Bayesian Genomic-Enabled Prediction Models for Ordinal and Count Data

Abstract

The purpose of this chapter is to present recent advances in models for genomic-enabled prediction developed for ordinal categorical and count data. For both models we provide details of their corresponding derivation and then apply them to a real data set. The proposed models were derived using a Bayesian framework. Bayesian logistic ordinal regression (BLOR) and Bayesian negative binomial regression (BNBR) make use of the Pólya-Gamma distribution to produce an analytic Gibbs, a sampler with similar full conditional distributions of a model with Gaussian response and can be used for complex data sets as those that arise in the context of genomic selection where the sample size usually is smaller than the number of covariates (markers). We illustrate the proposed models using simulation and a real data set. Results indicate that our models for ordinal categorical and count data are a good alternative for analyzing ordinal and count data in the context of genomic-enabled prediction.

Appendices

Appendix A: Derivation of Full Conditional Distributions for Model BLOR

Liabilities and ω _ijt . The fully conditional posterior distribution of liability l _ijt is

$$ P\left(\boldsymbol{l}| ELSE\right)\propto P\left(\boldsymbol{l}|\ \boldsymbol{\beta}, \boldsymbol{b}\right)P\left(\boldsymbol{y}|\boldsymbol{l},\boldsymbol{\gamma}\ \right) $$

$$ \propto \prod_{i=1}^I\prod_{j=1}^J\prod_{t=1}^{n_{ij}}f\left({l}_{ij t}\right)\sum_{c=1}^CI\left({y}_{ij t}=c\right)I\left({\gamma}_{c-1}<{l}_{ij t}<{\gamma}_c\right) $$

$$ \propto \prod_{i=1}^I\prod_{j=1}^J\prod_{t=1}^{n_{ij}}\frac{\exp \left(-{l}_{ij t}+{\boldsymbol{x}}_i^T\boldsymbol{\beta} +{b}_{1j}+{b}_{2 ij}\right)}{{\left[1+\exp \left(-{l}_{ij t}+{\boldsymbol{x}}_i^T\boldsymbol{\beta} +{b}_{1j}+{b}_{2 ij}\right)\right]}^2}\sum_{c=1}^CI\left({y}_{ij t}=c\right)I\left({\gamma}_{c-1}<{l}_{ij t}<{\gamma}_c\right) $$

$$ {\displaystyle \begin{array}{l}\propto \prod \limits_{i=1}^I\prod \limits_{j=1}^J\prod \limits_{t=1}^{n_{ij}}{2}^{-2}{\int}_0^{\infty}\exp \left[-\frac{\omega_{ij t}{\left(-{l}_{iljt}+{\boldsymbol{x}}_i^T\boldsymbol{\beta} +{b}_{1j}+{b}_{2 ij}\right)}^2}{2}\right]P\left({\omega}_{ij t};b=2,d=0\right) \\ {}\times d{\omega}_{ij t}\sum \limits_{c=1}^CI\left({y}_{ij t}=c\right)I\left({\gamma}_{c-1}<{l}_{ij t}<{\gamma}_c\right) \end{array}} $$

The last inequality was obtained using a technique called the Pólya-Gamma method (Scott and Pillow 2013), which is useful when working with logistic likelihoods, and has the form

$$ \frac{{\left({e}^{\psi}\right)}^a}{{\left(1+{e}^{\psi}\right)}^b}={2}^{-b}{e}^{\kappa \psi}{\int}_0^{\infty }{e}^{-\frac{\omega {\psi}^2}{2}}P\left(\omega; b,0\right) d\omega $$

where κ = a − b/2 and P(ω; b, d = 0) denotes the density of the random variable ω ~ PG(b, d = 0), where PG(b, d) denotes a Pólya-Gamma distribution l _ijt with parameters b and d and density

$P\left(\omega; b,d\right)=\left\{{cosh}^b\left(\frac{d}{2}\right)\right\}\frac{2^{b-1}}{\Gamma (b)}\sum_{n=0}^{\infty }{\left(-1\right)}^n\frac{\Gamma \left(n+b\right)\left(2n+b\right)}{\Gamma \left(n+1\right)\sqrt{2\pi {\omega}^3}}\exp \left(-\frac{{\left(2n+b\right)}^2}{8\omega }-\frac{d^2}{2}\omega \right),$where cosh denotes the hyperbolic cosine.

