Prawdopodobieństwo można określić na kilka sposobów, na przykład:
Funkcja z , który odwzorowuje do to znaczy .
funkcja losowa
moglibyśmy również wziąć pod uwagę, że prawdopodobieństwo to tylko „zaobserwowane” prawdopodobieństwo
w praktyce prawdopodobieństwo doprowadza informację o tylko do stałej multiplikatywnej, dlatego możemy uznać prawdopodobieństwo za klasę równoważności funkcji, a nie za funkcję
Kolejne pytanie pojawia się przy rozważaniu zmiany parametryzacji: jeśli jest nową parametryzacją, zwykle oznaczamy przez prawdopodobieństwo na i nie jest to ocena poprzedniej funkcji przy ale o . This is an abusive but useful notation which could cause difficulties to beginners if it is not emphasized.
What is your favorite rigorous definition of the likelihood ?
In addition how do you call ? I usually say something like "the likelihood on when is observed".
EDIT: In view of some comments below, I realize I should have precised the context. I consider a statistical model given by a parametric family of densities with respect to some dominating measure, with each defined on the observations space . Hence we define and the question is "what is ?" (the question is not about a general definition of the likelihood)
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Odpowiedzi:
Your third item is the one I have seen the most often used as rigorous definition.
The others are interesting too (+1). In particular the first is appealing, with the difficulty that the sample size not being (yet) defined, it is harder to define the "from" set.
To me, the fundamental intuition of the likelihood is that it is a function of the model + its parameters, not a function of the random variables (also an important point for teaching purposes). So I would stick to the third definition.
The source of the abuse of notation is that the "from" set of the likelihood is implicit, which is usually not the case for well defined functions. Here, the most rigorous approach is to realize that after the transformation, the likelihood relates to another model. It is equivalent to the first, but still another model. So the likelihood notation should show which model it refers to (by subscript or other). I never do it of course, but for teaching, I might.
Finally, to be consistent with my previous answers, I say the "likelihood ofθ " in your last formula.
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I think I would call it something different. Likelihood is the probability density for the observed x given the value of the parameterθ expressed as a function of θ for the given x . I don't share the view about the proportionality constant. I think that only comes into play because maximizing any monotonic function of the likelihood gives the same solution for θ . So you can maximize cL(θ∣x) for c>0 or other monotonic functions such as log(L(θ∣x)) which is commonly done.
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Here's an attempt at a rigorous mathematical definition:
LetX:Ω→Rn be a random vector which admits a density f(x|θ0) with respect to some measure ν on Rn , where for θ∈Θ , {f(x|θ):θ∈Θ} is a family of densities on Rn with respect to ν . Then, for any x∈Rn we define the likelihood function L(θ|x) to be f(x|θ) ; for clarity, for each x we have Lx:Θ→R . One can think of x to be a particular potential xobs and θ0 to be the "true" value of θ .
A couple of observations about this definition:
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