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## Selection of Parameters

There are several tunable parameters in the AIS-BN algorithm. We base the choice of these parameters on the Central Limit Theorem (CLT). According to CLT, if Z1, Z2, ..., Zn are independent and identically distributed random variables with E(Zi) = and Var (Zi) = , i = 1, ..., n, then = (Z1 + ... + Zn)/n is approximately normally distributed when n is sufficiently large. Thus, P(   . t) =  e-x2/2dx  . (11)

Although this approximation holds when n approaches infinity, CLT is known to be very robust and lead to excellent approximations even for small n. The formula of Equation 11 is an ( , ) Relative Approximation, which is an estimate of that satisfies

P(   )  .

If has been fixed, = . ( )  ,

where (z) =  e-x2/2dx. Since in our sampling problem, (corresponding to Pr( ) in Figure 2) has been fixed, setting to a smaller value amounts to letting / be smaller. So, we can adjust the parameters based on / , which can be estimated using Equation 3. It is also the theoretical intuition behind our recommendation wk 1/ in Section 3.1. While we expect that this should work well in most networks, no guarantees can be given here -- there exist always some extreme cases in sampling algorithms in which no good estimate of variance can be obtained.   Next: A Generalization of AIS-BN: Up: AIS-BN: Adaptive Importance Sampling Previous: Heuristic Initialization in AIS-BN
Jian Cheng 2000-10-01