Then the joint posterior distribution of l _ijt  and ω _ijt is equal to

$$ {\displaystyle \begin{array}{l}P\left(\boldsymbol{l},\boldsymbol{\omega} | ELSE\right)\propto \prod \limits_{i=1}^I\prod \limits_{j=1}^J\prod \limits_{t=1}^{n_{ij}}{2}^{-2}\exp \left[-\frac{\omega_{ij}{\left(-{l}_{ij t}+{\boldsymbol{x}}_i^T\boldsymbol{\beta} +{b}_{1j}+{b}_{2 ij}\right)}^2}{2}\right]P\left({\omega}_{ij t};2,0\right) \\ {}\times \sum \limits_{c=1}^CI\left({y}_{ij t}=c\right)I\left({\gamma}_{c-1}<{l}_{ij t}<{\gamma}_c\right) \end{array}} $$

Therefore, the fully conditional posterior distribution of liability l _ijt is a truncated normal distribution and its density is

$$ {\displaystyle \begin{array}{ll} & f\left({l}_{ijt}| ELSE\right)\\ {}& =\frac{\phi \left(\sqrt{\omega_{ijt}}\left({l}_{ijt}-{\boldsymbol{x}}_i^T\boldsymbol{\beta} -{b}_{1j}-{b}_{2 ij}\right)\right)}{\varPhi \left(\sqrt{\omega_{ijt}}\left({\gamma}_c-{\boldsymbol{x}}_i^T\boldsymbol{\beta} -{b}_{1j}-{b}_{2 ij}\right)\right)-\varPhi \left(\sqrt{\omega_{ijt}}\left({\gamma}_{c-1}-{\boldsymbol{x}}_i^T\boldsymbol{\beta} -{\boldsymbol{b}}_{1\boldsymbol{j}}-{\boldsymbol{b}}_{2\boldsymbol{ij}}\right)\right)} \end{array}} $$

For simplicity, ELSE is the data and the parameters, except for the one in question. ϕ and Φ are the density and distribution function of a standard normal random variable and the fully conditional posterior distribution l _ijt of ω _ijt is

$$ f\left({\omega}_{ijt}| ELSE\right)\propto {2}^{-2}\exp \left[-\frac{\omega_{ijt}{\left(-{l}_{ijt}+{\boldsymbol{x}}_i^T\boldsymbol{\beta} +{b}_{1j}+{b}_{2 ij}\right)}^2}{2}\right]P\left({\omega}_{ijt};2,0\right)\propto \exp \left[-\frac{\omega_{ijt}{\left(-{l}_{ijt}+{\boldsymbol{x}}_i^T\boldsymbol{\beta} +{b}_{1j}+{b}_{2 ij}\right)}^2}{2}\right]P\left({\omega}_{ijt};2,0\right) $$

From here and from Eq. (4.5) of Polson et al. (2013), we get that

$$ f\left({\omega}_{ijt}| ELSE\right)\sim PG\left(2,-{l}_{ijt}+{\boldsymbol{x}}_{\boldsymbol{i}}^{\boldsymbol{T}}\boldsymbol{\beta} +{b}_{1j}+{b}_{2 ij}\right) $$

Regression Coefficients ( β )

