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. It seems that one can accomplish the aims that the model adequacy assumption is meant to serve by making the Likelihood Principle a claim about the evidential meaning of an experimental outcome *with respect to the model*, rather than a claim about the evidential meaning of that outcome *simpliciter*. The advantage of relativizing the notion of evidential meaning in the Likelihood Principle to the model is that the Likelihood Principle formulated in this way can be regarded as true even when the model is false.

There are two questions I need to address in order to make this proposal compelling:

- Does it truly accomplish all of the aims that the model adequacy assumption is meant to serve?
- What does it even mean to say that the evidential meaning of an experimental outcome
*with respect to a model that is known to be false*depends only on its likelihood function?

To address Question 1, I’ll need to spend more time going back to the relevant sources than I have done so far. I doubt whether there is an adequate response to Question 2. The notion of “evidential meaning” that the Likelihood Principle invokes is obscure even when it is not relativized to an experimental model. The Likelihood Principle is an attempt to give an intuitively appealing partial explication of this undefined primitive notion. One might hope that our intuitions about the notion of evidential meaning will be a good guide for methodology, but this assumption should not be taken for granted. This latest refinement in the formulation of the Likelihood Principle will not address the fact that the Likelihood Principle itself might be a dead end as far as methodology is concerned.

This rather serious concern aside, are there any reasons for adopting the model adequacy assumption that relativizing the notion of evidential meaning in the Likelihood Principle to the experimental model does not address?

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Michael Lew says

Seems like a storm in a teacup to me. Given that likelihoods are only related to evidence in the form of a likelihood ratio and those ratios are only meaningful when they are within likelihood function ratios, it is not possible to use likelihoods as evidence in any way other than with respect to the model.

Greg Gandenberger says

Yes, I agree. The model adequacy assumption is only necessary for strong likelihoodist claims about what needs to be reported. Birnbaum (1962, 272) gets it right: “One basic consequence [of accepting the Likelihood Principle] is that reports of experimental results in scientific journals should in principle be descriptions of likelihood functions, when adequate mathematical-statistical models can be assumed, rather than reports of significance levels or interval estimates.” When “adequate mathematical-statistical models” cannot be assumed, a description of a likelihood function by itself is only sufficient relative to a model.