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Score based learning (SBL) is a promising approach for learning Bayesiannetworks. The initial step in the majority of the SBL algorithms consists ofcomputing the scores of all possible child and parent-set combinations for thevariables. For Bayesian networks with continuous variables, a particular scoreis usually calculated as a function of the regression of the child over thevariables in the parent-set. The sheer number of regressions models to besolved necessitates the design of efficient numerical algorithms. In thispaper, we propose an algorithm for an efficient and exact calculation ofregressions for all child and parent-set combinations. In the proposedalgorithm, we use QR decompositions (QRDs) to capture the dependencies betweenthe regressions for different families and Givens rotations to efficientlytraverse through the space of QRDs such that all the regression models areaccounted for in the shortest path possible. We compare the complexity of thesuggested method with different algorithms, mainly those arising in all subsetregression problems, and show that our algorithm has the smallest algorithmiccomplexity. We also explain how to parallelize the proposed method so as todecrease the runtime by a factor proportional to the number of processorsutilized.

Solving All Regression Models For Learning Gaussian Networks Using Givens Rotations