First note that the fully conditional posterior of l , β , ω is

$$ P\left(\boldsymbol{l},\boldsymbol{\beta}, \boldsymbol{\omega} | ELSE\right)\propto P\left(\boldsymbol{l}|\ \boldsymbol{\beta}, {\boldsymbol{b}}_1,{\boldsymbol{b}}_2\right)P\left(\boldsymbol{y}|\boldsymbol{l},\boldsymbol{\gamma}\ \right)P\left(\boldsymbol{\omega} \right)P\Big(\boldsymbol{\beta} \left|{\boldsymbol{\sigma}}_{\boldsymbol{\beta}}^2\Big)\right. $$

$$ \propto \mathit{\exp}\left(-\frac{1}{2}{\left(-\boldsymbol{l}+\boldsymbol{X}\boldsymbol{\beta } +\sum_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right)}^T{\boldsymbol{D}}_{\omega}\left(-\boldsymbol{l}+\boldsymbol{X}\boldsymbol{\beta } +\sum_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right)\right)P\left(\boldsymbol{\omega} \right)P\Big(\boldsymbol{\beta} \left|{\boldsymbol{\sigma}}_{\boldsymbol{\beta}}^2\Big)\right. $$

where $P\left(\boldsymbol{\omega} \right)={\prod}_{i=1}^I{\prod}_{j=1}^J\prod_{t=1}^{n_{ij}}P\left({\omega}_{ij t};2,0\right)$. Then, the full conditional posterior distribution of β is

$$ {\displaystyle \begin{array}{l}P\left(\boldsymbol{\beta} | ELSE\right)\hfill \\ {}\begin{array}{l}\hfill \\ {}\times \propto \mathit{\exp}\left(-\frac{1}{2}{\left(-\boldsymbol{l}+\boldsymbol{X}\boldsymbol{\beta } +\sum \limits_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right)}^T{\boldsymbol{D}}_{\omega}\left(-\boldsymbol{l}+\boldsymbol{X}\boldsymbol{\beta } +\sum \limits_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right) -\frac{1}{2}{\left(\boldsymbol{\beta} -{\boldsymbol{\beta}}_0\right)}^T\left({\boldsymbol{\varSigma}}_0^{-1}{\boldsymbol{\sigma}}_{\boldsymbol{\beta}}^{-2}\right)\left(\boldsymbol{\beta} -{\boldsymbol{\beta}}_0\right)\right)\hfill \end{array}\end{array}} $$

$$ \propto \exp \left(-\frac{1}{2}\left[{\boldsymbol{\beta}}^T\left({\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\beta}^{-2}+{\boldsymbol{X}}^T{\boldsymbol{D}}_{\omega}\boldsymbol{X}\right)\boldsymbol{\beta} -2{\left({\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\boldsymbol{\beta}}^{-2}{\boldsymbol{\beta}}_0-{\boldsymbol{X}}^T{\boldsymbol{D}}_{\omega}\left(\sum_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right)+{\boldsymbol{X}}^T{\boldsymbol{D}}_{\omega}\boldsymbol{l}\right)}^T\boldsymbol{\beta} \right]\right) $$

$$ \propto \mathit{\exp}\left(-\frac{1}{2}\left[{\left(\boldsymbol{\beta} -{\overset{\sim }{\boldsymbol{\beta}}}_0\right)}^T{\overset{\sim }{\boldsymbol{\varSigma}}}_0^{-1}\left(\boldsymbol{\beta} -{\overset{\sim }{\boldsymbol{\beta}}}_0\right)\right]\right) $$

where ${\overset{\sim }{\boldsymbol{\varSigma}}}_0={\left({\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\beta}^{-2}+{\boldsymbol{X}}^T{\boldsymbol{D}}_{\boldsymbol{\omega}}\boldsymbol{X}\right)}^{-1}$, ${\overset{\sim }{\boldsymbol{\beta}}}_0={\overset{\sim }{\boldsymbol{\varSigma}}}_0\left({\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\beta}^{-2}{\boldsymbol{\beta}}_0-{\boldsymbol{X}}^T{\boldsymbol{D}}_{\omega}\left(\sum_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right)+{\boldsymbol{X}}^T{\boldsymbol{D}}_{\omega}\boldsymbol{l}\right)$. It is important to point out that if we use a prior for β ∝ Constant (improper uniform distribution), then in ${\overset{\sim }{\boldsymbol{\varSigma}}}_0$ and ${\overset{\sim }{\boldsymbol{\beta}}}_0$ we need to make 0 the term ${\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\beta}^{-2}$. Finally, the full conditional) posterior of β is

$$ \boldsymbol{\beta} \left|\boldsymbol{ELSE}\right.\sim {N}_I\left({\overset{\sim }{\boldsymbol{\beta}}}_0,{\overset{\sim }{\boldsymbol{\varSigma}}}_0\right) $$

