Question

Consider the hierarchical Bayes model $$ \begin{aligned} Y & \sim b(n, p), \quad 00 \\ \theta & \sim \Gamma(1, a), \quad a>0 \text { is specified. } \end{aligned} $$ (a) Assuming squared-error loss, write the Bayes estimate of \(p\) as in expression (11.4.3). Integrate relative to \(\theta\) first. Show that both the numerator and denominator are expectations of a beta distribution with parameters \(y+1\) and \(n-y+1\). (b) Recall the discussion around expression (11.3.2). Write an explicit Monte Carlo algorithm to obtain the Bayes estimate in part (a).

Short answer

The Bayes estimate of p is given by \[p = \frac{E(B(y+1, n-y+1))}{E(B(n+2, n-y+1))}\] It can be obtained using a Monte Carlo algorithm that involves repeated random sampling to produce sequences of p. This algorithm proceeds by generating a random number \(\theta*\) from Γ(1, a), and then generating a random number p* from \(\theta* p^{\theta*-1}\), repeated over numerous iterations. The sequence of p* then is an approximation to the marginal posterior distribution of p.

Step-by-step solution

Finding the Bayes estimate of p

Remembering the postulate of Bayes’ rule, integrate with respect to \(\theta\) first. Given, \(Y \sim b(n, p)\) and \(p \mid \theta \sim h(p \mid \theta)=\theta p^{\theta-1}\), we can write the Bayes estimate of p as follows: \[p = \frac{\int_0^1 p(\theta p^{\theta-1})dp}{\int_0^1 (\theta p^{\theta-1})dp}\]

Evidencing numerator and denominator as expectations of a beta distribution

Using formula for Beta Distribution i.e., \(B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt\), we can determine the numerator and denominator as the expectation of a Beta Distribution function with parameters \(y+1\) and \(n-y+1\). Hence, \[\int_0^1 p(\theta p^{\theta-1})dp = E(B(y+1, n-y+1))\] as the numerator, and \[\int_0^1 (\theta p^{\theta-1})dp = E(B(n+2, n-y+1))\] as the denominator. Now, we can write the bayes estimate of p as: \[p = \frac{E(B(y+1, n-y+1))}{E(B(n+2, n-y+1))}\]

Writing a Monte Carlo algorithm

Monte Carlo algorithm is a method to generate sequences of random outcomes and approximate the statistical properties of the population. Let's build a Monte Carlo algorithm: \n1. Generate a random number \(\theta^{*}\) from Γ(1, a).\n2. Generate a random number p* from \(\theta* p^{\theta*-1}\).\nRepeating the process for a large number of iterations, p* will be an estimate of the marginal posterior of p.