Mathematically prove that a Beta prior distribution is conjugate to a Geometric likelihood functionI 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.

Question

Mathematically prove that a Beta prior distribution is conjugate to a Geometric likelihood functionI 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&#039;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  ).&#039;But I don&#039;t know how to prove it mathematically.

Kendrick Jacobs · Accepted Answer

Step 1Find   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 2Now 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.

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).

Answered question

Answer & Explanation

New Questions in High school geometry