Talk:Likelihood function

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I adjusted the wiktionary entry so it doesn't say that the mathematical definition is 'likelihood = probability'. Someone more mathematical than I may want to check to see if the mathematical definition I gave is correct. I defined "likelihood" in the parameterized-model sense, because that is the only way in which I have ever seen it used (i.e., not in the more abstract Pr(A | B=b) sense currently given in the Wikipedia article). 128.231.132.2 03:06, 21 March 2007 (UTC)

This article needs integrating / refactoring with the other two on the likelihood principle and maximum likelihood method, and a good going-over by someone expert in the field. -- The Anome

I emphatically agree. I've rewritten some related articles and I may get to this one if I ever have time. -- Mike Hardy

All was going well until I hit

In statistics, a likelihood function is a conditional probability function considered as a function of its second argument with its first argument held fixed, thus:

Would it be possible for someone to elaborate on that sentence of to given an example? FarrelIThink 06:12, 21 February 2007 (UTC)

Contents

[edit] Context tag

I added the context tag because the article starts throwing mathematical functions and jargon around from the very beginning with no explanation of what the letters and symbols mean. Rompe 04:40, 15 July 2006 (UTC)

The tag proposes making it more accessible to a general audience. A vernacular usage makes likelihood synonymous with probability, but that is not what is meant here. I doubt this topic can be made readily comprehensible to those not familiar at the very least with probability theory. So I question the appropriateness of the "context" tag. The article starts with the words "In statistics,...". That's enough to tell the general reader that it's not about criminology, church decoration, sports tactics, chemistry, fiction writing, etc. If not such preceeding words were there, I'd agree with the "context" tag. Michael Hardy 23:55, 16 July 2006 (UTC)

[edit] Which came first

Which came first? the common use as in "in all likelihood this will not occur" or the mathematical function?

[edit] Backwards

An earlier version of this page said "In a sense, likelihood works backwards from probability: given B, we use the conditional probability Pr(A|B) to reason about A, and, given A, we use the likelihood function L(A|B) to reason about B. ". This makes sense; i.e. it says it's backwards, and it is.

The current version uses L(B|A) instead, i.e. it says: "In a sense, likelihood works backwards from probability: given B, we use the conditional probability Pr(A|B) to reason about A, and, given A, we use the likelihood function L(B|A) to reason about B. " This does not make sense. It says it's backwards, but it talks as if Pr and L are interchangeable.

How about switching back to the earlier version, and providing a concrete example to help clarify it? Possible example: Given that a die is fair, we use the probability of getting 10 sixes in a row given that the die is fair to reason about getting 10 sixes in a row; or given that we got 10 sixes in a row, we use the likelihood of getting 10 sixes in a row given that the die is fair to reason about whether the die is fair. (Or should it say "the likelihood that the die is fair given that 10 sixes occur in a row"? What exactly is the definition of "likelihood" used in this sort of verbal context, anyway?) --Coppertwig 20:28, 24 August 2007 (UTC)

[edit] Likelihood of continuous distributions is a problem

The contribution looks attractive; however, it ignores several basic mathematical facts:

1. Usually likelihood is assessed using not one realization, but a series of observed random variables (independently identically distributed). Then the likelihood expands to a large product. Usually this is transformed by a logarithm to a sum. This transformation is not linear (like that mentioned in the entry), but it attains its maximum at the same point.

2. Likelihood can easily be defined for discrete distributions, where its values are values of some probabilities. A problem arises with an analogue for continuous distributions. Then the probability density function (pdf) is used instead of probability (probability function, pf). This is incorrect unless we use additional assumptions, e.g., continuity of the pdf. Without it, the notion of likelihood does not make sense, although this error occurs in most textbooks. (Do you know any which makes this correct? I did not find any, I did it in my textbook.) In any case, there are two totally different and incomaparable notions of likelihood, one for discerte, the other for continuous distributions. As a consequence, there is no notion of likelihood applicable to mixed distributions. (Nevertheless, the maximum likelihood method can be applied separately to the discrete and continuous parts.)

Mirko Navara, http://cmp.felk.cvut.cz/~navara —Preceding unsigned comment added by 88.146.54.129 (talk) 08:16, 22 February 2008 (UTC)

Just to clarify, by "the contribution" are you referring to the whole article or a particular section or edit? I assume the former.
On (1), well, the log-likelihood isn't mentioned in this article but clearly it isn't itself a likelihood. The invariance of maximum likelihood estimates to transformation is surely a matter not for this article but for the one on maximum likelihood. (I haven't checked that article to see what it says on the topic, if anything).
On (2), I think you've got a point that this article lacks a rigorous definition. I think the more accessible definition is needed too and should be given first. If you want to add a more rigorous definition, go ahead. I'm sure i've seen a measure-theoretic definition somewhere but I'm afraid i've never got to grips with measure theory myself.
When you say "I did it in my textbook", is that Teorie Pravděpodobnosti Na Kvantových a Fuzzy Logikách? I'm afraid i can't locate a copy to consult. Qwfp (talk) 09:34, 22 February 2008 (UTC)