Uncertainty and Decisions

In the seminal work of de Finetti (see the English translation of de Finetti 2017), the central idea for the Bayesian paradigm is to address decision-making in the face of uncertainty from a subjective viewpoint. Given the same set of uncertain circumstances, two decision-makers could differ in the following ways: How desirable different potential outcomes might seem to them. How likely they consider the various outcomes to be. How they feel their actions might affect the eventual outcome. The Bayesian decision-making paradigm is most easily viewed through the lens of an individual making choices (“decisions”) in the face of (personal) uncertainty. For this reason, certain illustrative elements of this section will be purposefully written in the first person. This decision-theoretic view of the Bayesian paradigm represents a mathematical ideal of how a coherentCoherence non-self-contradictory individual should aspire to behave. This is a non-trivial requirement, made easier with various mathematical formalisms which will be introduced in the modelling sections of this text. Whilst these formalisms might not exactly match my beliefs for specific decision problems, the aim is to present sufficiently many classes of models that one of them might adequately reflect my opinions up to some acceptable level of approximation. Coherence is also the most that will be expected from a decision-maker; there will be no requirement for me to choose in any sense the right decisions from any perspective other than my own at that time. Everything within the paradigm is subjective, even apparently absolute concepts such as truth. Statements of certainty such as “The true value of the parameter is x” should be considered shorthand for “It is my understanding that the true value of the parameter is x”. This might seem pedantic, but crucially allows contradictions between individuals, and between perspectives and reality: the decision-making machinery will still function.