A Decomposition Algorithm for Learning Bayesian Networks Based on Scoring Function

Algorithm 2

Algorithm for discovering Markov boundary.

(1) Input: Data set 𝐷; Variable set 𝑉=(𝑉₁,𝑉₂,…,𝑉_𝑛); Target variable 𝑇.

(2) Initialization: PC=𝑉⧵𝑇,SP=∅.

(3) Order-0 CI test: for each variable 𝑉_𝑖∈PC, if Ind(𝑉_𝑖;𝑇) is hold, then PC=PC⧵𝑉_𝑖,𝑆(𝑉_𝑖)=∅.

(4) Order-1 CI test: for each variable 𝑉_𝑖∈PC, if there is a variable 𝑉_𝑗∈PC⧵𝑉_𝑖 such that

 Ind(𝑉_𝑖;𝑇∣𝑉_𝑗), then PC=PC⧵𝑉_𝑖,𝑆(𝑉_𝑖)=𝑆(𝑉_𝑖)∪𝑉_𝑗.

(5) Find superset of spouses: for each variable 𝑉_𝑖∈PC, if there is a variable

 𝑉_𝑗∈𝑉⧵{𝑇∪PC}, such that ¬Ind(𝑉_𝑗;𝑇∣𝑆(𝑉_𝑗)∪𝑉_𝑖), then SP=SP∪𝑉_𝑗.

(6) Find parents and children of 𝑇: call the MMPC algorithm to get

 PC(𝑇)=MMPC(𝑇,𝐷(PC∪SP)). For each 𝑉_𝑖∈PC⧵PC(𝑇), if 𝑇∈MMPC(𝑉_𝑖,𝐷),

 then PC(𝑇)=PC(𝑇)∪𝑉_𝑖.

(7) Find spouses of 𝑇: for each variable 𝑉_𝑖∈PC(𝑇), if there is a variable

 𝑉_𝑗∈MMPC(𝑉_𝑖,𝐷)⧵{PC(𝑇)∪𝑇} and a subset 𝑍⊂PC∪SP⧵{𝑇∪𝑉_𝑗}, such that

 Ind(𝑇;𝑉_𝑗∣𝑍) and ¬Ind(𝑇;𝑉_𝑗∣𝑍∪𝑉_𝑖), then SP(𝑇)=SP(𝑇)∪𝑉_𝑗.

(8) Return MB(𝑇)=PC(𝑇)∪SP(𝑇).

(9) Output: A Markov boundary MB(𝑇) of 𝑇.