Decision Theory, Foundations of Probability, and Causality

Decision theory

It is not easy to disentangle measurement theory and decision theory because the measurement of subjective probability and utility has been such a central part of decision theory. The separation that I make will therefore be somewhat arbitrary. My really serious interest in psychology began with experimental research on decision theory in collaboration with my philosophical colleague Donald Davidson and a graduate student in psychology at that time, Sidney Siegel. Davidson and I had begun collaborative work with McKinsey in 1953 on the theory of value and also on utility theory. We continued this work after McKinsey’s death, and it is reflected in Davidson, McKinsey, and Suppes (1955a) and in the joint article with Davidson (1956c) on the finitistic axiomatization of subjective probability and utility, already mentioned. The article on the measurement of utility based on utility differences, with Muriel Winet, was also part of this effort.

Sometime during the year 1954, Davidson and I undertook, with the collaboration of Siegel, an experimental investigation of the measurement of utility and subjective probability. Our objective was to provide an explicit methodology for separating the measurement of the two and at the same time to obtain conceptually interesting results about the character of individual utility and probability functions. This was my first experimental work and consequently in a genuine sense my first real introduction to psychology. The earlier papers on the foundations of decision theory concerned with formal problems of measurement were a natural and simple extension of my work in the axiomatic foundations of physics. Undertaking experimental work was quite another matter. I can still remember our many quandaries in deciding how to begin, and seeking the advice of several people, especially our colleagues in the Department of Psychology at Stanford.

I continued a program of experimentation in decision theory as exemplified in the joint work with Halsey Royden and Karol Walsh (1959i) and the development of a nonlinear model for the experimental measurement of utility with Walsh (1959j). This interest continued into the sixties with an article (1960g) on open problems in the foundations of subjective probability. Then in 1961 I drew upon my interest in learning theory to try to create a behavioristic foundation for utility theory (1961a), and I also made an attempt in that same year to explain the relevance of decision theory to philosophy (196lb).

The most important effort in this period was the writing with Duncan Luce of a long chapter, ‘Preference, Utility and Subjective Probability’ (1965i), for Volume III of the Handbook of Mathematical Psychology. The organization of a large amount of material and the extensive interaction with Luce in the writing of this chapter taught me a great deal that I did not know about the subject, and I think the chapter itself has been useful for other people. It is also worth mentioning that large parts of the joint effort with Krantz, Luce and Tversky in writing our two-volume treatise on the foundations of measurement have been concerned with decision theory.

In the latter part of the sixties I wrote several articles in the foundations of decision theory, oriented more toward philosophy than psychology. Three of the articles appeared in a book on inductive logic edited jointly with Jaakko Hintikka, my part-time philosophical colleague at Stanford for many years.

One article dealt with probabilistic inference and the concept of total evidence (l966j). Here I advanced the argument that under a Bayesian conception of belief and decision there was no additional problem of total evidence, contrary to the view held by Carnap and also Hempel. According to this Bayesian view, which I continue to believe is essentially right on this matter, if a person is asked for the probability of an event at a given time, it will follow from the conditions of coherence on all of his beliefs at that time that the probability he assigns to the event automatically takes into account the total evidence that he believes has relevance to the occurrence of the event. The way in which total evidence is brought in is simple and straightforward; it is just a consequence of the elementary theorem on total probability.

A second article in the volume (1966e) set forth a Bayesian approach to the paradoxes of confirmation made famous by Hempel many years ago. I will not outline my solution here but much of the philosophical literature on the paradoxes of confirmation has taken insufficient account of the natural Bayesian solution, at least so I continue to think. A third article in the volume (1966f) dealt with concept formation and Bayesian decisions. Here I attempted to set forth the close relations between formal aspects of the psychology of concept formation and the theory of Bayesian decisions. I now think that the ideas I set forth here are the least interesting and the most transitory of those occurring in the three articles. The general idea of value in this article concerns the relation expressed between concept formation and the classical problem of induction. For those restricted settings in which no new concepts are needed but for which an induction about properties is required, a Bayesian approach is sound and can meet most, if not all, of the conceptual problems about induction that I regard as serious. On the other hand, a Bayesian viewpoint toward induction does not provide a general solution because it does not incorporate a theory of concept formation. Genuinely new inductive knowledge about the world requires not only a framework of inductive inference of the sort well worked out in the contemporary Bayesian literature but also a theory about how new concepts are to be generated and how their applicability is to be dealt with. This large and significant aspect of the general problem of induction seems to me still to be in a quite unsatisfactory state. In my own thinking, the problem of induction and the concept of rationality are closely tied together, and as I point out in the article on probabilistic inference mentioned above, the Bayesian approach still provides a very thin view of rationality, because the methods for changing belief as reflected in the introduction of new concepts or in the focus of attention are not at all adequately handled. The outlines of any future theory that will deal in even a partially satisfactory way with the central problem of concept formation are not at all visible, and it may even be that the hope for a theory that approaches completeness is mistaken.