Definitions and Notations 3/3

RJMCMC algorithm

Reversible Jump MCMC algorithm :

For a given state $X_t=\mathbf{x}$ :

1. Let $Q(\mathbf{x},.)=\sum_m Q_m(\mathbf{x},.)$ be the sum of several proposition kernels. Choose the kernel $m$ with probability $p_m(\mathbf{x})=Q_m(\mathbf{x},\mathbf{\Psi})$, and then generate $\mathbf{y} \sim \frac{Q_m(\mathbf{x},.)}{p_m(\mathbf{x})}$.

2. Compute Green's ratio :

$$R_m(\mathbf{x},\mathbf{y})=\frac{f_m(\mathbf{y},\mathbf{x})}{f_m(\mathbf{x},\mathbf{y})}$$

where $f_m(\mathbf{x},\mathbf{y})$ is detailed in the technical report.

3. With probability $\alpha_m=\min(1,R)$, accept the move and set $X_{t+1}=\mathbf{y}$. Otherwise, stay at $X_{t+1}=\mathbf{x}$.