An overview of the key works on probability (let me know if you disagree). Dale A History of Inverse Probability: from Thomas Bayes to Karl Pearson provides an alternative overview with selected quotes. But it seems to me that he understates the authors’ reservations about the generality of their theories.
Bayes and Laplace are often regarded as the founders of probability theory.
Bayes develops a numeric theory of probability, based on utility, including Bayes’ rule, and shows how to infer a probability interval. But he also cautions about extrapolating too far.
Laplace developed a version of Bayes’ rule based on utility, but recognizing that utility is typically ‘sub-linear’ or convex and that one should take account of any specific evidence that may conflict with, for example, Bayes’ rule. Also, Laplace recognizes that short-term regularities may not always be continued indefinitely.
Ramsey is often regarded as providing the mathematical foundations of subjective probability, based on ‘mathematical expectation’, but notes the limitations.
Jeffreys developed an ‘objective’ approach, based on selecting the ‘simplest’ of possible explanations.
Fisher notes some severe restrictions on the justifications for probability theory. (These are perhaps not always appreciated as they might be.)
Jaynes develops a probability theory based on entropy maximization. He notes that this implicitly assumes ‘an infinitely educated brain’.
Lindley is a well respected advocate of ‘Bayesianism’, but emphasises ‘the need to think’.
Keynes developed general, non-numeric, theories of probability, which he applied as the basis of his economic theories. Binmore has developed a theory based on the notion of ‘muddled’ sequences, as a generalization of conventional random sequences. Neither are well known.
Russell reviewed the then(1940s) mainstream theories, adapting Keynes’. He showed that conventional (numeric) probability was justified when normal science was, and discussed what these conditions might be. Perhaps confusingly, Russell uses the term ‘mathematical probability’ for the run-off-the-mill theory, only appropriate to run-of-the-mill circumstances. But it seems to me that the different theories espoused by Keynes in his Treatise for his mathematics fellwoship are also ‘mathematical’, albeit in need of a little development.
- Related material is under Economics, e.g. where Bernoulli , de Finetti and von Neuman & Morgenstern argue from economics to probability, or to the limits of probability theory.
- Many works under Rationality and Uncertainty argue for non-measureable uncertainty.
- Some works on ‘how people actually reason’, such as Ellsberg‘s, are also of interest.
- My blog has a browsable index, using mouseover. Pages on probability can be found under ‘bibliography/rationality and uncertainty/probability’. Sadly, it doesn’t work on most smartphones (2012).