A Derivation of full conditional distributions

It is straightforward to show that \[ \pi(\boldsymbol{\beta}\mid \boldsymbol{\Lambda}, \boldsymbol{y}, \boldsymbol{z}, \boldsymbol{X}) \propto \prod_{i=1}^k\prod_{\ell=1}^{p_s + p_\ell} \boldsymbol{\beta}_{i\ell}^{\sum_{j=1}^{n_i} z_{ij\ell} + a -1} (1- \boldsymbol{\beta}_{i\ell})^{n_i -\sum_{j=1} \boldsymbol{z}_{ij\ell} + b -1}, \] which means \[ \boldsymbol{\beta}_{i\ell} \mid \pmb{\Lambda}, \boldsymbol{y}, \boldsymbol{z}, \boldsymbol{X}\stackrel{iid}{\sim} \text{Beta}(\sum_{j=1}^{n_i} z_{ij\ell} + a, n_i - \sum_{j=1}^{n_i} z_{ij\ell} + b). \]

Consider the distribution of \(\boldsymbol{z}\mid \boldsymbol{\Lambda}, \boldsymbol{y}, \boldsymbol{\beta}, \boldsymbol{X}.\) We find that

Remark: \(\boldsymbol{z}_{ij\ell}\) are all independent conditional on \(\boldsymbol{\Lambda}, \boldsymbol{y}, \boldsymbol{\beta}, \boldsymbol{X}.\)

We now turn to the conditional distribution of \(\boldsymbol{y}\mid \boldsymbol{\Lambda}, \boldsymbol{y}, \boldsymbol{\beta}, \boldsymbol{X}.\) First, note that each \(\boldsymbol{y}_{j\prime \ell}\) takes values in the set \(S_\ell,\) which consists of all values for the \(\ell\)th field that appear anywhere in the data. Then the distribution of \(\boldsymbol{y}_{j\prime \ell} \mid \boldsymbol{\Lambda}, \boldsymbol{y}, \boldsymbol{\beta}, \boldsymbol{X}\) takes the form \(P(\boldsymbol{y}_{j\prime \ell} = w \mid \boldsymbol{\Lambda}, \boldsymbol{y}, \boldsymbol{\beta}, \boldsymbol{X}) = A_\phi \phi_w\) for all \(w \in S_\ell,\) where \(A_\phi = \left (\sum_{w \in S_\ell} \phi_w \right) ^{-1}.\)

Let \(R_j^\prime = \{(i,j) : \lambda_{ij} = j^\prime \}\) %denote be the set of all records that correspond to individual $ j^.$ Then if \(\ell \leq p_s,\)

\[\begin{align*} \phi_w = \mathop{ \prod_{(i,j) \in R_j^\prime} }_{z_{ij\ell}=1} h_\ell(w) \exp\left\{ -c \mathop{\sum_{(i,j) \in R_j^\prime}}_{z_{ij\ell}=1} d(X_{ij\ell},w) \right\} \alpha_\ell(w) \times \mathop{ \prod_{(i,j) \in R_j^\prime} }_{z_{ij\ell}=0} I(X_{ij\ell} = w). \end{align*}\]

Simplifying,

\[ \phi_w = \begin{cases} \displaystyle \alpha_\ell(w) \mathop{ \prod_{(i,j) \in R_j^\prime} }_{z_{ij\ell}=1} \left\{ h_\ell(w) \; \exp\left\{ -c \; d(X_{ij\ell},w) \right\} \right\} \text{if} \; X_{ij\ell} = w \; \forall (i,j) \in R_j^\prime \owns z_{ij\ell} = 0\\ 0 \; \text{otherwise.} \end{cases} \]

Hence, if \(\ell \leq p_s,\) then \(Y_{j^\prime \ell} \mid \boldsymbol{\Lambda}, \boldsymbol{z}, \boldsymbol{\beta}, \boldsymbol{X}\) has the distribution \[\begin{align*} P(Y_{j^\prime \ell} = w \mid \boldsymbol{\Lambda}, \boldsymbol{z}, \boldsymbol{\beta}, \boldsymbol{X}) = \frac{ \alpha_\ell(w) \mathop{ \prod_{(i,j) \in R_j^\prime} }_{z_{ij\ell}=1} \left\{ h_\ell(w) \; \exp\left\{ -c \; d(X_{ij\ell},w) \right\} \right\} } { \sum_{w \in S_\ell} \left( \alpha_\ell(w) \mathop{ \prod_{(i,j) \in R_j^\prime} }_{z_{ij\ell}=1} \left\{ h_\ell(w) \; \exp\left\{ -c \; d(X_{ij\ell},w) \right\} \right\} \right) }. \end{align*}\]

If instead, \(\ell > p_c,\) then we find that % $ P(Y_{j^} = w , , , ) =(w) ({w S_} _(w))^{-1}. $

Regarding, the linkage structure \(\boldsymbol{\Lambda},\) its full conditional is as follows: % $ P(_{ij} = v , , , ) = 0 $ if there exists \(\ell\) such that \(z_{ij\ell} = 0\) and \(X_{ij\ell} \neq Y_{v\ell}.\) Otherwise, % \[ P(\lambda_{ij} = v \mid \boldsymbol{y}, \boldsymbol{z}, \boldsymbol{\beta}, \boldsymbol{X}) \propto \mathop{ \prod_{\ell=1}^{p_s}}_{z_{ij\ell }= 1} \left\{ h_\ell(Y_{v \ell} ) \exp\left\{ -c \; d(X_{ij\ell},Y_{v \ell} ) \right\} \right\}. \]

Define \(\Omega_{ij} = \{j^\prime : X_{ij\ell} = Y_{j^\prime \ell} \; \forall \; \ell \owns z_{ij\ell} = 0\}.\) Then this implies \[\begin{align*} P(\lambda_{ij} = v \mid \boldsymbol{y}, \boldsymbol{z}, \boldsymbol{\beta}, \boldsymbol{X}) = \frac{ \displaystyle \mathop{ \prod_{\ell=1}^{p_s}}_{z_{ij\ell }= 1} \left\{ h_\ell(Y_{v \ell} ) \exp\left\{ -c \; d(X_{ij\ell},Y_{v \ell} ) \right\} \right\} } { \sum_{v^\prime \in \Omega_{ij}} \left\{ \displaystyle \mathop{ \prod_{\ell=1}^{p_s}}_{z_{ij\ell }= 1} \left\{ h_\ell(Y_{v \ell} ) \exp\left\{ -c \; d(X_{ij\ell},Y_{v \ell} ) \right\} \right\} \right\} }. \end{align*}\]