My conversations at the Munich Center for Mathematical Philosophy keep coming back to stopping rules, so I’ve decided to write a paper on the topic. Here is the general line that I plan to develop.

[Read more…]

# An Introduction to Likelihoodist, Bayesian, and Frequentist Methods (2/3)

The major virtues and vices of Bayesian, frequentist, and likelihoodist approaches to statistical inference.# Introduction

##### My goal in this post and the previous one is to provide a short, self-contained introduction to likelihoodist, Bayesian, and frequentist methods that is readily available online and accessible to someone with no special training who wants to know what all the fuss is about.

In the previous post, I gave a motivating example that illustrates the enormous costs of the failure of philosophers, statisticians, and scientists to reach consensus on a reasonable, workable approach to statistical inference. I then used a fictitious variant on that example to illustrate how likelihoodist, Bayesian, and frequentist methods work in a simple case. **In this post, I discuss a stranger case that better illustrates how likelihoodist, Bayesian, and frequentist methods come apart.** [Read more…]

# An Introduction to Likelihoodist, Bayesian, and Frequentist Methods (1/3)

# Introduction

I have been recommending the first chapter of Elliott Sober’s Evidence and Evolution to those who ask for a good introduction to debates about statistical inference. That chapter is excellent, but it would nice to be able to recommend something shorter that is readily available online. Here is my attempt to provide a suitable source. [Read more…]

# Wasserman’s Counterexample to the Likelihood Principle

# The Counterexample

Larry Wasserman presents the following purported counterexample to the Likelihood Principle in his lecture notes for a course on theoretical statistics. [Read more…]

# Do Frequentist Methods Violate the Likelihood Principle?

### The Sense in Which Frequentist Methods Violate the Likelihood Principle

**It is widely accepted that frequentist methods violate the Likelihood Principle.** After all, there can be two pieces of data A and B such that the Likelihood Principle implies that A and B are evidentially equivalent with respect to the set of hypotheses **H**, yet frequentist methods will yield different conclusions about **H** depending on whether A or B is fed into them.

### But What About Using Frequentist Considerations to Choose Among Priors?

There is another sense in which many frequentist methods do *not* violate the Likelihood Principle. A frequentist method is often equivalent (in a sense) to a Bayesian method with a particular prior probability distribution. From a Bayesian perspective, such frequentist methods involve updating a prior probability distribution in a way that does conform to the Likelihood Principle. **They violate the Likelihood Principle only by using “implied priors” that vary with the sampling distribution of the experiment to be performed**. [Read more…]

# Why It May Be Permissible to Violate the Likelihood Principle–Even if It’s True

### Where We’ve Been

I have *argued for* the **Likelihood Principle**, which says that the evidential meaning of a datum with respect to a partition depends on the probabilities that the elements of that partition ascribe to that hypothesis, up to a constant of proportionality. (Here)

From the Likelihood Principle, I have *argued for* the **Law of Likelihood**, which says that the degree to which a datum favors one element of a partition over another is given by the ratio of the probabilities that those hypotheses ascribe to that datum. (Here and here)

I have *argued against* **methodological likelihoodism**, which says that characterizing data as evidence in accordance with the Law of Likelihood is an adequate self-contained methodology for science (at least as a fallback option in cases in which prior probabilities are “not available”). (Here)

### Where We’re Going: The Methodological Likelihood Principle

The next claim I want to consider is the **Methodological Likelihood Principle**, which says that an adequate methodology for science respects evidential equivalence as characterized by the Likelihood Principle. [Read more…]

# Quote

It is not helpful to claim that likelihood provides a measure of evidential support, relevance, or confirmation. All of these terms have been used in so many ways and with such obscurity that unless some indication is given of how such assessments of support are to be employed in inquiry and deliberation, this characterization of the import of likelihood is hopeless.

–Isaac Levi, *The Enterprise of Knowledge *(1980), p. 344

# New Responses to Three Counterexamples to the Likelihood Principle

New Responses to Three Counterexamples to the Likelihood Principle: a revised draft of what is projected to be Chapter 2 of my dissertation.

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# Must a Proof of the Likelihood Principle Assume Model Adequacy?

Standard presentations of proofs of the Likelihood Principle include a warning to the effect that the proof assumes that the statistical model of the experiment in question is adequate in some sense. Many commenters have pointed out that statisticians typically if not always know that their models are literally false. Thus, the fact that proofs of the Likelihood Principle assume that the model is adequate casts doubt on the significance of those proofs.

Advocates of the Likelihood Principle have responded to this concern in several ways. The most popular response seems to be the *tu quoque* response that *every* school of statistics uses models. Thus, the fact that advocates of the Likelihood Principle need to make the literally false assumption that their model is adequate puts them in no worse position than anyone else in statistics.

A significant problem for this response is the problem that all *tu quoque* responses share: they do not make the problem the objection raises go away. The fact that no one else is in any better position than the advocate of the Likelihood Principle does not mean that he or she is in a good position.

I am entertaining the alternative response that proofs of the Likelihood Principle do not need to assume model adequacy after all. [Read more…]

# New Responses to Three Counterexamples to the Likelihood Principle

A draft of the second chapter of my dissertation is available here. This chapter provides new responses to purported counterexamples to the Likelihood Principle due to Fitelson, Armitage, and Stein that I take to be stronger in some respect than previous responses.

**Update: **The most recent version of this chapter is now here.

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