Table 4. Conditional distributions used for updating via Gibbs sampling for Q epidemic quantities.

Parameter	Conditional distribution
λ i	$\text{Gamma}(a_{\lambda }+Q,{\widehat{b}}_{\lambda i}+\sum _{q=1}^Q\frac{{\theta }_{\mu ,q}{(x_{i,q}-{\mu }_q)}^2-{\theta }_{\nu ,q}{(y_{i,q}-{\nu }_q-{\beta }_q(x_{i,q}-{\mu }_q\left)\right)}^2}{2})$ for ${\widehat{b}}_{\lambda i}=b_{\lambda }$ for all modeling assumptions i in the NHW method,${\widehat{b}}_{\lambda i}=a_{\lambda }/m_{\lambda }$ for all modeling assumptions i in the SHW method and ${\widehat{b}}_{\lambda i}=a_{\lambda }/{\widehat{m}}_{\lambda i}$ in the IHW method.
μ q	$\text{Normal}(\stackrel{-}{\mu },{\sigma }_{\mu }^2)$ with ${\sigma }_{\mu }^2={({\sum }^{}{\theta }_{\mu ,q}{\lambda }_i+{\theta }_{\nu ,q}{\beta }_q^2{\sum }^{}{\lambda }_i+{\tau }_{0,q})}^{-1}\stackrel{-}{\mu }={\sigma }_{\mu }^2({\sum }^{}{\theta }_{\mu ,q}{\lambda }_i{x}_{i,q}-{\theta }_{\nu ,q}{\beta }_q{\sum }^{}{\lambda }_i(y_{i,q}-{\nu }_q-{\beta }_q{x}_{i,q})+{\tau }_{0,q}{x}_{0,q})$
v q	$\text{Normal}(\stackrel{-}{\nu },{\left({\theta }_{\nu ,q}{\sum }^{}{\lambda }_i\right)}^{-1})$ with $\stackrel{-}{\nu }=\frac{{\sum }^{}{\lambda }_i(y_{i,q}-{\beta }_q(x_{i,q}-{\mu }_q\left)\right)}{{\sum }^{}{\lambda }_i}$
β q	$\text{Normal}(\stackrel{-}{\beta },{\sigma }_{\beta }^2)$ with ${\sigma }_{\beta }^2={\left({\theta }_{\nu ,q}{\sum }^{}{\lambda }_i{(x_{i,q}-{\mu }_q)}^2\right)}^{-1}$ $\stackrel{-}{\beta }={\sigma }_{\beta }^2{\theta }_{\nu ,q}^{-1}{\sum }^{}{\lambda }_i(y_{i,q}-{\nu }_q)(x_{i,q}-{\mu }_q)$
θ μ, q for q ≠ 1	$\text{Gamma}(a_{\theta }+\frac{N}{2},b_{\theta }+\frac{{\sum }^{}{\lambda }_i{(x_{i,q}-{\mu }_q)}^2}{2})$
θ v, q	$\text{Gamma}(a_{\theta }+\frac{N}{2},b_{\theta }+\frac{{\sum }^{}{\lambda }_i{(y_{i,q}-{\nu }_q-{\beta }_q(x_{i,q}-{\mu }_q\left)\right)}^2}{2})$
τ 0, q	$\text{Gamma}(a_{\tau ,q}+1/2,b_{\tau ,q}+\frac{{(x_{i,q}-{\mu }_q)}^2}{2})$
φ 0, q	Gamma(a ϕ, q, b ϕ, q)