Polygenic effects ( b _h )

Now the full conditional posterior of b _h is given as

$$ {\displaystyle \begin{array}{l}L\left({\boldsymbol{b}}_h| ELSE\right) \\ {}\propto \exp \left(-\frac{1}{2}{\left(-\boldsymbol{l}+\boldsymbol{X}\boldsymbol{\beta } +\sum \limits_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right)}^T{\boldsymbol{D}}_{\omega}\left(-\boldsymbol{l}+\boldsymbol{X}\boldsymbol{\beta } +\sum \limits_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right)\right)P\left({\boldsymbol{b}}_{\boldsymbol{h}}|{\sigma}_{b_h}^2\right) \end{array}} $$

$$ \propto \mathit{\exp}\left\{-\frac{1}{2}\ \left[{\boldsymbol{b}}_h^T\left({\sigma}_b^{-2}{\boldsymbol{G}}^{-1}+{\boldsymbol{Z}}_h^T{\boldsymbol{D}}_{\omega }{\boldsymbol{Z}}_h\right){\boldsymbol{b}}_h-2\ {\left({\boldsymbol{Z}}_h^T{\boldsymbol{D}}_{\omega}\boldsymbol{l}-{\boldsymbol{Z}}_h^T{\boldsymbol{D}}_{\omega}\boldsymbol{X}\boldsymbol{\beta } \right)}^T\ {\boldsymbol{b}}_h\right]\right\} $$

$$ \propto \mathit{\exp}\left\{-\frac{1}{2}\ {\left({\boldsymbol{b}}_h-{\overset{\sim }{\boldsymbol{b}}}_h\right)}^T{\boldsymbol{F}}_h^{-1}\left({\boldsymbol{b}}_h-{\overset{\sim }{\boldsymbol{b}}}_h\right)\right\} $$

This implies that the full conditional posterior of b _h is

$$ f\left({\boldsymbol{b}}_h| ELSE\right)\sim N\left({\overset{\sim }{\boldsymbol{b}}}_{\boldsymbol{h}}={\boldsymbol{F}}_{\boldsymbol{h}}\left({\boldsymbol{Z}}_h^T{\boldsymbol{D}}_{\omega}\boldsymbol{l}-{\boldsymbol{Z}}_h^T{\boldsymbol{D}}_{\omega }{\boldsymbol{\eta}}^h\right),{\boldsymbol{F}}_{\boldsymbol{h}}={\left({\sigma}_{b_h}^{-2}{\boldsymbol{G}}_h^{-1}+{\boldsymbol{Z}}_h^T{\boldsymbol{D}}_{\omega }{\boldsymbol{Z}}_h^T\right)}^{-1}\right) $$

with h = 1 , 2, η ¹ = Xβ + Z ₂ b ₂ and η ² = Xβ + Z ₁ b ₁.

Variance of polygenic effects $\left({\sigma}_{b_h}^2\right).$

Next, the conditional distribution of ${\sigma}_{b_h}^2$ is obtained. If ${\sigma}_{b_h}^2\sim {\chi}^{-2}\left({\nu}_h,{S}_h\right)\left( shape and\ scale\right)$, then

$$ P\left({\sigma}_{b_h}^2| ELSE\right)\propto \frac{1}{{\left({\sigma}_{b_h}^2\right)}^{\frac{\nu_h+{n}_h}{2}+1}}\exp \left(-\frac{{\boldsymbol{b}}_h^T{\boldsymbol{G}}_h^{-1}{{\boldsymbol{b}}_{\boldsymbol{h}}}_h+{\nu}_h{S}_h}{2{\sigma}_{b_h}^2}\right) $$

