where N , IG and SB denote normal, inverse-gamma and scaled-Beta distributions, respec- tively. The lower and upper bounds of the scaled-Beta distribution are obtained as men- tioned in Section 2. Combining the likelihood and the prior distributions, the augmented joint posterior density can be written as, where the likelihood, based on GAL (0 , σ, γ ), uses the fact that given z and δ , the observed y is independent of the remaining parameters, because (8) determines y$_{i}$ given ( z, δ ) with probability 1. The conditional density of latent data z can be obtained from (11) and is given by π ( z | β, σ, ν, h, γ ) = ∏ n i =1 N ( z$_{i}$ | x ′ $_{i}$β + Aν$_{i}$ + C | γ | h$_{i}$, σBν$_{i}$ ). Additionally, the prior distributions for ( β, σ, γ, δ ) are assumed to be independent in equation (14). Using the preceding explanations, the “complete data posterior” in equation (14) can be expressed as, The objects of interest i.e., ( z, β, ν, h, σ, γ, δ ) can be sampled by deriving the conditional pos- terior densities from the complete data posterior (15) and judiciously using the full likelihood (9) as presented in Algorithm 1. 11 π ( z, β, ν, h, σ, γ, δ | y ) ∝ { n ∏ i =1 1 { ξ$_{y}$$_{i}$$_{−}$$_{1}$ < z$_{i}$ < ξ$_{y}$$_{i}$ } N ( z$_{i}$ | x ′ $_{i}$β + Aν$_{i}$ + C | γ | h$_{i}$, σBν$_{i}$ ) × E ( ν$_{i}$ | σ ) N $^{+}$( h$_{i}$ | 0 , σ $^{2}$) } N ( β | β$_{0}$, B$_{0}$ ) IG ( σ | n$_{0}$/ 2 , d$_{0}$/ 2) × SB ( γ | L, U ) N ( δ | δ$_{0}$, D$_{0}$ ) . (15) π ( z, β, ν, h, σ, γ, δ | y ) ∝ f ( y | z, β, ν, h, σ, γ, δ ) π ( z | β, ν, h, σ, γ, δ ) π ( ν | σ ) π ( h | σ ) π ( β ) × π ( σ ) π ( γ ) π ( δ ) ∝ { n ∏ i =1 f ( y$_{i}$ | z$_{i}$, β, ν$_{i}$, h$_{i}$, σ, γ, δ ) π ( ν$_{i}$ | σ ) π ( h$_{i}$ | σ ) } π ( z | β, σ, ν, h, γ ) × π ( β ) π ( σ ) π ( γ ) π ( δ ) ∝ { n ∏ i =1 f ( y$_{i}$ | z$_{i}$, δ ) π ( ν$_{i}$ | σ ) π ( h$_{i}$ | σ ) } π ( z | β, σ, ν, h, γ ) × π ( β ) π ( σ ) π ( γ ) π ( δ ) , (14)