Taxonomy of Probability

Interpretations of Probability

Hierarchy of generality

Classical theory is restricted to equipossible cases
Frequency theory is restricted to repeatable, random phenomena
Subjectiveist theory applies to an event about which the agent is uncertain

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Taxonomy



Classical
(Laplace)


Frequency
(von Mises)


Propensity
(Popper)


                     Bayesian
                     /      \
                    /        \
                   /          \
                  /            \
             objective      subjective
              /    \     (Ramsey, de Finetti)
             /      \
            /        \
           /          \
          /            \
         /              \
     logical         empirical
(Carnap, Jaynes)   (Reichenbach)


Mathematical
(Kolmogorov)

Notes

Mathematical probability is distinct from the other five varieties which are "real world" probabilities.
The vast majority of statisticians are frequentists rather than Bayesians.
Frequentists are increasingly seeing the light and coming out as Bayesians.
Most Bayesians are subjective.
Good (1971) identified 46656 varieties of Bayesians.