 Solution

WriteMultiplying the likelihood by the improper prior gives the Bayesian posterior kernel

Sincethe first full conditional isHolding $\mu$ fixed and comparing powers and exponential rates givesThe assumed parity of $n$ makes $m$ a positive integer.

Initialize ${\mu }_0$ arbitrarily. At iteration $j$, putDraw $m$ independent uniform variables $U_{j1},\dots ,U_{jm}$ and setBy sampling an integer-shape gamma distribution, this has the required $\mathrm{Gamma}(m,r_j)$ law. Next draw $G_j\sim N(0,1)$ independently and setThese are exactly the two full-conditional updates of the normal mean-precision Gibbs sampler. After discarding burn-in, the pairs $({\mu }_j,{\omega }_j)$ are approximate samples from the posterior by Markov chain Monte Carlo.