However, only the method with αijk = 1/2 converges to zero error for a uniform distribution. From Fig. 6, for a small network, performances with de Campos’s method and MI are more adversely affected than those with the other methods because they have no strong consistency. 3 Experimentally Obtained Result with the Alarm Network To evaluate a large network, we first used the Alarm network because it is widely known as a benchmark structure for the evaluation of learning Bayesian networks. The Alarm network includes 37 variables and 46 edges.

If it is greater (smaller) than the critical value, then the null is rejected (cannot be rejected) (Agresti 2002; Spirtes et al. 2000). f=(|X|−1)(|Y |−1) , 2 = Xst (7) x∈X,y∈Y where Oxy (Exy ) is the number of records (expected to be if the null was correct) for which X = x, Y = y, and |X| and |Y | are the corresponding cardinalities. If the null is correct, P (x, y) = P (x) · P (y), ∀x ∈ X, y ∈ Y . We expect that Exy /N = (Ex /N ) · (Ey /N ), ∀x ∈ X, y ∈ Y and Exy = Ex · Ey /N for Ex and Ey , which are the numbers of records in which X = x and Y = y, respectively, and 2 is greater than a critical value where N is the total number of records.

Besides, algorithm showing good performance in the first experiment may performs terrible and require much more samples to achieve certain KL divergence. For example, QMAP can learn notably low KL divergence parameters with small data set. 2), extremely large number of samples (more than 100) are needed by QMAP algorithm. Learning Bayesian Network Parameters from Small Data Set 43 Fig. 8. 3 Time Consumption Analysis To show the time consumption of each algorithm, we continue the experiments and calculate the average running time of algorithm.