Consider random variable Y with a Poisson distribution: P ( y | θ ) = θ y e − θ y ! , y = 0 , 1 , 2 , … , θ &gt; 0 P ( y | θ ) = θ y e − θ y ! , y = 0 , 1 , 2 , … , θ &gt; 0Mean and variance of Y given θ are both equal to θ. Assume that ∑ i = 1 n y i &gt; 1.If we impose the prior p ∝ 1 θ , then what is the Bayesian posterior mode?I was able to calculate the likelihood and the posterior, but I'm having trouble calculating the mode so I'm wondering if I got the right posterior: P ( θ | y ) = l i k e l i h o o d ∗ p r i o r P ( θ | y ) ∝ ( θ ∑ i = 1 n y i e − n θ ) ( θ − 1 ) P ( θ | y ) ∝ θ ( ∑ i = 1 n y i ) − 1 e − n θ

Question

Consider random variable   Y with a Poisson distribution:  P  (  y  |  θ  )  =                    θ        y                    e                  −          θ                            y      !        ,  y  =  0  ,  1  ,  2  ,  …  ,  θ  &amp;gt;  0  P  (  y  |  θ  )  =                    θ        y                    e                  −          θ                            y      !        ,  y  =  0  ,  1  ,  2  ,  …  ,  θ  &amp;gt;  0Mean and variance of   Y given   θ are both equal to   θ. Assume that        ∑          i      =      1        n        y    i    &amp;gt;  1.If we impose the prior   p  ∝      1    θ  , then what is the Bayesian posterior mode?I was able to calculate the likelihood and the posterior, but I&#039;m having trouble calculating the mode so I&#039;m wondering if I got the right posterior:  P  (  θ  |  y  )  =  l  i  k  e  l  i  h  o  o  d  ∗  p  r  i  o  r  P  (  θ  |  y  )  ∝  (      θ                  ∑                  i          =          1                n                    y        i                  e          −      n      θ        )  (      θ          −      1        )  P  (  θ  |  y  )  ∝      θ          (              ∑                  i          =          1                n                    y        i            )      −      1            e          −      n      θ

Marisol Morton · Accepted Answer

The posterior mode is just the maximizing value of the posterior, so this is essentially just a calculus problem. You have already correctly derived the posterior kernel:  π  (  θ  |  y  )  ∝  exp  ⁡        (    (  n      y    ¯    −  1  )  ln  ⁡  θ  −  n  θ        )    .So the log-posterior can be written as:      F      y    (  θ  )  ≡  ln  ⁡  π  (  θ  |  y  )  =  (  n      y    ¯    −  1  )  ln  ⁡  θ  −  n  θ  +  const  .We can maximise this via ordinary calculus techniques. Differentiating with respect to   θ gives:            d              F              y                    d      θ        (  θ  )  =            n                  y        ¯            −      1        θ    −  n                              d        2                    F              y                    d              θ        2              (  θ  )  =  −            n                  y        ¯            −      1              θ      2        .You are told that   n      y    ¯    &amp;gt;  1 so the second derivative of the objective function is negative. This means that the objective function is strictly concave, so the maximizing value occurs at the unique critical point:  0  =            d              F              y                    d      θ        (      θ    ^    )  =            n                  y        ¯            −      1                θ      ^        −  n          ⟹              θ    ^    =      y    ¯    −      1    n    .So in this case we have mode   mode   π  (  θ  |  y  )  =      y    ¯    −  1  /  n (which is strictly positive since       y    ¯    &amp;gt;  1  /  n).

Consider random variable Y with a Poisson distribution: P ( y | θ<!--

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