Two Principles of Evidence and Their Implications for the Philosophy of Scientific Method

The notion of evidence is of great importance, but there are substantial disagreements about how it should be understood. One major locus of disagreement is the Likelihood Principle, which says roughly that an observation supports a hypothesis to the extent that the hypothesis predicts it. The Likelihood Principle is supported by axiomatic arguments, but the frequentist methods that are most commonly used in science violate it.\ud \ud This dissertation advances debates about the Likelihood Principle, its near-corollary the Law of Likelihood, and related questions about statistical practice. Chapter 1 provides a new axiomatic proof of the Likelihood Principle that avoids influential responses to previous proofs. Chapter 2 exhibits the close connection between the Likelihood Principle and the Law of Likelihood and responds to three purported counterexamples to them. Chapter 3 presents a new counterexample that is more difficult to avoid but argues that it does not speak against those principles in typical applications.\ud \ud The next two chapters turn to implications. It is motivated by tension among three desiderata for a method of evaluating hypotheses in light of data. We would like such a method to (1) respect the evidential meaning of data, (2) provide direct guidance for belief or action, and (3) avoid inputs that are not grounded in evidence. Unfortunately, frequentist methods violate (1) by violating the Likelihood Principle, likelihoodist methods violate (2) by directly addressing only questions about the evidential meaning of data, and Bayesian methods violate (3) by using prior probabilities. Chapter 4 sharpens this tension by arguing that no method that satisfies likelihoodist strictures can provide a genuine rival to frequentist and Bayesian methodologies. Chapter 5 argues that many frequentist violations of the Likelihood Principle are not required by basic frequentist commitments. Those that are may be permissible for the sake of enabling progress in the presence of highly indefinite prior beliefs, but they are not mandatory or even strongly motivated. These considerations support more widespread use of Bayesian methods despite difficulties in specifying prior probability distributions