By Glenn Shafer

Probabilistic specialist platforms emphasizes the fundamental computational rules that make probabilistic reasoning possible in professional platforms. the most important to computation in those structures is the modularity of the probabilistic version. Shafer describes and compares the relevant architectures for exploiting this modularity within the computation of earlier and posterior possibilities. He additionally shows how those related but diverse architectures observe to a large choice of alternative difficulties of recursive computation in utilized arithmetic and operations study.

The box of probabilistic professional platforms has persisted to flourish because the writer brought his lectures at the subject in June 1992, however the figuring out of join-tree architectures has remained lacking from the literature. This monograph fills this void by way of delivering an research of join-tree tools for the computation of past and posterior chances in trust nets. those equipment, pioneered within the mid to overdue Eighties, stay principal to the idea and perform of probabilistic professional structures. as well as merely probabilistic professional platforms, join-tree equipment also are utilized in professional structures according to Dempster–Shafer trust capabilities or on danger measures. adaptations also are used for computation in relational databases, in linear optimization, and in constraint delight.

This publication describes probabilistic specialist platforms in a extra rigorous and centred means than current literature, and gives an annotated bibliography that comes with tips to meetings and software program. additionally integrated are routines that might aid the reader start to discover the matter of generalizing from chance to broader domain names of recursive computation.

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**Example text**

Xi^^} for j = 2 , . . , k. Write w for {Xtl,... ,X^k}. 1. Q j j , . . , Qlk is a construction sequence for P^w. 2. Suppose c is a configuration of w. Suppose we modify the sequence Qi,---:Qn by deleting each Qi, and by slicing each of the other conditionals on w = c. Then the result is a construction sequence for P's posterior given w — c. Forward propagation in chains. As we have seen, it is trivial to reduce a belief net to a belief net for an initial segment. If the belief net is a chain, then with a bit of work we can also reduce it to a belief net for a final segment.

First domain. Right: two more head-tail pairs have been added. Each time, the new tail is contained in the existing domain, and the new head is disjoint from, it. and we say that the construction sequence represents this probability distribution. 1. 2 indicates one way that we can exploit a construction sequence. If we are interested only in the variables in di U • • • U di and not in the remaining variables—those in /ii+1 U • • • U hn—then we can simply omit the last n — i conditionals from the construction sequence: Q\- • • Qi is a construction sequence for the marginal probability distribution on d\ U • • • U ci,.

If such a join-tree cover does not exist, we will have to turn to alternative methods for marginalization, such as Markov-chain Monte Carlo. As we noted at the beginning of the chapter, there are heuristics that do produce reasonable choices for join-tree covers. Some of these heuristics do involve choosing an order for eliminating (summing out) the variables. This not only produces a join-tree cover; it also determines a placement of the factors in the join tree—each factor goes as close as possible to the root.