For each sampled set of knots, (k, tk, k′, tk′), do the following:

For t = 1 : L (we set L = 100) Sample the modified Cholesky parameters ψ and log (σ2) from $$\left(\begin{array}{c}\hfill {\psi }^t\hfill \\ \hfill \text{log}({\sigma }^{2,t})\hfill \end{array}\right)~MVN\left(\left(\begin{array}{c}\hfill \widehat{\psi }\hfill \\ \hfill \text{log}({\widehat{\sigma }}^2)\hfill \end{array}\right),-I{\left(\widehat{\psi },\text{log}({\widehat{\sigma }}^2)\right)}^{-1}\right),$$ where (ψ̂, σ̂2) are the maximum likelihood estimates of the Pourhamadi parameters and σ2 corresponding to the current set of knots, and I (ψ̂, log (σ̂2)) is their information matrix evaluated at these estimates. Accept this draw with probability $$\text{min}\left(1,\frac{\frac{p\left(y|k,{\mathbf{t}}_k,k\prime ,{\mathbf{t}}_k,{\psi }^t,{\sigma }^{2,t}\right)p\left({\psi }^t|k\prime ,{\mathbf{t}}_{k\prime }\right)p\left({\sigma }^{2,t}\right)}{\xi ({\psi }^t,\text{log}({\sigma }^{2,t}))}}{\frac{p(y|k,{\mathbf{t}}_k,k\prime ,{\mathbf{t}}_k,{\psi }^{t-1},{\sigma }^{2,t-1})p({\psi }^{t-1}|k\prime ,{\mathbf{t}}_{k\prime })p({\sigma }^{2,t-1})}{\xi ({\psi }^{t-1},\text{log}({\sigma }^{2,t-1}))}}\right)$$ where p (ψt|k′, tk′) is the prior distribution of ψ evaluated at ψt, p (σ2,t) is the prior distribution of σ2 evaluated at σ2,t, and ξ (ψt, log (σ2,t)) is the multivariate normal density given in step (a) evaluated at (ψt, log (σ2,t)). If the move is not accepted, let ψt = ψt–1, log(σ2,t) = log(σ2,t–1) Draw a fixed effect, α*, from the conditional posterior distribution $\alpha |k,{\mathbf{t}}_k,k\prime ,{\mathbf{t}}_{k\prime },{\mathrm{\Sigma }}_{\gamma }^t,{\sigma }^{2,t}\text{ where }\alpha |k,{\mathbf{t}}_k,k\prime ,{\mathbf{t}}_{k\prime },{\mathrm{\Sigma }}_{\gamma }^t,{\sigma }^{2,t}~MVN\left({(\mathbf{F}\prime {\mathbf{C}}^{-1}\mathbf{F})}^{-1}\mathbf{F}\prime {\mathbf{C}}^{-1}(Y,\widehat{\alpha })\prime ,{({\mathbf{F}\prime \mathbf{C}}^{-1}\mathbf{F})}^{-1}\right),\mathbf{F}=({\mathbf{B}\prime }_{F_1},{\mathbf{B}\prime }_{F_2},\dots ,{\mathbf{B}\prime }_{F_n},{\mathbf{I}}_{k+2})\prime$, C is a block-diagonal matrix composed of the n matrices ${\{({\mathbf{B}}_{R_i}{\mathrm{\Sigma }}_{\gamma }^t{\mathbf{B}\prime }_{R_i}+{\sigma }^{2,t}{\mathbf{I}}_{m_i})\}}_{i=1}^n$ and the matrix – ${\left(\frac{1}{n}{\displaystyle {\sum }_{i=1}^n\frac{1}{m_i}{I}_{\alpha ,i}(\widehat{\alpha })}\right)}^{-1}$, BFi, and BRi are defined in Section 3, Y is all of the observed data, and α̂ is the maximum likelihood estimate of α. Draw a random effect, γ*, from the conditional posterior distribution $\gamma |k,{\mathbf{t}}_k,k\prime ,{\mathbf{t}}_{k\prime },{\alpha }^{*},{\mathrm{\Sigma }}_{\gamma }^t,{\sigma }^{2,t}\text{ where }\gamma |k,{\mathbf{t}}_k,k\prime ,{\mathbf{t}}_{k\prime },{\alpha }^{*},{\mathrm{\Sigma }}_{\gamma }^t,{\sigma }^{2,t}~MVN(\lambda ,\mathrm{\Lambda }),\lambda ={\mathrm{\Sigma }}_{\gamma }^t{\mathbf{B}\prime }_{R_i}{({\sigma }^{2,t}{\mathbf{I}}_{m_i})}^{-1}(Y_i-{\mathbf{B}}_{F_i}{\alpha }^{*})$, and $$\mathrm{\Lambda }={\sum }_{\gamma }^t{\mathbf{B}\prime }_{R_i}\left\{{({\sigma }^{2,t}{\mathbf{I}}_{m_i})}^{-1}-{({\sigma }^{2,t}{\mathbf{I}}_{m_i})}^{-1}{\mathbf{B}}_{F_i}{\left(\frac{1}{{\sigma }^{2,t}}{\displaystyle \sum _{i=1}^n{\mathbf{B}\prime }_{F_i}{\mathbf{B}}_{F_i}}\right)}^{-1}{\mathbf{B}}_{F_i}{({\sigma }^{2,t}{\mathbf{I}}_{m_i})}^{-1}\right\}{\mathbf{B}}_{R_i}{\sum }_{\gamma }^t$$