Algorithm 1 PX-MCMC algorithm.

Sample $\mathbf{W},{\mathsf{\Gamma }}_W$ from: $[\mathbf{W},{\mathsf{\Gamma }}_W\mid \mathsf{\Theta },{\mathbf{Y}}_{1:T},{\tilde{\mathbf{x}}}_{1:T}]\propto$ $[{\mathbf{Y}}_{1:T}\mid \mathsf{\Theta },\mathbf{W},{\mathsf{\Gamma }}_W,{\tilde{\mathbf{x}}}_{1:T}][\mathbf{W}\mid {\mathsf{\Gamma }}_W][{\mathsf{\Gamma }}_W]$.   Generate ${\alpha }_{0,i,\ell }\sim \mathrm{Gau}(0,{\sigma }_{\alpha }^2)$ for $i=1,\dots ,n_h$ and $\ell =1,\dots ,n_h$.   Transform $\tilde{\mathbf{W}}=t_{{\mathit{\alpha }}_0}^{-1}(\mathbf{W})$.   Sample $\mathit{\alpha }$ from: $[\mathit{\alpha }\mid t_{\mathit{\alpha }}(\tilde{\mathbf{W}}),{\mathsf{\Gamma }}_W,\mathsf{\Theta },{\mathbf{Y}}_{1:T},{\tilde{\mathbf{x}}}_{1:T}]\propto$ $[{\mathbf{Y}}_{1:T}\mid t_{\mathit{\alpha }}(\tilde{\mathbf{W}}),{\mathsf{\Gamma }}_W,\mathsf{\Theta },\mathit{\alpha },{\tilde{\mathbf{x}}}_{1:T}][t_{\mathit{\alpha }}(\tilde{\mathbf{W}})\mid {\mathsf{\Gamma }}_W,\mathit{\alpha }][\mathit{\alpha }]\left|J_{\mathit{\alpha }}(\tilde{\mathbf{W}})\right|$.   Sample $\mathsf{\Theta }$ from: $[\mathsf{\Theta }\mid t_{\mathit{\alpha }}(\tilde{\mathbf{W}}),{\mathsf{\Gamma }}_W,\mathit{\alpha },{\mathbf{Y}}_{1:T},{\tilde{\mathbf{x}}}_{1:T}]\propto [{\mathbf{Y}}_{1:T}\mid t_{\mathit{\alpha }}(\tilde{\mathbf{W}}),\mathsf{\Theta },{\tilde{\mathbf{x}}}_{1:T}][\mathsf{\Theta }].$