Page 518 - Probability and Statistical Inference

10. Bayesian Methods 495

sufficient for θ given that = θ. We have been told that is positive and so
there is no point in assuming a normal prior for the parameter . For simplic-
ity, let us suppose that the prior distribution of on the space Θ = ℜ is
+
–1
exponential with known mean α (> 0). Now, let us first proceed to deter-
mine the marginal pdf m(x). The joint distribution of (v, X) is given by

Recall the standard normal pdf and the df
with y ∈ ℜ. Thus, for all x ∈ ℜ, the marginal pdf of X
can be written as

Now, combining (10.7.1)-(10.7.2), we obtain the posterior pdf of v given
that X = x as follows: For all x ∈ ℜ, θ ∈ ℜ ,
+

In this example, the chosen prior is not a conjugate one and yet we have been
able to derive the expression of the posterior pdf in a closed form. In the
Exercise 10.7.1 we ask the reader to check by integration that k(θ; x) is
indeed a pdf on the positive half of the real line. Also refer to the Exercises
10.7.2-10.7.3. !
Example 10.7.2 (Example 10.7.1 Continued) Let X be N(θ, 1) given that
= θ where v is the unknown population mean. Consider the statistic T = X
which is minimal sufficient for θ given that = θ. We were told that was
positive and we took the exponential distribution with the known mean α (>
–1
0) as our prior for .
Let us recall the standard normal pdf and the
df for y ∈ ℜ. Now, from the Example 10.7.1, we know
–1
that the posterior distribution of v is given by k(θ; x) = {Φ ([x – α])} φ(θ –
[x – α]) for θ > 0, –∞ < x < ∞.