how do they differ with respect to applications of the formal theory of probability?
Rather than the terms objective and subjective, I think it is more helpful to classify probabilities into two broad camps as physical and epistemic. Physical includes frequency and propensity theories, while epistemic includes logical probability and different versions of bayesianism. There is nothing inherently subjective about understanding probabilities as epistemic, it just means that a probability is a measure of information, so different people with different information will assign different probabilities. Epistemic probability can still be objective in that for a given specification of information, a unique conditional probability value will usually drop out.
The distinctive semantics of epistemic probability is that it is a credence, or degree of belief, or possibly a statement about information content. As such, it is not a property of the world itself, but of an individual's information about the world. Although the syntax and rules of probability are common, there is defintely an interaction between the mathematical theory and the interpretation. Kolmogorov uses the concept of a possibility space or event space, which is not an essential component of the epistemic approach. Consequently, advocates of the epistemic approach often prefer an alternative axiomatisation in which conditional probabilities are defined as primary, and categorical probabilities, if they are used at all, are just conditional probabilities that are conditional on tautologies. Cox showed how logical probability can be defined in a quite different way from Kolmogorov, and de Finetti showed how probability can be derived from Dutch book arguments about rational decisions.
As to applications, one consequence of the difference is that frequentist methods and bayesian methods often take quite an opposite approach. To the frequentist, the way the world is is a constant: so propositions about the way the world is don't have probabilities (except for being trivially either 0 or 1). The frequentist gathers data to test a hypothesis and it is the data that is random, and is modelled using random variables. The bayesian sees things the other way round: propositions about the way the world is have probabilities, because they are unknown, while the data is certain, because we possess it. The frequentist asks, how probable is it that I would see data like this given some hypothesis about the world? The bayesian asks, how probable is this hypothesis given the data I have? (I'm simplifying greatly here, and statisticians will be squirming, but broadly this difference holds.)
A quick intro to the main interpretations of probability discusses the following interpretations:
- classical (Bernoulli, Laplace, and most everyone up to the 1800’s)
- frequency (Venn, Reichenbach, von Mises)
- logical (Keynes, Jeffrey, Carnap)
- propensity (Popper)
- subjectivist (de Finetti, Jeffrey)
That intro was the first "non-encyclopedia" google hit, and SEP basically agrees:
Traditionally, philosophers of probability have recognized five leading interpretations of probability - classical, logical, subjectivist, frequentist, and propensity. But recently, so-called best-system interpretations of chance have become increasingly popular and important.
A prediction interpretation is added to this list by wikipedia (claiming this was once a dominant one):
An alternative account of probability emphasizes the role of prediction - predicting future observations on the basis of past observations, not on unobservable parameters. In its modern form, it is mainly in the Bayesian vein. This was the main function of probability before the 20th century, but fell out of favor compared to the parametric approach, which modeled phenomena as a physical system that was observed with error, such as in celestial mechanics.
Except for the propensity and the subjectivist interpretation, a common theme of the interpretations is to acknowledge that the concept of probability is needed, and then to suggest an interpretation with as few explicit commitments as possible, which is still explicit enough for the envisioned applications. Peter Whittle's Probability via Expectation approach appeared promising in this respect to Arnold Neumaier, to avoid undesired commitments to specific reality concepts when interpreting quantum mechanics. Avoiding commitments is different from a position like expressed by Wolfgang Schwarz, that such commitments are an inadequate interpretation of probabilistic theories in science.
For some applications like game theory, explicit interpretation of stochastic independence as a "successful" pokerface might be required for making probability theory applicable:
One theoretical weakness of a Turing machine is its predictability. An all powerful and omniscient opponent could exploit this weakness when playing some game against the Turing machine. So if a theoretical machine had access to a random source which its opponent could not predict (and could conceal its internal state from its opponent), then this theoretical machine would be more powerful than a Turing machine.
A nice explanation of various interpretations of probability are given here :5 interpretations of probability