Math Problems Vault

Problem 3m.1

Derive equations (3.60) and (3.61).

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Problem 3m.2

Using the noninformative prior for $\theta$ , show that the conditional posterior density $f(\beta \mid \mathbf{y}, \sigma )$ is multivariate normal. Hence deduce that the posterior mean of $\mathbf{\beta }$ is $\widehat{\mathbf{\beta }}$ .

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Problem 3m.3

(a)(b)(c)(d)

Suppose that we use the noninformative prior for $\mathbf{\theta }$ .

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(a)

If $v = \sigma^{2}$ , show that $f(v) \propto 1 / v$ .

(b)

Obtain an expression proportional to $f(\mathbf{\beta }, v \mid \mathbf{y})$ .

(c)

Using (3.62), integrate out $\beta$ to obtain

f(v \mid \mathbf{y}) \propto v^{-(\nu / 2+1)} \exp \left(-\frac{a}{v}\right)

where $\nu = n-p$ and $a = \left\| \mathbf{y}-\mathbf{X} \widehat{\mathbf{\beta }}\right\|^{2} / 2$ .

(d)

Find the posterior mean of $v$ .