9. In Example 4.9 (store use), introduce latent data via logistic sampling, namely Wi logistic(Xi, 1)I (0,) yi = 1, Wi logistic(Xi, 1)I

9. In Example 4.9 (store use), introduce latent data via logistic sampling, namely Wi ∼ logistic(Xiβ, 1)I (0,∞) yi = 1, Wi ∼ logistic(Xiβ, 1)I (−∞, 0) yi = 0,

and introduce variable weights as in Wi ∼ logistic(Xiβ, 1/λi )I (0,∞) yi = 1, Wi ∼ logistic(Xiβ, 1/λi )I (−∞, 0) yi = 0, λi ∼ Ga(ν/2, ν/2)
with ν = 4. Compare the pattern of weights λi to that of the Monte Carlo estimates of the CPO. Also apply the shifted intercept model to these data, namely yi ∼ Bern(πi ), logit(πi ) = bGi + δ2β2xi2 + δ3β3xi3 · · ·+δpβpxip, Gi ∼ Categorical(ω1, ω2, ω3), b1 = β1 − η (when Gi = 1), b2 = β1 (when Gi = 2), b3 = β1 + η (when Gi = 3), with ω1 = ω3 = 0.025, andη > 0 as an unknown parameter. The latter is best implemented with a mildly informative prior such as η ∼ Ga(1, 1) or η ∼ N(0, 1)I (0, )
in order to prevent numerical overflow in the regression.