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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 $ k$ in the opposite order, we can introduce a Markov kernel $ K$ 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 $ K$ 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).
$


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