This is the kernel of the scaled inverted χ ² distribution; therefore, the full conditional posterior is

$$ f\left({\sigma}_{b_h}^2| ELSE\right)\sim {\chi}^{-2}\left({\overset{\sim }{\nu}}_{\boldsymbol{h}}={\nu}_h+{n}_h,{\overset{\sim }{S}}_{\boldsymbol{b}}=\left({\boldsymbol{b}}_h^T{\boldsymbol{G}}_h^{-1}{\boldsymbol{b}}_h+{\nu}_h{S}_h\right)/{\nu}_b+{n}_h\right) $$

Threshold effects ( γ )

The density of the full conditional posterior distribution of the cth threshold, γ _c , is

$$ P\left(\boldsymbol{\gamma} |\boldsymbol{ELSE}\right)\propto P\left(\boldsymbol{y}|\boldsymbol{l},\boldsymbol{\gamma} \right)P\left(\boldsymbol{\gamma} \right) $$

$$ \propto \prod_{i=1}^I\prod_{j=1}^J\prod_{t=1}^{n_{ij}}\sum_{c=1}^CI\left({y}_{ij t}=c\right)I\left({\gamma}_{c-1}<{l}_{ij t}<{\gamma}_c\right)I\left(\boldsymbol{\gamma} \in \boldsymbol{T}\right) $$

(4.A.1)

If Eq. (4.A.1) is seen as a function of γ _c , it is evident that the value of γ _c must be larger than all the l _ijt |y _ijt  = c and smaller than all the l _ijt |y _ijt  = c + 1. Hence, as a function of γ _c , Eq. (4.A.1) leads to the uniform density

$$ P\left({\gamma}_c\left| ELSE\right.\right)=\frac{1}{\mathit{\min}\left({l}_{ijt}\left|{y}_{ijt}=c\right.+1\right)-\mathit{\max}\left({l}_{ijt}\left|{y}_{ijt}=c\right.\right)}I\left(\boldsymbol{\gamma} \in \boldsymbol{T}\right) $$

(4.A.2)

Equation (4.A.2) corresponds to a uniform distribution on the interval [min{min (l _ijt |y _ijt  = c + 1), γ _c + 1, γ _max}, max{max(l _ijt |y _ijt  = c), γ _c − 1, γ _min }] (Albert and Chib 1993; Sorensen et al. 1995).

Variance of location effects (${\sigma}_{\beta}^2\Big)$

If we give ${\sigma}_{\beta}^2\sim {\chi}^{-2}\left({\nu}_{\beta },{S}_{\beta}\right)\left( shape and\ scale\right)$, then

$$ P\left({\sigma}_{\beta}^2| ELSE\right)\propto P\left({\sigma}_{\beta}^2\right)P\left(\boldsymbol{\beta} |\ {\sigma}_{\beta}^2\right)=\frac{1}{{\left({\sigma}_{\beta}^2\right)}^{\frac{\nu_{\beta }}{2}+1}}\mathit{\exp}\left(-\frac{\nu_{\beta }{S}_{\beta }}{2{\sigma}_{\beta}^2}\right)P\left(\boldsymbol{\beta} |\ {\sigma}_{\beta}^2\right) $$

$$ \propto \frac{1}{{\left({\sigma}_{\beta}^2\right)}^{\frac{\nu_{\beta }+I}{2}+1}}\mathit{\exp}\left(-\frac{{\left(\boldsymbol{\beta} -{\boldsymbol{\beta}}_0\right)}^T{\boldsymbol{\varSigma}}_0^{-1}\left(\boldsymbol{\beta} -{\boldsymbol{\beta}}_0\right)+{\nu}_{\beta }{S}_{\beta }}{2{\sigma}_{\beta}^2}\right) $$

This is the kernel of the scaled inverted χ ² distribution; therefore, the full conditional posterior is

$$ {\sigma}_{\beta}^2\mid ELSE\sim {\chi}^{-2}\left({\overset{\sim }{\nu}}_{\beta }={\nu}_{\beta }+I,{\overset{\sim }{S}}_{\beta }=\left[{\left(\boldsymbol{\beta} -{\boldsymbol{\beta}}_0\right)}^T{\varSigma}_0^{-1}\left(\boldsymbol{\beta} -{\boldsymbol{\beta}}_0\right)+{\nu}_{\beta }{S}_{\beta}\right]/{\nu}_{\beta }+I\right) $$

