non-differentiable likelihood functions

[gusl]

Are likelihood functions ever non-differentiable?

Of course, you can always transform the parameters in such a way that you get a kink in the function... My question is whether this ever occurs naturally.

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UPDATE: clearly, if the likelihood for each data point is differentiable, then so is the likelihood for the whole data.

Comments

[bhudson.livejournal.com]

No delta functions?

[gustavolacerda.livejournal.com]

Likelihoods can't be infinite. In particular, they can't exceed 1.

Re: non-differentiable likelihood functions

[http://users.livejournal.com/cunctator_/]

Simulated discrete choice.

Re: non-differentiable likelihood functions

[gustavolacerda.livejournal.com]

Can you say more? Remember that it doesn't make sense to talk about continuity or differentiability if the parameter space isn't continuous.

[stepleton.livejournal.com]

I'm not sure what you mean here, but I think it may be due to some imprecision in the term "likelihood". For a continuous distribution, you would use the PDF as the likelihood function in most settings---so, for example, if you were considering 1-D normal distributions with stddev fixed to something small, the likelihood function for the mean parameter f(x | µ), given some single datum x, could indeed exceed 1.

Also, what does "naturally" mean in the original question?

[stepleton.livejournal.com]

PS: The "imprecision" here is not on your part, but on the part of users of the word all over.

[gustavolacerda.livejournal.com]

likelihood(theta) = P(D | theta). It is a probability, and as such it must be between 0 and 1. I am not aware of any other definitions of "likelihood".

[stepleton.livejournal.com]

But surely you have seen usages that imply that the value of a PDF as the function of the parameter is "the likelihood". Like this one here.

[gustavolacerda.livejournal.com]

I don't see any difference between that usage and my usage.