Likelihood function of a gamma distribution

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Homework Help Overview

The discussion revolves around the likelihood function of a gamma distribution, specifically in the context of a sample of random variables each following a Gamma(a,b) distribution, where both parameters a and b are unknown. The original poster expresses confusion regarding the nature of the likelihood function and its relationship to the probability density function (pdf).

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of the likelihood function and its numerical equivalence to the pdf in this context. Questions arise about why the likelihood function is not considered a pdf despite being derived from it. Some participants attempt to clarify the process of obtaining the likelihood function from the pdf.

Discussion Status

The discussion is ongoing, with participants sharing insights and clarifications. Some have expressed a better understanding of the concept, while others continue to seek clarity on specific aspects, particularly regarding the derivation and interpretation of the likelihood function.

Contextual Notes

There is a noted confusion about the distinction between the likelihood function and the pdf, as well as the implications of treating the likelihood function as a pdf. Participants also discuss the support of the likelihood function in relation to the parameters a and b.

Jamie_H
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Homework Statement


Hi all, I missed the day of class where we went over likelihood functions, and I'm quite confused!
For example, let's say I have n Xis, where each Xi ~ Gamma(a,b), where a and b are unknown.
I want to find the likelihood function of a and b, but I don't think I really understand what this represents. (Please note - I am not looking for the maximum likelihood estimator! Just the likelihood function - when I attempt to find an explanation this seems to be the only thing that comes up)


Homework Equations


The class notes for that day explain that the likelihood function is the same as the pdf in this case, so
[(b^a)/(Gamma(a)]*x(a-1)e^(-bx), a fact verified with wikipedia. No idea why though!


The Attempt at a Solution


Truthfully, I'm not sure where to start here, so I don't have much of an attempt yet. I imagine this is fairly straight-forward/there is an obvious explanation as to why the pdf and likelihood function are the same, and I'm just missing it.
Thanks in advance - I really appreciate it! :)
 
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Jamie_H said:

Homework Statement


Hi all, I missed the day of class where we went over likelihood functions, and I'm quite confused!
For example, let's say I have n Xis, where each Xi ~ Gamma(a,b), where a and b are unknown.
I want to find the likelihood function of a and b, but I don't think I really understand what this represents. (Please note - I am not looking for the maximum likelihood estimator! Just the likelihood function - when I attempt to find an explanation this seems to be the only thing that comes up)


Homework Equations


The class notes for that day explain that the likelihood function is the same as the pdf in this case, so
[(b^a)/(Gamma(a)]*x(a-1)e^(-bx), a fact verified with wikipedia. No idea why though!


The Attempt at a Solution


Truthfully, I'm not sure where to start here, so I don't have much of an attempt yet. I imagine this is fairly straight-forward/there is an obvious explanation as to why the pdf and likelihood function are the same, and I'm just missing it.
Thanks in advance - I really appreciate it! :)

The likelihood function is numerically equal to the pdf of x in this case, but is not a pdf itself; that is, the likelihood does not integrate to 1 when you integrate over a and/or b for fixed x! And, if you read the Wiki article *carefully* you will come to realize that. That same article also gives examples similar to your question.
 
Hi Ray,
Thanks for your help - I reread the page and I think I have a much firmer grasp of the concept. (This page https://onlinecourses.science.psu.edu/stat504/node/27 also does a nice job of offering some explanations with examples, in case anyone finds this with questions similar to mine.)
 
Sorry - I thought I understood this but I think I'm still a little confused. I understand why the likelihood function isn't a PDF, and I think I have a fairly good understanding of what it represents at this point. My only confusion lies in how the likelihood function is obtained. I don't see why it is necessarily numerically equal to the pdf?
Thanks again for your help!
 
Jamie_H said:
Sorry - I thought I understood this but I think I'm still a little confused. I understand why the likelihood function isn't a PDF, and I think I have a fairly good understanding of what it represents at this point. My only confusion lies in how the likelihood function is obtained. I don't see why it is necessarily numerically equal to the pdf?
Thanks again for your help!

Well, you compute it by substituting the value of x and calculating the pdf. For a sample x_1, x_2, ..., x_n you compute the multivariate pdf = product of individual pdf's at the different values. That is just a *definition*.
 
Ray Vickson said:
Well, you compute it by substituting the value of x and calculating the pdf. For a sample x_1, x_2, ..., x_n you compute the multivariate pdf = product of individual pdf's at the different values. That is just a *definition*.

Yeah that's the definition I had found, but what confuses me is that isn't each individual pdf a gamma distribution? So we would have a [gamma]^n for the range of x_1,...,x_n?
As a second (slightly related question) I understand its not a pdf in this case, but i am I correct in believing the likelihood function still has a pdf (and further that the support is a>0, b>0?)
Again, sorry I'm having so much trouble with this, i really appreciate the help
 
Jamie_H said:
Yeah that's the definition I had found, but what confuses me is that isn't each individual pdf a gamma distribution? So we would have a [gamma]^n for the range of x_1,...,x_n?
As a second (slightly related question) I understand its not a pdf in this case, but i am I correct in believing the likelihood function still has a pdf (and further that the support is a>0, b>0?)
Again, sorry I'm having so much trouble with this, i really appreciate the help

I have nothing more to say that is new or different from what went before. I am now signing off.
 
So I don't know why this was so hard for me to grasp, but in case anyone has a similar question, the answer I've arrived at is that the likelihood function is indeed {pdf of gamma}^n,
with the support x1...xn>0, a>0, b>0.
 

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