sampling, the test results reported in this paper do not include this improvement of the AIS-BN algorithm.

Figure 3 lists an algorithm that implements Step 7 of the basic AIS-BN algorithm listed in Figure 2. When we estimate Pr'( x i | pa(X i ), e), we only use the samples obtained at the current stage. One reason for this is that the information obtained in previous stages has ben absorbed by Pr k (X\E). The other reason is that in principle, each successive iteration is more accurate than the previous one and the importance function is closer to the optimal importance function. Thus, the samples generated by P 1 k +1 (X\E) are better than those generated by Pr k (X\E). Pr'(X, lpa(X, ), e) - Prk(X, lpa(X,), e) corresponds to the vector of first partial derivatives in the direction of the maximum decrease in the error. η( k ) is a positive function that determines the learning rate. When η(k) = 0 (lower bound), we do not update our importance function. When η( k ) = 1 (upper bound), at each stage we discard the old function. The convergence speed is directly related to η( k ). If it is small, the convergence will be very slow due to the large number of updating steps needed to reach a local minimum. On the other hand, if it is large, convergence rate will be initially very fast, but the algorithm will eventually start to oscillate and thus may not reach a minimum. There are many papers in the field of neural network learning that discuss how to choose the learning rate and let estimated importance function converge quickly to the destination function. Any method that can improve learning rate should be applicable to this algorithm. Currently, we use the following function proposed by Ritter et al. (1991)

$$\eta(\,k)=a\,\left({\frac{b}{a}}\right)^{k\,/k_{\mathrm{max}}}\,,$$
$$\left(10\right)$$

where a is the initial learning rate and b is the learning rate in the last step. This function has been reported to perform well in neural network learning (Ritter et al., 1991).

## 3.3 Heuristic Initialization In Ais-Bn

The dimensionality of the problem of Bayesian network inference is equal to the number of variables in a network, which in the networks considered in this paper can be very high. As a result, the learning space of the optimal importance function is very large. Choice of the initial importance function Pr 0 (X\E) is an important factor affecting the learning - an initial value of the importance function that is close to the optimal importance function can greatly affect the speed of convergence. In this section, we present two heuristics that help to achieve this goal.

Due to their explicit encoding of the structure of a decomposable joint probability distribution, Bayesian networks offer computational advantages compared to finite-dimensional integrals. A possible first approximation of the optimal importance function is the prior probability distribution over the network variables, Pr(X). We propose an improvement on this initialization. We know that the effect of evidence nodes on a node will be attenuated as the path length of that node to evidence nodes is increased (Henrion, 1989) and the most affected nodes are the direct ancestors of the evidence nodes. Initializing the ICPT tables of the parents of the evidence nodes to uniform distributions in our experience improves the convergence rate. Furthermore, the CPT tables of the parents of an evidence node E may be not favorable to the observed state e if the probability of E = e without