Bayesian Network Construction for Contingent Probabilistic Plan Evaluation

Bayesian Networks We can see in Figure 3 that a plan fragment m ay be repeated many times in a CPP. We can speed up the reasoning process by abstracti ng away the "unnecessary" nodes in the BN rep resenting a plan fragment a nd storing the abstract B:.J. In the abstraction process we need to retain the interface and eviden ce variables. The interface variables consist of input variables which furm the inte rface' with preceeding BNs a nd the output variables , included in thi s class are goal va riables, which form the intprface with the succeeding BT\l's or for querying purpose. The evidence variables are usrd for probability updating . For our current purposes, the'y a re observation variables whose values are used in later branching conditions. Int ernal variables afE~ the vari a bles which are not classified as interface or evidence. They can be abstracted away withuut affecting th e reasoning on th e whole BN-tree. 81 Definition 4 An abstract BN of a ronditi onal EN N is a conditional network M such that: (1) ,V and M have the sam e int erface and evidenct vanables. (2) Any node in N is in M (3) For any probability distribution assign ed to th e input variables, the resulting marginal probability distribution of tllP random varia bles com.mon to th! two networks arc the sa m. e. In (Lam 1994) the author propOSe's a techniqu e to abstract away a portion of a B;\! . That technique' tu rns out to be unsound : the resu lting network does not represent the margin alized probability distribution of thr original proba.bility distribution. Wp propose a less ex tensive but sound approach of abstract ion. For exampl e, th r t ra nsfurm a tion in Figures 6.(a.) is useful in temp oral reasoning. Th e sequence' of nodes (1) -.. (2) -.. ... (n ) can occur bpcause of the persistence rul es. Assume nod e i has mi values . The clustering algori thm would create n 1 clusters of ::;ize rno x mi x mi+l, i = 1, ... , n l. The network after transformation has onl y one clus ter of sizp ma x ml x m n · In most cases, the new neb·vork is more efficil'nt than the original. In Figure 6.( a) we assume that nodes (2) , ... , (n 1) are inte rnal nodes, thpre are no a rrows connected to them other than the ones shown in the figure and node (n) has only t wo incoming arcs. The new link m atrix of node (11) contains Pr(xnlxl, xa), wh ere Xi is the RV a.sso ciaterl wi th node (i) . Two other tra nsforrllati ons are g i ven in Figurps 6. (b) and (c ). In Figure 6.(b) we assu me th at there a re no arrows conn!xted tu (2) other than the unes shown in the figure. In ordrl' to use the t ransform a tion in Figure 6.(c), we require th at the a rc departing from (0) in the figure is t he unly one conn ected to (0) . We also can use soml' tr ansform at ions , which arE' described by Shachter (Shachter 1988) , for abs tr action purpose . In the full pap er, we prove the suundness of our abstraction [,ules.