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

[prado]

Thanks, Rubén

My point with this topic was to clarify that the likelihood-based 
approach is a distinct paradigm in statistical inference, and there is 
people in biology applying it successfully.

I agree with you that this point should be better stressed, specially 
for biologists. Taper & Lelle  "The Nature of Scientific Evidence" 
(Chigago Univ Press, 2007) is a great help in this respect.

Could you indicate the best works by Nelder Lindsey that could 
contribute to this point?

Abraços

Paulo

Rubén Roa-Ureta escreveu:
> 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
> 

-- 
Paulo Inácio de Knegt López de Prado
Depto. de Ecologia - Instituto de Biociências - USP
Rua do Matão, travessa 14, nº 321
Cid. Universitária, São Paulo - SP - Brasil
CEP 05508-900
+55-11-30917599 (sala)
+55-11-30917600 (Secretaria)