### Papers

- “Differences Among Noninformative Stopping Rules Are Often Relevant to Bayesian Decisions.” a
*rXiv*:1707.00214 - “Why I Am Not a Likelihoodist.”
*Philosophers’ Imprint*16, 7 (2016): 1-22 - “A New Proof of the Likelihood Principle.”
*The British Journal for the Philosophy of Science*66, 3 (2015): 475-503.- Published version (paywalled)
- Preprint

- “Producing a Robust Body of Data with a Single Technique.”
*Philosophy of Science*77, 3 (2010): 381-399.- Published version (paywalled)
- Preprint

- “New Responses to Three Purported Counterexamples to the Likelihood Principle”
- “What Likelihoodists Should (And Should Not) Say About Borel’s Paradox”
- “Should You Proportion Your Beliefs to Your Evidence?”

### Recorded Talk

How to think about Bayesian, Frequentist, and Likelihoodist Methods

### Thesis Project: Two Principles of Evidence and Their Implications for the Philosophy of Scientific Method

#### Background

**Norms of scientific research influence life-and-death decisions.** For instance, frequentist norms led the UK Collaborative ECMO Trial Group to perform a randomized trial comparing a new treatment called ECMO for severe respiratory problems in infants to then-conventional therapy, even though previous trials provided convincing evidence for ECMO’s superiority according to Bayesian and likelihoodist standards. The result was predictable: the trial had to be terminated after fifty-four infants died under conventional therapy.

**My thesis project is about two principles of evidence and their implications for the philosophy of scientific method. Bayesian and likelihoodist methods conform to those principles, while frequentist methods do not.** All three types of methods use likelihood functions, where the likelihood function for a set of hypotheses **H** on a datum $E$ is the probability of $E$ given $H$ (written $\Pr(E|H)$) as a function of $H$ as it varies over the set **H**. However, they use them in different ways and for different immediate purposes.

**Frequentist** methods are most commonly used in science. They use likelihood functions to design experiments that are guaranteed to perform relatively well in a certain sense in the indefinite long run, no matter what the truth may be. **Bayesian** methods require supplying a “prior probability distribution” over the relevant set of hypotheses without reference to the data. That distribution might represent one’s subjective degrees of belief, or it might be chosen according to one of several formal rules that are intended to be “noninformative.” The Bayesian then uses the likelihood function to update that distribution in accordance with Bayes’s theorem. **Likelihoodists** use likelihood functions to characterize the data as evidence. (See this article for a more thorough introduction to frequentist, Bayesian, and likelihoodist methods.)

The two principles I investigate are the Likelihood Principle and the Law of Likelihood. They accord with one of the leading ideas in the philosophy of induction, that “saving the phenomena is a mark of truth” (Norton 2005). In other words, an observation confirms a hypothesis to the extent that the hypothesis predicts it. The *Likelihood Principle* says, roughly, that the evidential meaning of $E$ with respect to **H** depends only on the likelihood function of $E$ on **H**. Frequentist methods violate this principle by drawing different conclusions from data that have the same likelihood function. The *Law of Likelihood* goes further in that it says something about *how* the evidential meaning of the data depends on the likelihood function: it says that $E$ favors $H_1$ over $H_2$ if and only if the likelihood ratio $\mathcal{L}=\Pr(E|H_1)/ \Pr(E|H_2)$ is greater than one, with $E$ measuring the degree of favoring. The Law of Likelihood is the likelihoodist’s primary interpretive tool.

**All three types of methods have advantages and disadvantages.** As shown in the figure at the top of this page, frequentist methods violate the Likelihood Principle, which leads its advocates to claim that it fails to respect the evidential meaning of the data. Bayesian methods require supplying prior probability distributions that must ultimately be without evidential support. Likelihoodist methods directly address only questions about evidence, and not about what we should believe or do in light of the evidence.

#### My Contribution

##### Part 1: A Defense of the Likelihood Principle and the Law of Likelihood

**The first question I address is whether or not we should accept the Likelihood Principle and the Law of Likelihood.** The strongest prior argument for the Likelihood Principle was Birnbaum’s 1962 proof that it follows from two principles that seem to be compelling from both frequentist and pre-theoretic perspectives. That proof has been influential, but it has also been controversial. Arguments due to Durbin (1970) and Kalbfleisch (1975) for restricting its premises in ways that would block the proof continue are cited in widely used textbooks (e.g. Casella and Berger 2002), and Mayo has recently argued that the “so-called proof” is not even valid.

I aim to lay this controversy to rest in “A New Proof of the Likelihood Principle” (preprint, published version). **I give a new proof of the Likelihood Principle** that improves on Birnbaum’s proof in that restrictions on its premises analogous to those Durbin and Kalbfleisch propose placing on Birnbaum’s premises are not enough to block the new proof. **I also argue that Mayo’s attempted refutation of Birnbaum’s proof rests on a misreading of his premises.**

Advocates of frequentist methods have attempted not only to undermine arguments for the Likelihood Principle and the Law of Likelihood, but also to defeat them by presenting examples in which they give seemingly counterintuitive results. Many of those counterexamples have been adequately addressed elsewhere. **I identify three purported counterexamples that have not been adequately addressed and respond to them** in “New Responses to Three Counterexamples to the Likelihood Principle.” I also argue in that paper that the Law of Likelihood is compelling if the Likelihood Principle is true. Two of those counterexamples were originally presented in the 1960s but have only been addressed as threats to Bayesian positions and not as threats to the Likelihood Principle and the Law of Likelihood that can be used to support those positions. The third is a recent example from Fitelson (2007). In “How Likelihoodists Should (and Should Not) Respond to Borel’s Paradox,” **I address a fourth apparent counterexample that I developed in response to a suggestion from Teddy Seidenfeld.**

##### Part 2: An Argument for the Pro-Bayesian Implications of the Likelihood Principle

**In Part 2 of my project, I turn from assessing the Likelihood Principle and the Law of Likelihood to evaluating their implications.** In “Why I Am Not a Likelihoodist” (open-access published version), **I argue against the methodological likelihoodist view that it is possible to provide an adequate, closed methodology for science based on likelihood functions alone.** Given the Likelihood Principle, it follows that it is impossible to provide an adequate, closed methodology for science based on the evidential meaning of the data alone. Something else is needed—either additional ingredients such as the prior probability distributions to which Bayesians appeal, or an approach such as the frequentist one that does not respect the evidential meaning of the data but claims to be justified on other grounds.

In “Why Frequentist Violations of the Likelihood Principle Are Permissible at Best,” **I argue against the idea that frequentist methods should be used despite the fact that the violate the Likelihood Principle because of the guarantees about long-run performance that they provide.** I argue that frequentist violations of likelihoodist principles are at best permissible when the opinions of the relevant doxastic agent are sufficiently indefinite. They are not clearly preferable to imprecise Bayesian methods in those circumstances. They are not in fact more objective than those methods in any important sense. They do control worst-case performance, but an imprecise Bayesian can do the same thing in a more discriminating way.

#### Implications

The Bayesian approach for which I argue faces strong opposition because it requires prior probability distributions that are not themselves based on evidence. If my arguments are correct, however, then **this need for prior probability distributions is a burden to be shouldered rather than an obstacle to be avoided.** There is an urgent need to find ways to make the use of Bayesian methods more palatable to practicing scientists. As the ECMO case illustrates, these issues are not idle, purely intellectual concerns. In some cases, they are a matter of life or death.