Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 40 / 5 / v / Solution

Introduce a model indicator $M\in \{0,1\}$. In $M=0$, set $\mu =0$, hence $\theta =1$, and retain the two group effects. In $M=1$, include $\mu$ with the prior induced by the gamma baseline. Choose positive prior model probabilities $p_0,p_1$ summing to one; the supplied within-model parameter priors alone do not specify these probabilities.

Let $r=(a_2,b_2)$, use the same proper prior $p(r)$ in both models, and denote their likelihoods by $L_0(r)$ and $L_1(\mu ,r)$. The unnormalized joint model targets areFor a birth move, draw an auxiliary $u$ from a positive probability density function $q(u)$ and map $(r,u)$ to $({\mu }^{\prime }=u,r^{\prime }=r)$. The reverse death move simply deletes $\mu$. This is a dimension-matching bijection with absolute Jacobian one. If the birth and death selection probabilities are $b_0$ and $d_1$, respectively, the birth and death moves for Bayesian variable selection rule givesThe reverse acceptance is the reciprocal ratio, capped at one, evaluated at the current $\mu$. Choosing $q=p_{\mu }$ cancels the added-parameter prior; a data-informed proposal may mix better but must retain its proposal factor. Within each model, also update its continuous parameters to ensure exploration.

For an explicit likelihood ratio, with $E$ and $T$ from part (ii),This supplies a complete reversible-jump Markov chain Monte Carlo move. Its acceptance probabilities enforce detailed balance between the two dimensions. The lower-dimensional state is a separate point-mass model, not a zero-probability equality test inside the continuous full-model prior.