Methodological likelihoodism is the claim that it is possible to provide an adequate self-contained methodology for science on the basis of likelihood functions alone. My argument against methodological likelihoodism is as follows.

- (1) An adequate self-contained methodology for science provides good norms of commitment vis-á-vis hypotheses.
- (2) Some rule of the following form is a good purely likelihood-based norm of commitment vis-á-vis hypotheses if anything is, where $T$ is one’s total relevant evidence and $f$ is a nondecreasing function such that $f(1)=1$ and $f(x)>1$ for some $x$:

ProportionRelativeAcceptance to (aFunction of) theEvidence (PRAFE): Accept $H_1$ over $H_2$ to the degree $f(\mathcal{L}) = f(\Pr(T|H1)/\Pr(T|H2))$.

- (3) A good norm of commitment vis-á-vis hypotheses is compatible with the following rules:
- (3A) If $H_1$ is logically equivalent to $H_2$, and $H_3$ is logically equivalent to $H_4$, do not prefer $H_1$ to $H_3$ and $H_4$ to $H_2$ (where preferring $H_1$ to $H_2$ means accepting $H_1$ over $H_2$ to a degree greater than one).
- (3B) If $H_1$ and $H_2$ are mutually exclusive, accept $H_1\cup H_2$ over $H_3$ to a degree greater than that to which you accept $H_1$ over $H_3$, for any $H_3$ such that you accept $H_1$ over $H_3$ to some definite degree and accept $H_2$ over $H_3$ to a degree greater than zero.

- (4) A good norm of commitment vis-á-vis hypotheses is compatible with the following rule:
- (4A) If $H_1$ and $H_2$ are logically equivalent given your total evidence, do not prefer $H_1$ to $\sim H_1$ and $\sim H_2$ to $H_2$.

(3) and (4) each entail that no rule of the form given in (2) is a good norm of commitment vis-á-vis hypotheses. Thus, the conjunction of (1), (2), and either (3) or (4) entails that methodological likelihoodism is false.

See this paper for the relevant proofs, a defense of each premise, and a response to attempts to defend methodological likelihoodism on reliabilist grounds.

**Question:** What do you think is the weakest link in this argument? Why?

To share your thoughts about this post, comment below or send me an email.

Comments support Markdown formatting and $\LaTeX$ mathematical expressions (surround expressions with single dollar signs for in-line math or double dollar signs for display math).

Michael Lew says

(Welcome back.)

To my mind the weakest link in your argument is the fact that it treats hypotheses as independent objects rather than the set of parameter values that a likelihood function can sensibly deal with.

As you know from my previous comments, I think that there are good reasons to suppose that likelihoods are useful for evaluating the support for hypotheses that can be expressed as points on a single likelihood function. If that is true then the hypotheses in point 3 of your argument that are logically equivalent are in fact identical, and those that are not mutually exclusive are not expressible as points on the same likelihood function and so the law of likelihood is irrelevant to consideration of their support by the evidence.

From my viewpoint your argument is not successful as an argument against the use of methodological likelihood in circumstances where it is appropriate, but may be useful for delineation of what those circumstances actually are.

Greg Gandenberger says

Thanks, Michael. I’m glad you think my argument is potentially useful for something!

Note that in (3A) the Law of Likelihood is applied directly only to the comparison between H1 and H3 and to the comparison between H2 and H4. It is never applied to logically equivalent or even non-mutually exclusive pairs of hypotheses. The issue is with the consistency (in some sense) of judgments produced by (PRAFE) when applied to distinct but logically related questions.

lotharson says

Hello. this is an interesting post!

First of all, I should say where I am coming from.

I am a frequentist believing that any event has a physical probability, even putative future ones such as a technological singularity.

I only resort to likelihoods for the

evaluation of scientific theorieswhich are propositions but no events.So I am not a typical likelihoodist, rather a frequentist embracing parts of this approach for his scientific work.

With that mind, let us see your premises.

” (1) An adequate self-contained methodology for science provides good norms of commitment vis-á-vis hypotheses.”

While it might be very desirable, what warrant do we have for supposing that such a “self-contained” methodology exists at all?

I am a pluralist.