Appendix B: Derivation of Full Conditional Distributions for Model BNBR

Full conditional for β ^∗

$$ f\left(\ {\boldsymbol{\beta}}^{\ast}| ELSE\right)=\prod_{i=1}^I\prod_{j=1}^J\prod_{t=1}^{n_{ij}}\mathit{\Pr}\left({Y}_{ij t}={y}_{ij t}|{\boldsymbol{x}}_i^T,r,{\omega}_{ij t},{b}_{1j},{b}_{2 ij}\right)f\left(\ {\boldsymbol{\beta}}^{\ast}\right) $$

$$ {\displaystyle \begin{array}{l}\propto \mathit{\exp}\left({\boldsymbol{\kappa}}^T\boldsymbol{X}\ {\boldsymbol{\beta}}^{\ast}+{\boldsymbol{\kappa}}^T\sum \limits_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h-\frac{1}{2}{\left(\boldsymbol{X}\ {\boldsymbol{\beta}}^{\ast}+\sum \limits_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right)}^T\right.\\ {}\left.{\boldsymbol{D}}_{\omega}\left(\boldsymbol{X}\ {\boldsymbol{\beta}}^{\ast}+\sum \limits_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h\right)-\frac{1}{2}{\left(\ {\boldsymbol{\beta}}^{\ast}-{\boldsymbol{\beta}}_0\right)}^T{\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\beta}^{-2}\left(\ {\boldsymbol{\beta}}^{\ast}-{\boldsymbol{\beta}}_0\right)\right)\end{array}} $$

$$ \propto \mathit{\exp}\left(-\frac{1}{2}\left[{\boldsymbol{\beta}}^{\ast T}\left({\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\beta}^{-2}+{\boldsymbol{X}}^T{\boldsymbol{D}}_{\omega}\boldsymbol{X}\right)\ {\boldsymbol{\beta}}^{\ast}-2{\left({\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\beta}^{-2}{\boldsymbol{\beta}}_0-{\boldsymbol{X}}^T{\boldsymbol{D}}_{\omega} \sum_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h+{\boldsymbol{X}}^T\boldsymbol{\kappa} \right)}^T\ {\boldsymbol{\beta}}^{\ast}\right]\right) $$

$$ \propto \mathit{\exp}\left(-\frac{1}{2}\left[{\left(\ {\boldsymbol{\beta}}^{\ast}-{\overset{\sim }{\boldsymbol{\beta}}}_0\right)}^T{\overset{\sim }{\boldsymbol{\varSigma}}}_0^{-1}\left(\ {\boldsymbol{\beta}}^{\ast}-{\overset{\sim }{\boldsymbol{\beta}}}_0\right)\right]\right)\propto N\left({\overset{\sim }{\boldsymbol{\beta}}}_0,{\overset{\sim }{\boldsymbol{\varSigma}}}_0\right) $$

where ${\overset{\sim }{\boldsymbol{\varSigma}}}_0={\left({\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\beta}^{-2}+{\boldsymbol{X}}^T{\boldsymbol{D}}_{\boldsymbol{\omega}}\boldsymbol{X}\right)}^{-1},$ ${\overset{\sim }{\boldsymbol{\beta}}}_0={\overset{\sim }{\boldsymbol{\varSigma}}}_0\left({\boldsymbol{\varSigma}}_0^{-1}{\sigma}_{\beta}^{-2}{\boldsymbol{\beta}}_0-{\boldsymbol{X}}^T{\boldsymbol{D}}_{\omega}\sum_{h=1}^2{\boldsymbol{Z}}_h{\boldsymbol{b}}_h+{\boldsymbol{X}}^T\boldsymbol{\kappa} \right)$.

