A Classification of Types of Probability and Theories of Probability

There are three fundamental conceptual divisions in the way that probability theory has been interpreted:

(1) the Classical interpretation;

(2) the epistemological (or epistemic) probability theory, further divided into

(i.) the logical interpretation;
(ii.) the subjective interpretation (personalism, subjective Bayesianism);
(iii.) the intersubjective interpretation;

(3) objective probability theory, further divided into

(i.) the frequency interpretation;
(ii.) the propensity interpretation.

(Gillies 2000: 2).

These are basically overarching philosophical interpretations of probability. The Classical interpretation is probably of historical interest only.

Keynes and Harold Jeffreys held the logical interpretation (2.i), which nevertheless seems widely rejected by modern philosophers of probability.

Frequency theorists include John Venn, A. N. Kolmogorov and Richard von Mises.

A subjective personalist theory of probability was developed by Bruno de Finetti, Frank P. Ramsey, and Leonard J. Savage. A decision making theory was developed from this that is still fundamental in neoclassical economics.

Regarding actual types of probability as a property and not a philosophy theory, although there are different classifications (see Appendix 1), perhaps there are two types, as argued by Rudolf Carnap and Ian Hacking:

(1) Epistemic/epistemological probability
A property of inferred propositions in inductive arguments, depending on the validity and soundness of the inductive arguments and evidence offered in support of it. It is thus a partial logical entailment. This is basically inductive probabilism.

(2) Aleatory probability
Long-run, relative frequency probabilities that are numerical values and that pertain to properties of elements of sets, classes or kinds. (McCann 1994: 27).

Basic notation to express probability is

P(h|e),
where P is the probability,
h is some hypothesis or conclusion, and
e is the evidence or premises.

This is usually read as “the probability of h given evidence e.” Numerical values for probability lie between 0 and 1.

0 denotes impossibility and 1 certainty.

Epistemic/epistemological probability is obviously strongly connected with induction, generally the following types of argument:

(1) induction by simple enumeration;
(2) argument by analogy;
(3) statistical syllogism, and
(4) induction to a particular.

While aleatory probabilities are capable of having numerical values, it seems that many types of inference from inductive arguments are not.

But even many events that look like they might have aleatory probabilities cannot yield them:

“In games of chance, scientific inference is possible because … an aggregate regularity (in fair games) is readily apparent; chance affords an objective, homogeneous, stationary series. In empirically observable series, on the other hand, series chosen from a potentially unstable natural environment, such homogeneity and regularity may not be in evidence. One cannot a priori assume stability; rather one must be alert to the possibly chaotic nature of any empirical series which may, over the short and the long run, generate patterns for which a probability distribution does not exist or one which generates no discernible pattern whatsoever.” (McCann 1994: 32–33).

For example, what use is the time series data on the average daily selling price of a stock in providing an objective numerical value for the probability that this stock will have value y on the 15 July, 2017? The answer is: it is useless.

The assumption of an objective, homogeneous, stationary process producing events or variables over time, in the past, present and future, is the ergodic hypothesis or ergodic axiom, familiar from neoclassical economics. If the relative frequencies of outcomes of some process converge over a long-run time series, then the process is ergodic (Glickman 2003: 368). But many economic phenomena are non-ergodic, and, for example, non-stationarity is a sufficient condition for non-ergodicity. Therefore objective probabilities do not exist in such processes: past and present time series data are of limited value or just useless for strict prediction or forecasts in terms of numerical value probabilities.

But probabilities – whether (1) objectively numerical or (2) inductive and non-numerical – only form a basis and criterion for decision and action, and decision making theory must be concerned with how people actually make decisions in particular situations, and avoid highly abstract, logically incoherent, and empirically false theories.

APPENDIX 1
Other systems of classifying types of probability as a property include the following:

(I.) Wesley Salmon (1967):

(1) Classical.
(2) subjective;
(3) frequency
(4) logical, and
(5) personal.

(II.) Roy Weatherford (1982):

(1) Classical.
(2) subjective/personal;
(3) frequency, and
(4) logical.

(III.) Leonard J. Savage (1972):

(1) necessarian;
(2) personalist, and
(3) frequentist.

Further Reading
Abrams, Marshall. 2012. “Mechanistic Social Probability: How individual Choices and Varying Circumstances produce Stable Social Patterns,” in Harold Kincaid (ed.), The Oxford Handbook of Philosophy of Social Science, Oxford University Press, Oxford. 184-226.

Galavotti, M. C. 2010. “Probability,” in Stathis Psillos and Martin Curd (eds.), The Routledge Companion to Philosophy of Science. Routledge, London and New York. 414-424.

Hartmann, S. and J. Sprenger, 2010. “Bayesian Epistemology,” in Sven Bernecker and Duncan Pritchard (eds.), The Routledge Companion to Epistemology. Routledge, London. 609-620.

Humphreys, Paul. 1998. “Probability, Interpretations of,” in Edward Craig (ed.), Routledge Encyclopedia of Philosophy. Volume 7, Nihilism - Quantum Mechanics. Routledge, London.

Interpretations of Probability, Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/

Loewer, Barry. 1998. “Probability Theory and Epistemology,” in Edward Craig (ed.), Routledge Encyclopedia of Philosophy. Volume 7, Nihilism - Quantum Mechanics. Routledge, London. 705-711.

Skyrms, B. 2010. “Probability, Theories of,” in Jonathan Dancy, Ernest Sosa, and Matthias Steup (eds.), A Companion to Epistemology (2nd edn.). Wiley-Blackwell, Oxford. 622–626.


BIBLIOGRAPHY
Gillies, Donald. 2000. Philosophical Theories of Probability. Routledge, London and New York.

Glickman, M. 2003. “Uncertainty,” in J. E. King (ed.), The Elgar Companion to Post Keynesian Economics. E. Elgar Pub., Cheltenham, UK and Northhampton, MA. 366–370.

McCann, Charles R. 1994. Probability Foundations of Economic Theory. Routledge, London.

Salmon, Wesley Charles. 1967. Foundations of Scientific Inference. University of Pittsburgh Press, Pittsburgh.

Savage, Leonard Jimmie. 1972. The Foundations of Statistics (2nd rev. edn.). Dover, New York.

Weatherford, Roy. 1982. Philosophical Foundations of Probability Theory. Routledge & Kegan Paul, London.