The name is absent

24

conditional posterior distribution. That is, from the posterior distribution for
the linear regression:

G

υ_ij -r_i= x[β_j +         = ∂)μ_jg + ⅛ for i = 1,... ,n,

s=ι

with response Pi = υ_ij - ri. The residuals are eɪ,... e_n ~ N(0, σ²) and the prior
is the multivariate normal distribution resulting from the product of (2.10)
and the prior distribution of (μ_jι,   , μ_jβ) when marginalizing with respect to

φ. The latter is a multivariate normal distribution of dimension G with mean
zero and covariance matrix σ²_μIo + 1g1g^cγ⅛> where Ig is the identity matrix of
dimension G and lɑ is the column vector of length G with all entries equal to
1.

(b) For each i = 1,... ,n, we sample r⅛ from the complete conditional posterior dis-
tribution, or, equivalently, from the posterior distribution for a linear regression
with response y*_j* = υ_ij - (χTβ_j + μ_t,w_ij),

y*_j* = r_i + e_j, for; = 1,...,7,

with eɪ,... Ej ~ ΛΓ(O, σ∣) and prior r_i ~ ΛΓ(O, σ²).

(c) We update β_j with a random walk Metropolis-Hasting transition probability. We
generate β ~ N(β_j,c²) where c > 0 and compute the posterior distribution of
β_j (marginalizing with respect to w):

p(βj I μ, r, p, z) ex f_βj(∕3₇) n”₌₁ ^ι⅛.,_σ2 (‰ι,_j∙ - r_i - ∑°=₁ Pj_gμ_ja)
~ ^χTβ_j,σl (θzij,j ~^ri~ Σg=lPjgμjg^,)

where Φ_μσ2 (ɪ) represents the normal cdf with mean μ and variance σ² evaluated
at x. Let A = p(β ∣ μ,r,p, z)∕p(∕¾ ∣ μ,r, p,z). With probability min{l,Λ} we
replace β_j by β.

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