Taking a Gamble

Gambling is an ancient human activity. We indulge ourselves by allowing ourselves to experience the dangers and thrills of chance in a somewhat controlled way. The history of lotteries and related games is explored. The chances of drawing various poker hands are laid out. The role of probability in horse racing is described, and how the odds quoted are not strictly statements of probability, but terms on which business is to be done. Political prediction betting markets give us a further interpretation of probability as a way of expressing strength of opinion in a quantifiable, albeit flawed way. Wagers encourage boasters to put their money where their mouth is, and so to quantify their degree of belief.

How Not to Know the Principle of Induction

In The Problems of Philosophy, Bertrand Russell presents a justification of induction based on a principle he refers to as “the principle of induction.” Owing to the ambiguity of the notion of probability, the principle of induction may be interpreted in two different ways. If interpreted in terms of the subjective interpretation of probability, the principle of induction may be known a priori to be true. But it is unclear how this should give us any confidence in our use of induction, since induction is applied to the external world outside our minds. If the principle is interpreted in light of the objective interpretation of induction, it cannot be known to be true a priori, since it applies to frequencies that occur in the world outside the mind, and these cannot be known without recourse to experience. Russell’s principle of induction therefore fails to provide a satisfactory justification of induction.

Quantum Retrodiction: Foundations and Controversies

Prediction is the making of statements, usually probabilistic, about future events based on current information. Retrodiction is the making of statements about past events based on current information. We present the foundations of quantum retrodiction and highlight its intimate connection with the Bayesian interpretation of probability. The close link with Bayesian methods enables us to explore controversies and misunderstandings about retrodiction that have appeared in the literature. To be clear, quantum retrodiction is universally applicable and draws its validity directly from conventional predictive quantum theory coupled with Bayes’ theorem.

The Limits of Numerical Probability: Frank H. Knight and Ludwig von Mises and the Frequency Interpretation

In the following I will (1) briefly restate the principles of the fre-quency interpretation of probability as originally formulated by Richard von Mises; (2) show why Frank H. Knight and Ludwig von Mises must be considered frequency theorists; and (3) discuss and evaluate the arguments provided by F. H. Knight and L. v. Mises against the possibility of applying probability theory in the area of economic forecasting (whether on the micro or the macro level). Key words: probability theory, economic forecasting, frequency distribution. Clasificación JEL: B41, B53. Resumen: En este trabajo, 1) volveré a plantear brevemente los principios de la interpretación frecuencialista de la probabilidad tal y como fueron formulados originalmente por Richard von Mises; 2) mostraré por qué Frank H. Knight y Ludwig von Mises deben ser considerados teóricos de la interpretación frecuencialista, y 3) discutiré y evaluaré los argumentos proporcionados por F. H. Knight y L. v. Mises en contra de la posibilidad de aplicar la teoría de la probabilidad en el área de la previsión económica (ya sea a nivel microeconómico o macroeconómico). Palabras clave: teoría de la probabilidad, previsión económica, distribución de frecuencias.

