A Bayesian framework for risk perception

Abstract

We present here a Bayesian framework of risk perception. This framework encompasses plausibility judgments, decision making, and question asking. Plausibility judgments are modeled by way of Bayesian probability theory, decision making is modeled by way of a Bayesian decision theory, and relevancy judgments are modeled by way of a Bayesian information theory. These theories are discussed in Parts I, II, and III, respectively, of this thesis. Bayesian probability theory is fairly well known and well established. Bayesian probability theory is not only a powerful tool of data analysis, but it also may function as a model for the way we (implicitly) do induction, that is, the way we make plausibility judgments on the basis of incomplete information. In Part I of this thesis we will make the case that Bayesian probability theory is nothing but common sense quantified. The Bayesian decision theory, as proposed in this thesis, derives directly from Bayesian probability theory. In this decision theory we compare utility probability distributions, which are constructed by way of assigning utilities, that is, subjective worths, to the objective outcomes of our outcome probability distributions, which are derived by way of Bayesian probability theory. When the outcomes under consideration are monetary, then we may use the Weber-Fechner law of psychophysics, or, equivalently, Bernoulli's utility function, to assign utilities to these outcomes. This mapping of outcomes to utilities, transforms our outcome probability distributions to their corresponding utility probability distributions. That utility probability distribution which is located more to the right on the utility axis will tend to be, depending on the context of our problem of choice, either more profitable or less disadvantageous than the utility probability distribution that is more to the left. So, we will tend to prefer that decision which `maximizes' our utility probability distributions. This then, in a nutshell, is the whole of our Bayesian decision theory. In Part~II of this thesis, we will apply the Bayesian decision theory to both investment and insurance problems. Not all questions are equal, some questions, when answered, may give us more information than others. Stated differently, questions may differ in their relevancy, in relation to some issue of interest we wish to see resolved. This is borne out by the well known adage that, 'to know the question, is to have gone half the journey'. Bayesian information theory, by way of a mathematical operationalization of the concept of a question, allows us to determine which question, when answered, will be the most informative in relation to some issue of interest. The Bayesian information theory does this by assigning relevancies to the questions under consideration. These relevancies are then operated upon, by way of the information theoretical product and sum rules, in order to determine the relevancy of some question in relation to the issue of interest. The Bayesian information theory constitutes an expansion of the 'canvas of rationality', and, consequently, of the range of psychological phenomena which are amenable to mathematical analysis. For example, we may assign relevancies not only to questions, but also to the messages that are communicated to us by some source of information. The relevancy of a message represents the usefulness of that message, when received, in determining some issue of interest. By assigning a relevancy to the message, we indirectly assign a relevancy to the sources of information itself; possible examples of sources of information being the media, scientists, and governmental institutions. In Part~III of this thesis, we will give an information theoretical analysis of a simple risk communication problem. Bayesian probability has its axiomatic roots in lattice theory, as the product and sum rule of Bayesian probability theory may be derived by way of consistency requirements on the lattice of statements. One may derive, likewise, by way of consistency requirements on the lattice of questions, the product and sum rules of Bayesian information theory. So, if we choose rationality, that is, consistency requirements on lattices, as our guiding principle in the derivation of our theories of inference, then we get on the one hand a Bayesian probability theory, with as its specific application a Bayesian decision theory, and on the other hand we get a Bayesian information theory. In doing so, we obtain a comprehensive, coherent, and powerful framework with which to model human reasoning, in the widest sense.