In existing models of decision under uncertainty where the decision maker may have multiple beliefs (vague beliefs) the different beliefs are not distinguished by their degree of reliability. This paper extends the model of vague beliefs in this direction: the decision maker may have multiple beliefs, but also has a complete order over them (a belief over his own beliefs, or higher order belief...

3 Basics of information theory 3 3.1 Entropy and uncertainty . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Joint and conditional entropy . . . . . . . . . . . . . . . . . . . . 4 3.3 Mutual information and relative entropy . . . . . . . . . . . . . . 5 3.4 Some inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.5 Entropy of discrete distributions . . . . . . . . . . . ...

Probabilistic independence has proved to be a fundamental tool that can dramatically simplify the task of eliciting, representing, and computing with probabilities. We advance the position that notions of utility independence can serve similar functions when reasoning about preferences and utilities during decision making. In this paper we first summarize existing results and definitions concer...

A decision problem is defined in terms of an outcome space, an action space and a loss function. Starting from these simple ingredients, we can construct: Proper Scoring Rule; Entropy Function; Divergence Function; Riemannian Metric; and Unbiased Estimating Equation. We illustrate these for the case of a Riemannian outcome space. From an abstract viewpoint, the loss function defines a duality b...

We briefly introduce and discuss some criteria for optimal decisions. For the majority of this module, we consider a person’s utility for different options (or ‘rewards’ or ‘actions’) and, with uncertainty about the reward which will follow a specific decision, we aim at maximisation of the expected utility, which is actually the utility of the optimal decision (note that a decision is a gamble...

This paper explores the possibility that causal decision theory can be formulated in terms of probabilities of conditionals. It is argued that a generalized Stalnaker semantics in combination with an underlying branching time structure not only provides the basis for a plausible account of the semantics of indicative conditionals, but also that the resulting conditionals have properties that ma...

In a recent paper, Deutsch [1] claims to derive the “probabilistic predictions of quantum theory” from the “non-probabilistic axioms of quantum theory” and the “non-probabilistic part of classical decision theory.” We show that his derivation fails because it includes hidden probabilistic assumptions.

At the 1976 Oxford Conference, Aanderaa introduced a new class of machines which he called F machines (later renamed as modular machines). Using these he gave two remarkably short and easy examples of finitely presented groups with unsolvable word problem. Both of these examples, together with an exposition of modular machines, will be found in a paper by Aanderaa and Cohen [2]. The aim of this...

Many social scientists other than psychologists try to account for the behavior of individuals. Economists and a few psychologists have produced a large body of theory and a few experiments that deal with individual decision making. The kind of decision making with which this body of theory deals is as follows: given two states, A and B, into either one of which an individual may put himself, t...

This paper provides new foundations for Bayesian Decision Theory based on a representation theorem for preferences de…ned on a set of prospects containing both factual and conditional possibilities. This use of a rich set of prospects not only provides a framework within which the main theoretical claims of Savage, Ramsey, Je¤rey and others can be stated and compared, but also allows for the po...