Let X be a random variable with density function fX (x; ) = exp {ix i K ()}f0 (x) depending on the unknown parameter

Let X be a random variable with density function fX (x; θ) = exp {θix i − K (θ)}f0 (x)

depending on the unknown parameter θ. Let θ have Jeffreys’s prior density

π (θ) = |Krs (θ)|

1/2

.

Using Bayes’s theorem, show that the posterior density for θ given x is approximately π (θ|x) ≃ c exp {θix i − K (θ) − K∗ (x)}|Krs (θ)|
1/2 , whereas the density of the random variable is approximately p (θˆ θ) ≃ c exp {θix i − K (θ) − K∗ (x)} Krs (θˆ)
1/2 .
In the latter expression x is considered to be a function of .
Find expressions for the constants in both cases.