On R. Muth’s Impossible Dream: Trying to turn Subjective Probabilities into Objective Probabilities

Abstract

R. Muth’s 1961 Econometrica paper is based on a fundamental and basic confusion on his part about the nature of subjective and objective probability that is impossible to remedy. A bifurcation occurred among scientists, mathematicians, and statisticians, starting in the mid 17th century with the work of Pascal, Fermat,and Cardano, about whether probability was subjective or objective. It could not be both. It had to be one or the other.

Subjective probability always dealt with probability as being inside or internal to the human mind. It was a degree of belief of an individual based on his mental, personal, and psychological assessment of the data and factors he believed were involved in determining the behavior of the particular phenomena. On the other hand, Objective probability was always outside or external to the human mind. It was a part of the empirical, real world of phenomena that could be observed and was completely independently from any personal, emotional, mental, psychological or subjective factors. Objective probability was always a relative frequency in occurrence that would approach the true or correct answer in the limit as the number of observations of the particular phenomena approached infinity or some very large number that was an approximation of infinity. Historically, there are no adherents of the subjective approach to probability who accept the objective approach to probability and there are no adherents of the objective approach to probability who accept a subjective approach to probability. There is not, nor could there ever be, a subjective –objective hybrid theory because the definitions of subjective and objective probability directly contradict and conflict with each other.

Muth had no historical knowledge of these two completely different definitions, forms, concepts and approaches to probability. Neither did the referees of his paper for Econometrica nor any proponent, advocate, supporter or user of what Muth claimed was the rational expectations hypothesis, which was that, for a given information set, the subjective estimates of probability (subjective probability distributions) were distributed around one single, objective, true, correct objective probability distribution. Muth apparently never realized that his definition was an oxymoron. It is impossible by definition. Nowhere in any of his references are there any that deal with the completely different theories of subjective and objective probability; there are no references to any sources in the history of science, history of probability, history of statistics, philosophy of science, philosophy of probability, or philosophy of statistics. There are no references to methodology or epistemology or to the philosophy of social science.

Muth used the 1944 Von Neumann-Morgenstern approach to Expected Utility as the foundation for his theory of consumer and producer maximization of utility and profit in his 1961 article. However, it was made very clear by Von Neumann and Morgenstern that they were going to assume the use of objective frequencies only, because they were only going to deal with risk, where risk was defined to be based on known, objective probability. No subjective probability was allowed. Von Neumann had a completely different theory to deal with (complete) uncertainty when no probabilities were known, which he called min-max. Thus, the individual consumer-producer probability distributions would have to be objective, not subjective, in Muth’s paper in order to be consistent with the theory of risk analyzed by Von Neumann-Morgenstern in 1944. Out of nowhere, Muth starts talking about subjective probabilities (distributions) being distributed, for a given information set, around a true, correct, objective probability distribution. This is impossible and can never occur. There is absolutely no foundation at all in any theory of probability for Muth’s definition of rational expectations.

The entire literature on the rational expectations hypothesis needs to be completely and totally reformulated and rewritten so as to be consistent with a foundation in either the subjective theory of probability or the objective theory of probability. Otherwise, rational expectationists need to state that they are not dealing with probability and statistics at all. They are simply assuming the very extreme position of complete knowledge from the Greenspan or Keynesian Continuum or spectrum that ranges from situations of complete ignorance to situations of complete knowledge. Currently, there simply is no foundation in any theory of probability for the rational expectations hypothesis, as defined by Muth in 1961 and accepted in the literature in economics since the Robert Lucas paper of 1972, which simply repeats all of the unsupportable claims made by Muth in his definition of rational expectations in 1961 that confused subjective and objective probability.