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Can maximum log likelihood take a value of 0 ? I am using nlminb() in R to minimize a function that returns -sum(log(x)), and I am getting the $objective from nlminb() of my full model as 0 and null model as 0.47.

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    $\begingroup$ Why shouldn't it take such value..? $\endgroup$ – Tim Dec 2 '16 at 23:03
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    $\begingroup$ @Tim Since a likelihood is often defined as a probability and all probabilities are $1$ or less, the logarithm must not be positive. Thus, a positive "log likelihood" can only be reported when shortcuts are taken to avoid computing a normalizing constant for the probability distribution or else probability densities are involved. Therefore it seems like there's good motivation for this question. $\endgroup$ – whuber Dec 2 '16 at 23:08
  • $\begingroup$ @whuber agree, I asked so that the question is stated directly. $\endgroup$ – Tim Dec 2 '16 at 23:12
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Can maximum log likelihood take a value of 0 ?

This boils down to "can a log-likelihood take a value of 0?", to which the answer is "certainly, yes"; log-likelihoods may be positive, negative or zero. Consider, for example, that densities can be greater than 1 and a product of things above and below 1 could happen to come out to be essentially 1.

However - and I suspect this might be the case here - it's common to shift log-likelihoods by arbitrary constants* (as long as it's the same arbitrary constant each time if you're making comparisons). In this situation it's quite possible that they're making it such that the full model has likelihood 0 (besides simplifying things like subsequent likelihood ratio tests, it may well be useful for numerical reasons).

* indeed Fisher defined likelihood up to a multiplicative constant, so some authors at least would say that's not just a common thing to do but actually part of how likelihoods work.

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  • $\begingroup$ you said that the log likelihood can be "positive, negative or zero". But it is only positive if you set an offset, right? $\endgroup$ – Meep Aug 28 '19 at 10:31
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    $\begingroup$ No. Even if you define log-likelihood as exactly the log of the joint density it can be positive, since (obviously) density can be greater than 1. $\endgroup$ – Glen_b -Reinstate Monica Aug 28 '19 at 16:12

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