Reversible jump MCMC
Reversible jump MCMC
Reversible jump MCMC is a Bayesian algorithm to infer the number of components/ clusters from a set of data. For this illustration we shall consider a two component model at most.
Model
The likelihoods can be represented as: \[ \begin{align} p(y_i|\lambda_{11},k=1)=&\lambda_{11}\exp(-\lambda_{11}y_i)\\ p(y_i|\lambda_{12},\lambda_{22},k=2,z_i)=&\prod_j (\lambda_{j2}\exp(-\lambda_{j2}y_i))^{1(z_i=j)} \end{align} \]
The priors on the latent variables are:
\[ \begin{align} p(\lambda_{jk})\propto & \frac{1}{\lambda_{jk}}\qquad \lambda_{jk}\in[a,b]\\ p(z_i=1)=&\pi\\ p(\pi) = & \text{Dir}(\alpha) p(k=j)= & 1/K \end{align} \]
Jumping dimensions
We need to consider a Metropolis-Hastings (MH) step to consider going from one component to two components. The MH step in general is as follows:
\[ \begin{align} \alpha = & \frac{p(y,\theta_2^{t+1})}{p(y,\theta_1^t)}\frac{q(\theta_1^t|\theta_2^{t+1})}{q(\theta_2^{t+1}|\theta_1^{t})}\\ A = & \text{min}\left(1,\alpha\right) \end{align} \]
where,
\[ \begin{align} p(y_i,\theta_2)=& p(y|\lambda_{12},\lambda_{22},\pi)p(\lambda_{12})p(\lambda_{22})p(\pi)\\ =&\pi p(y_i|\lambda_{12})+(1-\pi) p(y_i|\lambda_{22}) \end{align} \]
Jumping from 1 dim to 2
In this case let the parameters \[\theta=\{\cup_j\lambda_{jk},k,\pi\}\] . As we can let the proposal distribution be anything, we let \[q(\theta_1\to\theta_2)\] as follows: \[ \begin{align} q(\lambda_{j2},\pi,k=2|k=1,\lambda_{11})=q(\lambda_{j2}|k=2,\lambda_{11})q(\pi|k=2)q(k=2|k=1) \end{align} \]
We let the proposal \[q(k=2\vert k=1)=1\]. We also have the following dimensional jump:
\[ \begin{align} \mu_1,\mu_2\sim & U(0,1)\\ \lambda_{12}=&\lambda_{11}\frac{\mu_1}{1-\mu_1}\\ \lambda_{22}=&\lambda_{11}\frac{1-\mu_1}{\mu_1}\\ \pi=&\mu_2 \end{align} \]
Thus, in order to find the distribution (q({j2}k=2,{11})) we use the change of variable identity that \[q(\lambda_{j2}\vert k=2,\lambda_{11})=q(\mu_1)\vert J\vert\] where, \[J\] is the jacobian \[\frac{\partial(\lambda_{11},\mu_1)}{\partial(\lambda_{12},\lambda_{22})}\]. The Jacobian determinant is found to be \[\frac{\mu_1(1-\mu_1)}{2\lambda_{11}}\] while \[q(\mu_1)=q(\mu_2)=1\] since they are sampled from standard uniform distributions. Also (q(_2)=q(k=2)).
Since we need the ratio of proposed states ( ) we are also required to find ( q({11},k=2{2j},,k=1) = q({11}{2j},k=2) q(k=1 k=2) ). We again take ( q(k=1k=2)=1 ). (q(_{11}=)=1)
Jumping from 2 to 1
The MH step is conducted using the reciprocal of \(\alpha\) in the equation above.