John Maynard Keynes and Ludwig von Mises on Probability

The economic paradigms of Ludwig von Mises on the one hand and of John Maynard Keynes on the other have been correctly recognized as antithetical at the theoretical level, and as antagonistic with respect to their practical and public policy implications. Characteristically they have also been vindicated by opposing sides of the political spectrum. Nevertheless the respective views of these authors with respect to the meaning and interpretation of probability exhibit a closer conceptual affinity than has been acknowledged in the literature. In particular it is argued that in some relevant respects Ludwig von Mises’ interpretation of the concept of probability exhibits a closer affinity with the interpretation of probability developed by his opponent John Maynard Keynes than with the views on probability espoused by his brother Richard von Mises. Nevertheless there also exist significant differences between the views of Ludwig von Mises and those of John Maynard Keynes with respect to probability. One of these is highlighted more particularly: where John Maynard Keynes advocated a monist view of probability, Ludwig von Mises embraced a dualist view of probability, according to which the concept of probability has two different meanings each of which is valid in a particular area or context. It is concluded that both John Maynard Keynes and Ludwig von Mises presented highly nuanced views with respect to the meaning and interpretation of probability. JEL codes: B00; B40; B49; B53; C00. Key words: General Methodology; Austrian Methodology; Keynesian Methodology; Quantitative and Qualitative Probability Concepts: Meaning and Interpretation; Frequency Interpretation; Logical Interpretation; John Maynard Keynes; Ludwig von Mises; Richard von Mises. Resumen: Los paradigmas económicos de Ludwig von Mises por una parte, y de John Maynard Keynes por otra, han sido correctamente reconocidos como contradictorias a nivel teórico, y como antagonistas, con respecto a sus implicaciones políticas prácticas y públicas. Aún así, las respectivas visiones de estos autores con respecto al significado e interpretación de la probabilidad, muestra una afinidad conceptual más estrecha que los que se ha reconocido en la literatura. Se ha argumentado especialmente que en algunos aspectos importantes, la interpretación de Ludwig von Mises del concepto de probabilidad, muestra una más estrecha afinidad con la interpretación de probabilidad desarrollada por su oponente John Maynard Keynes, que con las maneras de ver la probabilidad respaldadas por su hermano Richard von Mises. Sin embargo, también existen grandes diferencias entre los puntos de vista de Ludwig von Mises y aquellos de John Maynard Keynes con respecto a la probabilidad. Uno de ellos destaca principalmente: cuando John Maynard Keynes aboga por un punto de vista monista de la probabilidad, Ludwig von Mises defiende un punto de vista dualista de la probabilidad, de acuerdo con el cual el concepto de probabilidad recibe dos significados diferentes, y en donde cada uno de ellos es válido en un área o contexto en particular. Se concluye que tanto John Maynard Keynes como Ludwig von Mises presentan puntos de vista claramente diferenciados con respecto al significado e interpretación de la probabilidad. Códigos JEL: B00; B40; B49; B53; C00. Palabras clave: Metodología General; Metodología austríaca; Metodología Keynesiana; Conceptos de probabilidad cuantitativos y cualitativos: Significado e Interpretación; Interpretación frecuencialista; Interpretación lógica; John Maynard Keynes; Ludwig von Mises; Richard von Mises.

Two Senses of “Probability”

The word “probability” has long been used in (at least) two distinct senses. One sense has to do with a rational agent’s degree of belief, commonly called credence in the philosophical literature. The other sort of probability is thought to be characteristic of a physical system, such as a roulette wheel; these are “in the world” rather than in our heads. This concept is called chance. It would be a mistake to think of these as rivals for the title of the single correct interpretation of probability. Rather, they are both useful concepts, with different roles to play. This chapter is an introduction to these concepts and their relations. It includes a discussion of the proper formulation and justification of a principle that links the two concepts, the Principal Principle. It is argued that neither of these concepts is dispensable. This raises the question of whether there is a notion of probability that can play the role of objective chance and is compatible with deterministic laws of physics.

The Interpretation of Probability in the Tractatus Logico-Philosophicus

In this paper, I propose an assessment of the interpretation of the mathematical notion of probability that Wittgenstein presents in TLP (1963: 5.15 – 5.156). I start by presenting his definition of probability as a relation between propositions. I claim that this definition qualifies as a logical interpretation of probability, of the kind defended in the same years by J. M. Keynes. However, Wittgenstein’s interpretation seems prima facie to be safe from two standard objections moved to logical probability, i. e. the mystic nature of the postulated relation and the reliance on Laplace’s principle of indifference. I then proceed to evaluate Wittgenstein’s idea against three criteria for the adequacy of an interpretation of probability: admissibility, ascertainability, and applicability. If the interpretation is admissible on Kolmogorov’s classical axiomatisation, the problem of ascertainability brings up a difficult dilemma. Finally, I test the interpretation in the application to three main contexts of use of probabilities. While the application to frequencies rests ungrounded, the application to induction requires some elaboration, and the application to rational belief depends on ascertainability.

The Errors of Art and Nature

This chapter analyzes the law of facility of errors. All the early applications of the error law could be understood in terms of a binomial converging to an exponential, as in Abrahan De Moivre's original derivation. All but Joseph Fourier's law of heat, which was never explicitly tied to mathematical probability except by analogy, were compatible with the classical interpretation of probability. Just as probability was a measure of uncertainty, this exponential function governed the chances of error. It was not really an attribute of nature, but only a measure of human ignorance—of the imperfection of measurement techniques or the inaccuracy of inference from phenomena that occur in finite numbers to their underlying causes. Moreover, the mathematical operations used in conjunction with it had a single purpose: to reduce the error to the narrowest bounds possible. With Adolphe Quetelet, all that began to change, and a wider conception of statistical mathematics became possible. When Quetelet announced in 1844 that the astronomer's error law applied also to the distribution of human features such as height and girth, he did more than add one more set of objects to the domain of this probability function; he also began to break down its exclusive association with error.