Bayesian Variable Selection for Multi-response Linear Regression - Technologies and Applications of Artificial Intelligence - page 84

probabilities of ʷ 8 , 6 , ʷ 9 , 5 , ʷ 9 , 6 ,and ʷ 9 , 9 are still larger than 0.5. Thus the proposed two-
layer Gibbs sampler can successfully recover supports correctly. The posterior means
of the coefficients for the selected variables are also shown in Table 2.

posterior probability of δ j

0 20 40 60 80 100 120 140 160 180 200

variable

Fig. 1. The posterior probabilities of ʴ j : P ( ʴ j =1 | Y ) estimated by the two-layer Gibbs sampler
in the simulated example

posterior probability of η jm

eta_[7,m] eta_[8,m] eta_[9,m]

eta_[11,m] eta_[12,m] eta_[13,m]

Fig. 2. The posterior probabilities of ʷ j,m : P ( ʷ j,m =1 |ʴ j =1 , Y ) obtained by the two-layer
Gibbs sampler in the simulated example

To compare with the other approaches, first the group-wise Gibbs sampler, Algo-
rithm 1, is used for the same simulation data. In the group-wise Gibbs sampler, indi-
cator variables, ʴ j ,j =1 , ,p, are only adopted in the model. To implement this

algorithm, the prior parameters set-up are chosen the same as these in the two-layer
Gibbs sampler, and the median probability criterion is also adopt for the posterior

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