Prior and Likelihood Representation

The first chapter introduced the philosophy of Bayesian statistics: when making individual decisions in the face of uncertainty, probability should be treated as a subjective measure of beliefs, where all quantities unknown to the individual should be treated as random quantities. Eliciting individual probability assessments is a non-trivial endeavour. Even if I have a relatively well-formed opinion about some uncertain quantity, coherently assigning precise numerical values (probabilities) to all potential outcomes of interest for that quantity can be particularly challenging when there are infinitely many possible outcomes. To counter these difficulties, it can be helpful to consider mathematical models to represent an individual’s beliefs. There is no presumption that these models should be somehow correct in terms of representing true underlying dynamics; nonetheless, they can provide structure for representing beliefs coherently to a good enough degree of approximation to enable valid decision making. The main simplification which will be considered, exchangeabilityExchangeable, occurs in contexts where a sequence of random quantities are to be observed and a joint probability distribution for the sequence is required. Symmetries in one’s beliefs about sequences lead to familiar specifications of probability models which are often considered to be the hallmark of Bayesian thinking: a likelihood distribution combined with a priorPriordistribution distribution.