In my previous post I presented some reasons to resist a clever counterexample to the Law of Likelihood developed by Mike Titelbaum. In that post I chose to stay at the level of intuitions about the example and about what kinds of features we might want a measure of evidential favoring to have. In this post I go deeper by examining Mike’s example in light of the purpose of the Law of Likelihood. [Read more…]
Last week’s post in which I presented a purported counterexample to the Law of Likelihood (due to Mike Titelbaum) generated a lot of interest. In this post I comment on the counterexample. Next week I plan to “go meta” by commenting on the purpose of the Law of Likelihood and of counterexamples such as Mike’s.
A talk I recently gave prompted Mike Titelbaum to develop a purported counterexample to the Law of Likelihood. In this post I present the counterexample with minimal comments for the sake of eliciting reactions to it that are not influenced by my thoughts. I plan to follow up with my comments next week. [Read more…]
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…]
How My Work Provides an Argument for Subjective Bayesianism
Subjective Bayesianism seems to be the only serious alternative to methodological likelihoodism that conforms to the Likelihood Principle. (Objective Bayesian methods violate the Likelihood Principle in their rules for selecting priors.) My arguments for the Likelihood Principle and against methodological likelihoodism can thus be used as an argument for subjective Bayesianism.
Common Criticisms of Subjective Bayesianism
Subjective Bayesianism is often criticized for being too subjective and too permissive. It’s about what you believe rather than, say, what your evidence supports. On standard formulations, it only requires that one’s beliefs be synchronically and diachronically coherent–that is, to conform to the axioms of probability at a time and to update by Bayesian conditioning over time. It doesn’t tell you what you ought to believe about a given hypothesis on the basis of your evidence except as a function of what you believed about it before and what you believe about other related hypotheses.
I don’t find these objections persuasive. Subjective opinions are important. What am I to use in making decisions if not my own values and opinions (appropriately influenced by my evidence and my interactions with others, of course)? I’d like to be able to give an account of “what one’s evidence supports” in an objective sense that provides good norms of belief or action, but I’m not aware of any such account (likelihoodism does not provide it), and skeptical arguments such as Hume’s problem of induction lead me to doubt that such an account is possible.
My Main Concern About Subjective Bayesianism
The main in-principle problem with subjective Bayesianism, it seems to me, is not that it uses subjective beliefs when the agent in question has them, but that it doesn’t give much guidance to the agent who doesn’t have them. (It faces many additional problems in practice, but I will leave those for another day.) [Read more…]
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…]
My argument against methodological likelihoodism has three steps:
- Argue that an adequate self-contained methodology for science provides good norms of commitment vis-à-vis hypotheses.
- Argue that if there are any good purely likelihood-based norm of commitment, then they have a particular form.
- Show that any norm of that form has undesirable consequences.
I focus in this post on Step 2. [Read more…]