MCMC

MCMC scheme

A data augmentation algorithm is formed by two successive probability transitions--one from $\mathcal{Q}$ to $\mathcal{Z}$ and the other from $\mathcal{Z}$ to $\mathcal{Q}$ . Given a current state $\theta\in\mathcal{Q}$ we can generate the transition probability

$\displaystyle k(\theta, z) = \prod_{i=1}^n \left[ \frac{\displaystyle \theta_{... ...ta_1 f_1(x_i) + \cdots + \theta_k f_k(x_i)} \right], \quad z\in\mathcal{Z}.$

Successively given a state $z \in\mathcal{Z}$ we can update a state in $\mathcal{Q}$ from the Dirichlet distribution $k^\star(z,\cdot)$ with parameter $(m_1(z)+1,\ldots,m_k(z)+1)$ . Combined together,

$\displaystyle K^\star(\theta, \cdot) = \sum_{z \in \mathcal{Z}} k(\theta, z) k^\star(z, \cdot)$

becomes a Markov transition kernel on $\mathcal{Q}$ .

Gibbs sampler. Let

$\displaystyle D(z) = \frac{\Gamma(m_1(z)+1)\cdots\Gamma(m_k(z)+1)} {\Gamma(m_1(z)+ \cdots +m_k(z) + k)}$

be the normalizing constant for the density $k^\star(z,\cdot)$ , and let

$\displaystyle \pi(z) \propto D(z) \prod_{i=1}^n f_{z_i}(x_i)$

be the marginal density on $\mathcal{Z}$ . Then we obtain the joint density $\pi(\theta,z)$ by

$\displaystyle \pi^\star(\theta) k(\theta, z) = \pi(z) k^\star(z, \theta) \propto \prod_{i=1}^n \theta_{z_i} f_{z_i} (x_i)$

Thus, the kernel $K^\star$ is a Gibbs sampler using the conditional density $\pi(\theta \vert z)$ and $\pi(z \vert \theta)$ .

Alternative Markov chain. Combining the transition probabilities $k^\star$ and in the opposite order, we can introduce a Markov kernel on the state space $\mathcal{Z}$ by

$\displaystyle K(z, \cdot) := \int_{\mathcal{Q}} k^\star(z, \theta) k(\theta, \cdot) d\theta ,$

where $d\theta$ is a surface integral over $\mathcal{Q}$ . This Markov kernel

is also time-reversible with stationary distribution $\pi$ . Hence, we can instead materialize a Gibbs sampler for $\pi(z)$ in a discrete setting, and obtain the target observation from $\pi^\star(\theta)$ by way of sampling it from

$\displaystyle \pi^\star(\theta) = \sum_{z \in \mathcal{Z}} \pi(z) k^\star(z, \theta).$