William Peden

How should we make decisions when we do not know the relevant physical probabilities? In these ambiguous situations, we cannot use our knowledge to determine expected utilities or payoffs. The traditional Bayesian answer is that we should create a probability distribution using some mix of subjective intuition and objective constraints. Imprecise Bayesians argue that this approach is inadequate for modelling ambiguity. Instead, they represent doxastic states using credal sets. Generally, insofar as we are more uncertain about the physical probability of an event, there is more divergence in the credal set. Hence, their approach can represent these ambiguities via the extent of the divergence. Imprecise Bayesianism has mostly been advocated for its epistemological features. In this article, we examine its properties for decision-making. We develop a model for comparing standard and Imprecise Bayesianism by testing their performances in a classic decision problem. We find that the representational tools of Imprecise Bayesianism also cause it to underperform in our tests. This issue has been overlooked, because previous research on Imprecise Bayesianism has not utilised agent-based modelling to provide information about its performance in the short-run. Overall, we reveal the Ambiguity Dilemma for Imprecise Bayesianism: To what extent should one value representational power or decision-making performance?

Do imprecise beliefs lead to worse decisions under uncertainty? This BJPS Short Reads article provides an informal introduction to our use of agent-based modelling to investigate this question. We explain the strengths of imprecise probabilities for modelling evidential states. We explain how we used an agent-based model to investigate the relative performance of Imprecise Bayesian reasoners against a standard Bayesian who has precise credences. We found that the very features of Imprecise Bayesianism which give it representational strengths also cause relative underperformance in this decision problem. We explain the practical relevance of this research to applications of Bayesian reasoning in areas like artificial intelligence.

Many philosophers argue that Keynes’s concept of the “weight of arguments” is an important aspect of argument appraisal. The weight of an argument is the quantity of relevant evidence cited in the premises. However, this dimension of argumentation does not have a received method for formalisation. Kyburg has suggested a measure of weight that uses the degree of imprecision in his system of “Evidential Probability” to quantify weight. I develop and defend this approach to measuring weight. I illustrate the usefulness of this measure by employing it to develop an answer to Popper’s Paradox of Ideal Evidence.

According to John D. Norton's Material Theory of Induction, all reasonable inductive inferences are justified in virtue of background knowledge about local uniformities in nature. These local uniformities indicate that our samples are likely to be representative of our target population in our inductions. However, a variety of critics have noted that there are many circumstances in which induction seems to be reasonable, yet such background knowledge is apparently absent. I call such absences ‘the frontiers of science', where background scientific theories do not provide information about such local uniformities. I argue that the Material Theory of Induction can be reconciled with our intuitions in favour of these inductions. I adapt an attempted justification of induction in general, the Combinatoric Justification of Induction, into a more modest rationalisation at the less foundational level that the critics discuss. Subject to a number of conditions, we can extrapolate from large samples using our knowledge of facts about the minimum proportions of representative subsets of finite sets. I also discuss some of Norton's own criticisms of his theory and argue that he is overly pessimistic. I conclude that Norton's theory at least performs well at the frontiers of science.