In Bayesian probability theory, a class of distribution of prior distribution theta is said to be the conjugate to a class of likelihood function f(x| theta) if the resulting posterior distribution is of the same class as of f(theta).

Almintas2l 2022-07-17 Answered
Mathematically prove that a Beta prior distribution is conjugate to a Geometric likelihood function
I have to prove with a simple example and a plot how prior beta distribution is conjugate to the geometric likelihood function. I know the basic definition as
'In Bayesian probability theory, a class of distribution of prior distribution θ is said to be the conjugate to a class of likelihood function f ( x | θ ) if the resulting posterior distribution is of the same class as of f ( θ ).'
But I don't know how to prove it mathematically.
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Answers (1)

Kendrick Jacobs
Answered 2022-07-18 Author has 16 answers
Step 1
Find f ( x | θ ). Using Bayes theorem we know that f ( x | θ ) = C f ( θ | x ) f ( θ ). C is just a normalisation constant to make it integrate to 1.
f ( θ ) is the PDF of the prior distribution. I.e. beta distribution (with some parameters ( α , β )). Here f ( θ ) = C θ α 1 ( 1 θ ) β 1
f ( θ | x ) is the likelihood function for θ given that the data x is distributed by a geometric distribution with parameter θ. Our geometric likelihood function is f ( θ | x ) = i = 0 n ( 1 θ ) x i θ = ( 1 θ ) i = 0 n x i θ n .
Step 2
Now were going to find the product of these and we expect it will have the same form as the beta prior but with new parameters α , β , and we will find the parameters.
So f ( θ | x ) f ( θ ) = C θ α + n 1 ( 1 θ ) i = 0 n x i + β 1 . We can see the new parameters are α = α + n, and β = i = 0 n x i + β. Mission accomplished.

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