Suppose that X 1 , X 2 , … , X n are IID Bernoulli random variables with success probability equal to an unknown parameter θ ∈ [ 0 , 1 ].Let A and B be nonnegative constants. If we impose the prior π ( θ ) ∝ ( θ a ) ( ( 1 − θ ) b ), then what is the Bayesian posterior mode?

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Suppose that       X    1    ,      X    2    ,  …  ,      X    n   are IID Bernoulli random variables with success probability equal to an unknown parameter   θ  ∈  [  0  ,  1  ].Let   A and   B be nonnegative constants. If we impose the prior   π  (  θ  )  ∝  (      θ    a    )  (  (  1  −  θ      )    b    ), then what is the Bayesian posterior mode?

Braylon Perez · Accepted Answer

To find the Posterior mode, we need to maximize the posterior distribution with respect to   θ. So first we find the joint density:      f    (    X      |    θ    )    =          ∏              i        =        1            n              θ                  x        i              (    1    −    θ          )              1        −                  x          i                      =          θ              ∑                  x          i                      (    1    −    θ          )              n        −        ∑                  x          i                    Then we can get the posterior distribution:      π    (    θ      |    x    )    ∝    f    (    x      |    θ    )    π    (    θ    )    ∝          θ              ∑                  x          i                      (    1    −    θ          )              n        −        ∑                  x          i                            θ      a        (    1    −    θ          )      b        =          θ              ∑                  x          i                +        a              (    1    −    θ          )              n        +        b        −        ∑                  x          i                      =    exp    ⁡          (      (      n      −      n                  X        ¯            +      b      )      log      ⁡      (      1      −      θ      )      +      (      n                  X        ¯            +      a      )      log      ⁡      θ      )      To find the posterior mode, we need to maximize the posterior with respect to   θ:Let   g  (  θ  )  =  (  n  −  n      X    ¯    +  b  )  log  ⁡  (  1  −  θ  )  +  (  n      X    ¯    +  a  )  log  ⁡  θ            g          ′        (    θ    )    =    −                  n        −        n                        X          ¯                +        b                    1        −        θ              +                  n                        X          ¯                +        a            θ        =    0  Solving for   θ, we get               θ      ^        =                  n                        X          ¯                +        a                    n        +        a        +        b            , which you can check is the maximum by making sure       g          ′      ′        (  θ  )  &amp;gt;  0.

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