Page 517 - Probability and Statistical Inference

494 10. Bayesian Methods

real valued. Let us continue to denote the prior of and the pmf or pdf of T
given = θ by h(θ) and g(t; θ) respectively.
Next, one obtains the posterior distribution k(θ; t) as in (10.2.3) and pro-
ceeds to evaluate the posterior probability of the null and alternative spaces
Θ , Θ respectively as follows:
0 1

We assume that both α , α are positive and view these as the posterior
1
0
evidences in favor of H , H respectively.
0 1
We refrain from giving more details. One may refer to Ferguson (1967) and
Berger (1985).
Example 10.6.1 (Example 10.3.5 Continued) Let X , ..., X be iid N(θ, 4)
1
5
given that = θ where (∈ ℜ) is the unknown population mean. Let us
suppose that the prior distribution of v on the space Θ = ℜ is N(3, 1). We wish
to test a null hypothesis H : < 1 against an alternative hypothesis H : >
1
0
2.8. Consider the statistic which is minimal sufficient for θ given
that = θ. Suppose that the observed value of T is t = 6.5.
6.5
-1
5
With µ = (-- + 3) (- + 1) ≈ 2. 0556 and the
4 4
posterior distribution of turns out to be Let us denote a random
variable Y which is distributed as and Z = (Y – µ)/σ .
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Now, from (10.6.1), we have α = P(Y < 1) = P(Z < -1. 5834) ≈ .0 5 6665
0
and α = P(Y > 2.8) = P(Z > 1. 1166) ≈ . 13208. Thus, the Bayes test from
1
(10.6.2) will reject H . !
0
So far, we have relied heavily upon conjugate priors. But, in some situa-
tions, a conjugate prior may not be available or may not seem very appealing.
The set of examples in the next section will highlight a few such scenarios.
10.7 Examples with Non-Conjugate Priors

The following examples exploit some specific non-conjugate priors. Certainly
these are not the only choices of such priors. Even though the chosen priors
are non-conjugate, we are able to derive analytically the posterior and Bayes
estimate.
Example 10.7.1 Let X be N(θ, 1) given that =θ where is the
unknown population mean. Consider the statistic T = X which is minimal