Full conditional for ω _ijt

$$ f\left({\omega}_{ijt}| ELSE\right)\propto \exp \left[-\frac{\omega_{ijt}{\left({\boldsymbol{x}}_i^T{\boldsymbol{\beta}}^{\ast }+{b}_{1j}+{b}_{2 ij}\right)}^2}{2}\right]f\left({\omega}_{ijt};{y}_{ijt}+r,0\right) $$

$$ \propto \exp \left[-\frac{\omega_{ijt}{\left({\boldsymbol{x}}_i^T{\boldsymbol{\beta}}^{\ast }+{b}_{1j}+{b}_{2 ij}\right)}^2}{2}\right]f\left({\omega}_{ijt};{y}_{ijt}+r,0\right)\propto PG\left({y}_{ijt}+r,{\boldsymbol{x}}_i^T{\boldsymbol{\beta}}^{\ast }+{b}_{1j}+{b}_{2 ij}\right) $$

Full conditional for b ₁

Defining η ¹ = X β ^∗ + Z ₂ b ₂, the conditional distribution of b ₁ is given as

$$ f\left({\boldsymbol{b}}_1| ELSE\right)\propto \mathit{\exp}\left({\boldsymbol{\kappa}}^T{\boldsymbol{Z}}_1{\boldsymbol{b}}_1-\frac{1}{2}{\left({\boldsymbol{Z}}_1{\boldsymbol{b}}_1+{\boldsymbol{\eta}}^1\right)}^T{\boldsymbol{D}}_{\omega}\left({\boldsymbol{Z}}_1{\boldsymbol{b}}_1+{\boldsymbol{\eta}}^1\right)\right)\ f\left({\boldsymbol{b}}_1|{\sigma}_{b_1}^2\right) $$

$$ \propto \mathit{\exp}\left\{-\frac{1}{2}\ \left[{\boldsymbol{b}}_1^T\left({\sigma}_{b_1}^{-2}{\boldsymbol{G}}_1^{-1}+{\boldsymbol{Z}}_1^T{\boldsymbol{D}}_{\omega }{\boldsymbol{Z}}_1\right)\boldsymbol{u}-2\ {\left({\boldsymbol{Z}}_1^T\boldsymbol{\kappa} -{\boldsymbol{Z}}_1^T{\boldsymbol{D}}_{\omega }{\boldsymbol{\eta}}^1\right)}^T\ {\boldsymbol{b}}_1\right]\right\} $$

$$ \propto \exp \left\{-\frac{1}{2}{\left({\boldsymbol{b}}_1-{\overset{\sim }{\boldsymbol{b}}}_1\right)}^T{\boldsymbol{F}}_1^{-1}\left({\boldsymbol{b}}_1-{\overset{\sim }{\boldsymbol{b}}}_1\right)\right\}\sim N\left({\overset{\sim }{\boldsymbol{b}}}_1,{\boldsymbol{F}}_1\right) $$

where${\boldsymbol{F}}_1={\left({\sigma}_{b_1}^{-2}{\boldsymbol{G}}_1^{-1}+{\boldsymbol{Z}}_1^T{\boldsymbol{D}}_{\omega }{\boldsymbol{Z}}_1\right)}^{-1}\ \mathrm{and}\ {\overset{\sim }{\boldsymbol{b}}}_1={\boldsymbol{F}}_1\left({\boldsymbol{Z}}_1^T\boldsymbol{\kappa} -{\boldsymbol{Z}}_1^T{\boldsymbol{D}}_{\omega }{\boldsymbol{\eta}}^1\right)$.

Full conditional for ${\sigma}_{b_h}^2$

$$ f\left({\sigma}_{b_h}^2| ELSE\right)\propto \frac{1}{{\left({\sigma}_{b_h}^2\right)}^{\frac{\nu_{b_h}+{n}_{b_h}}{2}+1}}\mathit{\exp}\left(-\frac{{\boldsymbol{b}}_h^T{\boldsymbol{G}}_h^{-1}{\boldsymbol{b}}_h+{\nu}_{b_h}{S}_{b_h}}{2{\sigma}_{b_h}^2}\right) $$

$$ \propto {\chi}^{-2}\left({\overset{\sim }{\nu}}_b={\nu}_{b_h}+{n}_{b_h},{\overset{\sim }{S}}_b=\left({\boldsymbol{b}}_h^T{\boldsymbol{G}}_h^{-1}{\boldsymbol{b}}_h+{\nu}_{b_h}{S}_{b_h}\right)/{\nu}_{b_h}+{n}_{b_h}\right) $$

with ${n}_{b_1}=J$ and ${n}_{b_2}= IJ$.

