For estimating above model, Chib and Greenberg (1994) divided it into two linear regression models. First, by using notations, $ X_{k,t}^{*} = X_{k,t} - \Psi_{kk}X_{k,t-1} $ , and $ S_{k,t}^{*} = S_{k,t} - \Psi_{kk}S_{k,t-1} $ where subscript k is k-th indicator of data set $ X_{t} $ , above equation is represented as

$$ \begin{array}{r}{\mathbf{X}_{t}^{*}=\mathbf{\Lambda}S_{t}^{*}+\mathbf{\nu}_{t},\qquad\mathbf{\nu}_{t}\sim i.i.d.N(\mathbf{0},\mathbf{R}),}\end{array} $$

Second, by using notation $ e_{k,t} = X_{k,t} - \Lambda_{k}S_{t} $ which means measurement errors, the equation is also rewritten as

$$ \boldsymbol{e}_{k}=\boldsymbol{\Psi}_{k k}\boldsymbol{e}_{k,-1}+\boldsymbol{\nu}_{k}, $$

where $ e_{k}=[e_{k,2},\ldots,e_{k,T}]^{\prime} $ , $ e_{k,-1}=[e_{k,1},\ldots,e_{k,T-1}]^{\prime} $ . We sample parameter $ (\mathbf{\Lambda},\mathbf{R}) $ given parameter $ \Psi $ from the first equation, and parameter $ \Psi $ given $ (\mathbf{\Lambda},\mathbf{R}) $ from the second equation sequentially based on the following two-step algorithm.

Step 1. Sampling ( $ \Lambda_{k}, R_{kk} $ ) from conditional posterior distribution $ p(\Lambda_{kk}, R_{kk} \mid \Psi_{kk}, S^{T}, \theta, X^{T}) $ for estimating equation

$$ \mathbf{X}_{t}^{*}=\mathbf{\Lambda}S_{t}^{*}+\boldsymbol{\nu}_{t},\qquad\boldsymbol{\nu}_{t}\sim i.i.d.N(\mathbf{0},\mathbf{R}). $$

The posterior density of $ (\Lambda, R_{kk}) $ given the unobserved state variables $ S^{T} $ and deep parameters $ \theta $ is represented as

$$ p(\mathbf{\Lambda}_{k},R_{kk}|\Psi_{kk},\boldsymbol{S}^{T},\boldsymbol{X}^{T})\quad\propto\quad p(\boldsymbol{X}^{T}|\boldsymbol{S}^{T},\mathbf{\Lambda}_{k},R_{kk},\boldsymbol{\Psi}_{kk},\boldsymbol{\theta})p(\boldsymbol{\Lambda}_{kk},R_{kk}), $$

where $ p(X^{T}|S^{T}, \Lambda_{k}, R_{kk}, \Psi_{kk}, \theta) $ is likelihood function and $ p(\Lambda_{kk}, R_{kk}) $ is prior density.

As shown by Chib and Greenberg (1994), the above likelihood function is proportional to a Normal -Inverse-Gamma density as

$$ p(\boldsymbol{X}^{T}|\boldsymbol{S}^{T},\boldsymbol{\Lambda}_{k},R_{kk},\boldsymbol{\Psi}_{kk},\boldsymbol{\theta})\quad\propto\quad p_{NIG}(\boldsymbol{\Lambda}_{k},R_{kk}\mid\hat{\boldsymbol{\Lambda}}_{k},(\boldsymbol{S}^{*\prime}\boldsymbol{S}^{*}),s,T-N-2) $$

where

$$ \begin{array}{rcl}\hat{\boldsymbol{\Lambda}}_{k}&=&\boldsymbol{(S^{*}{}^{\prime}\boldsymbol{S}^{*})^{-1}\boldsymbol{S}^{*}{}^{\prime}\boldsymbol{X}^{*}}_{k}\\s&=&\boldsymbol{X}^{*}{}_{k}^{\prime}\Big(\mathbf{I}_{T}-\boldsymbol{S}^{*}(\boldsymbol{S}^{*}{}^{\prime}\boldsymbol{S}^{*})^{-1}\boldsymbol{S}^{*}{}^{\prime}\Big)\boldsymbol{X}^{*}_{k}=\boldsymbol{X}^{*}{}_{k}^{\prime}\Big(\boldsymbol{X}^{*}{}_{k}^{*}-\boldsymbol{S}^{*}\hat{\boldsymbol{\Lambda}}_{k}\Big).\end{array} $$

Since above prior $ p(\Lambda_{kk}, R_{kk}) $ is assumed to be Normal-Inverse-Gamma $ p_{NIG}(\mathbf{\Lambda}_{k}, R_{kk} | \mathbf{\Lambda}_{k,0}, \mathbf{M}_{k,0}, s_{0}, \nu_{0}) $ , the resulting conditional posterior density is also Normal-Inverse-Gamma as following.

$$ \begin{aligned}p(\mathbf{\Lambda}_{k},R_{kk}|\Psi_{kk},\mathbf{S}^{T},\mathbf{X}^{T})\quad&\propto\quad p_{NIG}(\mathbf{\Lambda}_{k},R_{kk}\mid\hat{\mathbf{\Lambda}}_{k},(\mathbf{S}^{*\prime}\mathbf{S}^{*}),s,T-N-2)\\&\times\quad p_{NIG}(\mathbf{\Lambda}_{k},R_{kk}\mid\mathbf{\Lambda}_{k,0},\mathbf{M}_{k,0},s_{0},\nu_{0})\\&\propto\quad p_{NIG}(\mathbf{\Lambda}_{k},R_{kk}\mid\bar{\mathbf{\Lambda}}_{k},\bar{\mathbf{M}}_{k},\bar{s},\bar{\nu})\end{aligned} $$

where