Saturday, August 30, 2014

Gillies’ Philosophical Theories of Probability, Chapter 2

Chapter 2 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with the Classical interpretation of probability theory.

As Gillies notes, the “Classical” interpretation was the earliest theory of probability and its most important statement was by Pierre-Simon Laplace (1749–1827) in his Essai Philosophique sur les Probabilit├ęs [A Philosophical Essay on Probabilities] (1814). However, it is largely of historical interest now, and has no supporters today (Gillies 2000: 3).

Laplace’s Essai Philosophique sur les Probabilit├ęs (1814) made the assumption of universal determinism on the basis of Newtonian mechanics (Gillies 2000: 16). Laplace argued that an agent with perfect knowledge of Newtonian mechanics and all matter could predict the future state of the universe. It is only human ignorance that prevents perfect forecasting, and leads us to calculate probabilities (Gillies 2000: 17). Thus probability, according to Laplace, is a measure of human ignorance (Gillies 2000: 21).

Laplace’s formula for calculating probabilities is the familiar one where the probability P(E) of any event E in a finite sample space S, where all outcomes are equally likely, is the number of outcomes for E divided by the total number of outcomes in S.

But there is an obvious limitation with this, as pointed out by the later advocates of the frequency theory of probability like Richard von Mises: what if our outcomes are not equiprobable? (Gillies 2000: 18). Thus the Classical interpretation of probability has a serious shortcoming.

As probability theory came to be increasingly applied to phenomena in the natural and social sciences in the 19th century, its limiting assumption of equiprobable outcomes was exposed as a problem, and the relative frequency approach was developed as a new and alternative theory.

BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.

Friday, August 29, 2014

Gillies’ Philosophical Theories of Probability, Chapter 1

Donald Gillies’ Philosophical Theories of Probability (2000) is an excellent overview of probability theory.

The book is of great interest, because Gillies (2000: xiv) has knowledge of Post Keynesian work on probability and uncertainty, and also sees his “intersubjective” theory of probability as a compromise between the theories of Keynes and Ramsey.

Probability has both a mathematical and philosophical/epistemic aspect.

The earliest “Classical” interpretation of probability of Pierre-Simon Laplace (1749–1827), which was based on earlier work from the 1650 to 1800 period, is now of historical interest only, and has no supporters today (Gillies 2000: 3).

Gillies (2000: 1) identifies five major modern interpretations of probability, which are in turn divided into two broad categories, as follows:
(i) Epistemological/Epistemic probability theories
(1) the logical interpretation;
(2) the subjective interpretation (personalism, subjective Bayesianism);
(3) the intersubjective view.
(ii) Objective probability theories
(4) the frequency interpretation;
(5) the propensity interpretation.
The “intersubjective” interpretation of probability is developed by Gillies (2000: 2) himself.

The epistemological/epistemic group of probability theories take probability to be a degree of belief, whether rational or subjective (Gillies 2000: 2).

The objective probability theories take probabilities to be an objective aspect of certain things or processes in the external world (Gillies 2000: 2).

Gillies (2000: 2–3) argues that all the major theories of probability may be compatible, as long as they are limited to their appropriate domains: for example, objective probabilities are usually appropriate for the natural sciences and epistemological/epistemic probabilities for the social sciences.

Serious study of probability began with mathematical theories of probability, often inspired by interest in gambling games (Gillies 2000: 4, 10), and these mathematical theories emerged in the 17th and 18th centuries, and famously in the correspondence between Blaise Pascal (1623–1662) and Pierre de Fermat (1601/1607–1665) in 1654 (Gillies 2000: 3), Jacob Bernoulli’s (1655–1705) treatise Ars Conjectandi (1713), the work of Abraham de Moivre (1667–1754), and of Thomas Bayes (c. 1701–1761) (Gillies 2000: 4–8).

BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.