Richard Price, Miracles and the Origins of Bayesian Decision Theory Geoffrey Poitras, Simon Fraser University HES Notre Dame U., June 20,

2 Who was the Rev’d Richard Price? Treatments of Richard Price ( ) in the history of economic thought are almost invisible (Morgan 1815, Pearson 1978 and Thomas 1977 useful for biography) Price has contributions still of relevance today: – Social security reform –Proper management of insurance companies –Inter-generational implications of government debt No substantive mention in HET vs. voluminous modern efforts on contemporaries in political economy, esp., Adam Smith and David Hume

HES Notre Dame U., June 20, Price and the ‘Early History of Financial Economics’ What is financial economics? HET timeline differs from classical political economy (Poitras 2000, Poitras and Jovanovic 2010) Price (1771) made seminal developments to life insurance by using sophisticated pricing formulae for financial securities developed before Adam Smith ( ) was born Building on work of Christian Huygens ( ), Jan de Witt ( ) solved the price for a life annuity Edmond Halley ( ) and Abraham de Moivre ( ) also were seminal

HES Notre Dame U., June 20, Early History Precursors of Price

The Reverend Richard Price ( ) A ‘dissenting’ (non-Anglican) English minister Observations on Reversionary Payments (1776) is the founding work of modern insurance mathematics – took mathematical contributions of de Moivre and applied to problems of insurance and social security design – Also was ‘father of modern public pension plans’ – the first actuary (at the Equitable) HES Notre Dame U., June 20,

Important Contributions of Richard Price Price (1758), A Review of the Principle Questions and Difficulties of Morals – produced while Price served as a family chaplin Price is responsible for the posthumous publication of Bayes (1763) Price (1768), Four Dissertations (much of this written prior to Bayes 1763) – Dissertation IV of central concern to this talk Price (1771), Observations on Reversionary Payments with further editions from (1772) containing scheme for old age pensions Price, R. (1776), Observations on the Nature of Civil Liberty, the Principles of Government, and the Justice and Policy of the War with America -- marked Price for infamy HES Notre Dame U., June 20,

Price and Bayes’s Theorem Rev’d Thomas Bayes (1702?-1761) earned the eponym ‘Bayes’s theorem’ for results appearing posthumously in Bayes (1763) –Price was mentioned in Bayes’s will and requested “by the relatives of that truly ingenious man, to examine the papers which he had written on different subjects, and which his own modesty would never suffer him to make public” (Morgan 1815, p.25) In a letter dated Nov. 10, 1763, Price communicated to the Royal Society the contents of a theorem unearthed in Bayes’s papers. That letter, published in Philosophical Papers contained the now famous Bayes’s Theorem. Price provided an Appendix to the paper HES Notre Dame U., June 20,

Bayes’s Theorem and Dissertation IV Price (1768) contains Dissertation IV, The Importance of Christianity, the Nature of Historical Evidence and Miracles – Dissertation IV still has modern relevance due to the seminal application of Bayesian decision theory. Following Morgan (1815) the main text of Dissertation IV was completed about 1760 –Argument using Bayes’s theorem appears as lengthy footnotes consistent with timeline for publication of Bayes (1763) –Main thesis of Dissertation IV does not require Bayes’s theorem HES Notre Dame U., June 20,

David Hume and the Miracles Debate Hume (1751, p.203) provides the basic issue in Hume’s skeptical attack: “we may establish it as a Maxim, that no human Testimony can have such Force as to prove a Miracle, and make it a just Foundation for any such System of Religion.” HES Notre Dame U., June 20,

Excerpts from Hume (1751) Hume (1751, p.180, 181, 182) on Hume’s attack: –“A Miracle is a Violation of the Laws of Nature; and as a firm and unalterable Experience has establish’d these Laws, the Proof against a Miracle, from the very Nature of the Fact, is as entire as any Argument against Experience can possibly be imagin’d... But ‘tis a Miracle, that a dead man should come to Life; because that has never been observ’d in any Age or Country... The plain Consequence is (and ‘tis a general Maxim worthy of our Attention), ‘that no Testimony is sufficient to establish a miracle, unless the Testimony be of such a Kind, that its Falsehood would be more Miraculous than the Fact which it endeavours to establish’.” HES Notre Dame U., June 20,

Epistemology of Hume’s Attack on Christian Miracles An integral part of a much larger philosophical project, Hume’s attack is about the use of inductive empiricism to infer causes from effects, a problem that inspired Bayes (1763) and still generates intellectual debate. Hume was a skeptical empiricist –As an empiricist, sensory perception was the primary source of knowledge; reason is subordinate to observation –As a skeptic, Hume recognized the fallibility of observation Skepticism has difficulties with assigning probabilities to seemingly certain events (e.g., nearly zero for a miracle) –Use of the ‘rising sun’ example in discussions up to Laplace where ‘Laplace’s rule of succession’ provides the probability for the next occurrence of an event that has never failed HES Notre Dame U., June 20,

Price’s Appendix and Laplace’s Rule of Succession Following Dale (1991), let x denote the probability associated with the next occurrence of an event S, where S i is the occurrence of S on trial i, then in modern notation Price’s Appendix initially gives results for: HES Notre Dame U., June 20,

Price’s Use of Bayes’s Theorem in Diss. IV Price uses a Bayesian argument to “mathematically demonstrate” that, even though a particular event has never been seen to occur in ten previous trials, calculating “the probability of its happening in a single trial [that] lies somewhere between any two degrees of probability that can be named”. This produces an unexpected (uniform prior) result: HES Notre Dame U., June 20,

Miracles and Bayes’s Theorem After Price Modern interpretation initially advanced by Condorcet. Letting ~ indicate negation, this produces two ‘prior’ probabilities which are relevant: P[M] and P[~M] where P[M | T[M]] is the probability that a miracle occurred, given that there was testimony for a miracle. Using the inverse probability form of Bayes’s theorem, this conditional probability can be solved as: HES Notre Dame U., June 20,