Full conditional for ${\sigma}_{\beta^{\ast}}^2$

$$ f\left({\sigma}_{\beta^{\ast}}^2| ELSE\right)\propto \frac{1}{{\left({\sigma}_{\beta^{\ast}}^2\right)}^{\frac{\nu_{\beta^{\ast }}+I}{2}+1}}\exp \left(-\frac{{\left({\boldsymbol{\beta}}^{\ast }-{\boldsymbol{\beta}}_0\right)}^T{\boldsymbol{\varSigma}}_0^{-1}\left({\boldsymbol{\beta}}^{\ast }-{\boldsymbol{\beta}}_0\right)+{\nu}_{\beta^{\ast }}{S}_{\beta^{\ast }}}{2{\sigma}_{\beta^{\ast}}^2}\right) $$

$$ \propto {\chi}^{-2}\left({\overset{\sim }{\nu}}_{\beta^{\ast }}={\nu}_{\beta^{\ast }}+I,{\overset{\sim }{S}}_{\beta }=\left[{\left({\boldsymbol{\beta}}^{\ast }-{\boldsymbol{\beta}}_0\right)}^T{\boldsymbol{\varSigma}}_0^{-1}\left({\boldsymbol{\beta}}^{\ast }-{\boldsymbol{\beta}}_0\right)+{\nu}_{\beta^{\ast }}{S}_{\beta^{\ast }}\right]/{\nu}_{\beta^{\ast }}+I\right) $$

Full conditional for r

To make the inference of r, we first place a gamma prior on it as r ~ G(a ₀, 1/b ₀). Then we infer a latent count L for each Y ∼ NB(μ, r) conditional on Y and r. Since L ~ Pois(−r log(1 − π)), by construction we can use the Gamma-Poisson conjugacy to update r. Therefore, 

$$ f\left(r| ELSE\right)\propto f(r)\prod_{i=1}^I\prod_{j=1}^J\prod_t^{n_{ij}}f\left({y}_{ij t}|{L}_{ij t}\right)f\left({L}_{ij t}\right) $$

$$ \propto {r}^{a_0-1}\mathit{\exp}\left(-r{b}_0\right)\prod_{i=1}^I\prod_{j=1}^J\prod_t^{n_{ij}}{\left(- rlog\left(1-{\pi}_{ij}\right)\right)}^{L_{ij t}}\mathit{\exp}\left(r\ \mathit{\log}\left(1-{\pi}_{ij}\right)\right) $$

$$ \propto {r}^{a_0+{\sum}_{i=1}^I{\sum}_{j=1}^J{\sum}_{t=1}^{n_{ij}}{L}_{ij t}-1}\exp \left[-\left({b}_0-\sum_{i=1}^I\sum_{j=1}^J{\sum}_{t=1}^{n_{ij}}\log \left(1-{\pi}_{ij}\right)r\right)\right] $$

$$ \propto G\left({a}_0-\sum_{i=1}^I\sum_{j=1}^J{\sum}_{t=1}^{n_{ij}}\mathit{\log}\left(1-{\pi}_{ij}\right),\frac{1}{b_0+{\sum}_{i=1}^I{\sum}_{j=1}^J{\sum}_{t=1}^{n_{ij}}{L}_{ij t}}\right) $$

(4.A.5)

According to Zhou et al. (2012), the conditional posterior distribution of L _ijt is a Chinese restaurant table (CRT) count random variable . That is, L _ijt  ~ CRT(y _ijt , r) and we can sample it as ${L}_{ijt}={\varSigma}_{l=1}^{y_{ijt}}{d}_l,$ where ${d}_l\sim Bernoulli\left(\frac{r}{l-1+r}\right).$