[R-sig-eco] Are likelihood approaches frequentist?

[Rubén Roa-Ureta]

Paulo Inácio de Knegt López de Prado wrote:
> Dear r-sig-ecology users
>
> Here follow the messages I exchanged with Ben Bolker last week about the
> likelihood and frequentist approaches. We both would like to open this topic
> for discussion in the list.
>
> Best wishes
>
> Paulo
>
> ----------------------------------------------------------------------------
> Dear Dr. Bolker, 
>
> I am puzzled why some authors treat likelihood approaches as 
> frequentist, as it seems you did in page 13 of your book 'Ecological Models
> and Data'. 
> This sounds odd to me because  what brought my attention to likelihood was
> Richard Royall's book 'Statistical Evidence'. His framing of a paradigm
> based on the likelihood principle, and the clear distinction he makes
> between this paradigm and frequentist and Bayesian approaches looks
> quite convincing to me.
>   
Paulo,
The likelihood function is the central concept of statistical inference, 
so working with the likelihood you can have Bayesian, frequentist 
(better called sampling-distribution inference), or likelihoodist 
inference, depending on what do you do with your likelihoods. In 
Bayesian inference the likelihood function updates prior opinion by 
bringing the data into the inference, in sampling-distribution inference 
(a.k.a. frequentist) it allows the building of better confidence 
intervals by finding in the sample space likelihood values that could 
have occurred if data similar to the data you have had been obtained, 
and in the direct-likelihood approach the likelihood is directly used to 
compare two hypotheses or equivalently to build direct-likelihood 
intervals. For example, the likelihood ratio test (not to be confused 
with the pure likelihood ratio, or differences in support) based on a 
limiting Chi-square distribution is a likelihood-based frequentist 
method. Frequentist statisticians evaluate the likelihood from the 
sample, and then proceed to evaluate the likelihood for other potential 
samples, thus building their confidence intervals and p-values. On the 
other hand Bayesian and likelihoodist statisticians only use the 
likelihood evaluated at the actual sample that was obtained. From that 
point of view one can say that Bayesian and likelihoodist are closer to 
each other than to frequentists, however both Bayesian and frequentists 
base their inference on probabilities (posterior probabilities or error 
rates) whereas likelihoodists base their inference on, well, likelihood 
only.
Royall's points are very convincing indeed, at least they were for me 
too. Royall's concept of evidence in the sample about competing 
hypotheses and on approximate likelihoods for problems with nuisance 
parameters, plus Edwards' mathematical proofs of the properties of the 
support function, plus Jim Lindsey's arguments about Akaike's index in 
model selection, provide a complete theory of statistical inference, 
based exclusively on the likelihood, IMHO.
> I agree with him that we use likelihood criteria to identify, among
> competing hypotheses, which one attribute the highest probability to  a
> given dataset. If I understood correctly, this is what Royal calls the
> 'evidence value' of a data set to a hypothesis 'vis a vis' other
> hypotheses. I also like his idea that the role of statistics in science
> is just to gauge this evidence value, no less, no more.
>
> This approach differs from the frequentist because the sampling
> space is irrelevant, that is, other datasets that might be observed do not
> affect the evidence value of the observed data set. My favourite example is
> the comparison of binomial and negative binomial experiments on coin
> tossing, in the sections 1.11 and 1.12 of his book.
>
> I am not an "orthodox likelihoodist"; on the contrary, I agree with the
> pragmatic view you express in your book. I'd just like to understand
> the key differences among the available statistical tools, in order to make
> a good pragmatic use of them. I'd really appreciate if you can help me
> with this.
>
> Best wishes
>
> Paulo
>   
"There is nothing more practical than a good theory". I'm not sure who 
was the original author of that quote (in a book I read long ago it was 
said that the author was Einstein) but it applies here. Likelihoodist, 
frequentist, and Bayesian inferences are not compatible. Especially 
likelihoodist and Bayesian versus frequentist, so the pragmatst who 
change allegiance is making an error at some point.

>>   Very well put.  Royall, and Edwards (author of _Likelihood_, Johns
>> Hopkins 1992) are what I would call "pure", or "hard-core",
>> or "orthodox", likelihoodists. They are satisfied with a statement
>> of relative likelihood, and don't feel the need to attach a p-value
>> to the result in order to have a decision rule for hypothesis rejection.
>>
>>   Far more commonly, however, people impose (? add ?) an additional
>> layer of frequentist procedure on top of this basic structure, namely
>> using the likelihood ratio test to assess the statistical significance
>> of a given observed likelihood ratio and/or to set a cutoff value
>> for profile confidence intervals.  Using the LRT puts the inference
>> back squarely into the frequentist domain, although the sample space
>> we are now dealing with (sample space of likelihoods derived from
>> coin-tossing experiments) is quite different from the one
>> we started with (sample space of outcomes of coin-tossing experiments).
>> As far as I can see, Edwards and Royall are almost alone in their
>> adherence to "pure" likelihood -- most of the rest of us pander
>> to the desire for p-values (or, less cynically, to the desire
>> for a probabilistically sound decision rule).
>>     
Two other great statisticians that subscribe to the likelihoodist school 
of inference are Jim Lindsey
and John Nelder.
At least once a year I hear someone at a meeting say that there are two 
modes of inference:
frequentist and Bayesian. That this sort of nonsense should be so 
regularly propagated shows how
much we have to do. To begin with there is a flourishing school of 
likelihood inference, to which I
belong.

>>  I would also add that different scientists have different
>> goals (belief, prediction, decision, assessing evidence). I too
>> think Royall makes a good case for the primacy of
>> assessing strength-of-evidence, and he gives the clearest
>> explanation I have seen, but I wouldn't completely
>> rule out the other frameworks.
>>     
I tend to think there is a place for Bayesian inference in prediction.

Rubén