Reversible Jump MCMC

Algorithm Reversible Jump MCMC

1. Initialise $k$ and ${\theta }_k$ at iteration $t=1$.
2. For iteration $t\ge 1$ perform
   1. Within-model move: with a fixed model $k$, update the parameters ${\theta }_k$ according to any MCMC updating scheme.
   2. Between-models move: simultaneously update model indicator $k$ and the parameters ${\theta }_k$ according to the general reversible proposal/acceptance mechanism.
3. Increment iteration $t=t+1$. If $t<N$,go to Step 2.

Dimension Matching

Fact In order to match dimensions between the two model states, a random vector $u$ of length $d_{k\to k^{\prime }}=n_{k^{\prime }}-n_k$ is generated from a known density $q(u)$. The current state ${\theta }_k$ and the random vector $u$ are then mapped to the new state ${\theta }_{k^{\prime }}^{\prime }=g_{k\to k^{\prime }}({\theta }_k,u)$ through a one-to-one mapping function $g_{k\to k^{\prime }}:{\mathbb{R}}^{n_k}\times {\mathbb{R}}^{d_k}\to {\mathbb{R}}^{n_{k^{\prime }}}$.

e.g. Suppose that model $M_1$ has states $(k=1,{\theta }_1\in \mathbb{R})$ and model $M_2$ has states $(k=2,{\theta }_2\in {\mathbb{R}}^2)$. Let $(1,{\theta }^{\star })$ denote the current state in $M_1$ and $(2,({\theta }^{(1)},{\theta }^{(2)}))$ denote the proposed state in $M_2$. Under dimension matching with a simple split/merge move, we might generate a random scalar $u$, and let ${\theta }^{(1)}={\theta }^{\star }+u$ and ${\theta }^{(2)}={\theta }^{\star }-u$, with the reverse move given deterministically by ${\theta }^{\star }=\frac{1}{2}({\theta }^{(1)}+{\theta }^{(2)})$. For the same setup but with a simple birth/death move, we might specify ${\theta }^{(1)}={\theta }^{\star }$ and ${\theta }^{(2)}={\theta }^{\star }+u$, with the reverse move given deterministically by ${\theta }^{\star }={\theta }^{(1)}$.