Objective Bayesianism

alonecannot be such an epistemology, since it cannot make a difference between ignorance and knowledge: in many situations utter ignorance and warranted knowledge produces the very same probability distribution.Therefore objective Bayesianism is

insufficientfor making sense of reality.“2) Some rule of the following form is a good purely likelihood-based norm of commitment vis-á-vis hypotheses if anything is, where T is one’s total relevant evidence and f is a nondecreasing function such that f(1)=1 and f(x)>1 for some x:

Proportion Relative Acceptance to (a Function of) the Evidence (PRAFE): Accept H1 over H2 to the degree f(L)=f(Pr(T|H1)/Pr(T|H2)).”

This already assumes a fundamental Bayesian assumption, namely that there is such a thing as a

degreeof acceptance of a theory.But what is it supposed to be? Is that an ideal psychological state that no real human (brain) could ever reach?

If I find that p(Data| A) = 0.4 , p(Data| B) = 0.3 , p(Data| C) = 1E-12) I would

pragmaticallydismiss theory C and just consider A and B for the pursuit of my investigations.But I don’t believe there is an ideal psychological state that every rational agent ought to have in such a situation.

” (3) A good norm of commitment vis-á-vis hypotheses is compatible with the following rules:

(3A) If H1 is logically equivalent to H2, and H3 is logically equivalent to H4, do not prefer H1 to H3 and H4 to H2 (where preferring H1 to H2 means accepting H1 over H2 to a degree greater than one).

(3B) If H1 and H2 are mutually exclusive, accept H1∪H2 over H3 to a degree greater than that to which you accept H1 over H3, for any H3 such that you accept H1 over H3 to some definite degree and accept H2 over H3 to a degree greater than zero.

(4) A good norm of commitment vis-á-vis hypotheses is compatible with the following rule:

(4A) If H1 and H2 are logically equivalent given your total evidence, do not prefer H1 to ∼H1 and ∼H2 to H2.”

Since I reject the existence of degrees of belief (being something more than a subjective state of mind), I am rather unmoved by this.

What we have are

degrees of agreement with the real worldwhich can be either represented as the discrepancies between measurements and predictions of the theorydor as the probability of the measurements givent the theory, which areobjectivequantities.I don’t believe there is such a thing as p(String theory) or p(Universal gravitation) and the psychological state of minds of real humans is going to keep fluctuating owing to a great number of non-rational factors and should not be the ground of our decisions.

So, your arguments are rather unlikely (pun intended!) to convince a frequentist only interested in p(Data | theory) and who rejects the existence of degrees of acceptance.

I should add that likelihoodism and Bayesianism are by no means mutually exclusive if you are a pluralist.

If unlike me someone believes in the existence of degrees of acceptance and probabilities of scientific theories, I would advise her to

use likelihood ratiosfor computing pseudo-prior probabilities after a sufficiently large number of measurements instead of using the principle of indifference.This would allow her to start her Bayesian calculations with values

sticking to the real worldinstead of a probability distribution having no physical meaning.I am sure this would also be much more fruitful in terms of practical results.

Friendly greetings from Europe.

Greg Gandenberger says

Thanks for the comment, Marc. You have brought up some good points for discussion.

Let me explain what I mean by “self-contained.” Methodological likelihoodists need not believe that methods based on likelihood functions alone are appropriate for all scientific problems. However, they must hold more than that (1) such methods are appropriate when they would give the same answer as a reasonable Bayesian or frequentist method and (2) the outputs of purely likelihood-based methods are useful as inputs for some other method, such as Bayesian updating. Those claims could indicate nothing more than a roundabout form of Bayesianism or frequentism. A common pluralist view that qualifies as a form of methodological likelihoodism is that Bayesian methods are appropriate when prior probabilities are “available,” while likelihoodist methods are appropriate when they are not.

I take it that the purpose of a methodology is to provide some kind of explicit rules for regulating one’s commitments vis-à-vis hypotheses in light of one’s data. By providing a rule that gives degrees of acceptance, I’m not thereby assuming that for an arbitrary pair of hypotheses there is such a thing as the degree to which I accept one of those hypotheses over the other.

Frequentist typically want to be able to talk about rejection and acceptance (or non-rejection) even though they decline to assign posterior probabilities to hypotheses. They too attempt to provide good norms of commitment vis-à-vis hypotheses.

lotharson says

“I take it that the purpose of a methodology is to provide some kind of explicit rules for regulating one’s

commitmentsvis-à-vis hypotheses in light of one’s data. ”But what is such a degree of commitment?

If it is more than a brain state, could it be defined as the amount of efforts a scientist invests in the various theories?

P.S: you don’t need to answer me everywhere, one place suffices 🙂