M. Peot and R. Shachter. Fusion and propogation with multiple observations in belief networks. Artificial Intelligence, 48:299--318, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the computation is a single marginal, P (X i jX j ) The general inference algorithms are de ned in terms of message passing on a tree. If the original graph has undirected cycles (loops) it must be converted to a socalled junction tree using triangulation [Kja90] or cutset conditioning [Pea88, PS91] The tree may be directed or undirected, the messages may be passed in parallel or sequentially 3 , and the computation of the messages may or may not involve a division operation. For instance, Pearl s algorithm [Pea88] was formulated for directed trees without division; the Hugin JLO ....

....to a single Gaussian using moment matching. The implementation of this in [Lau92] is numerically unstable, and has been improved in [LJ99] See also [Min01] 3 If messages are passed sequentially, the scheduling usually uses two passes, often called collect distribute or forwards backwards [PS91] 5 Undirected models are already parameterized in terms of potentials on cliques, so no conversion is necessary. Directed models are parameterized in terms of CPDs, but we can simply de ne the potentials as (X i ; Pa i ) P (X i jPa i ) However, this is only possible for some kinds of CPDs: ....

M. Peot and R. Shachter. Fusion and propogation with multiple observations in belief networks. Articial Intelligence, 48:299-318, 1991.

.... is in fact an HMM, then a single FB iteration (2TN message computations) will result in the exact posteriors, whereas it requires T iterations of the decentralized protocol (each iteration computing 2TN messages in parallel) to reach the same result; hence the centralized algorithm is more ecient [14]. For loopy graphs, it is not clear which protocol is better; it depends on whether local or global information is more important for computing the posteriors. It is also easy to see that the fully factorized version of BK is equivalent to a single FB pass of LBP applied to a modi ed DBN, as ....

M. Peot and R. Shachter. Fusion and propogation with multiple observations in belief networks. Arti- cial Intelligence, 48:299-318, 1991.

....marginals at the same time, we can use dynamic programming to avoid the redundant computation that would be involved if we used variable elimination repeatedly. If the underlying undirected graph of the BN is acyclic (i.e. a tree) we can use a local message passing algorithm due to Pearl [Pea88, PS91] This is a generalization of the well known forwards backwards algorithm for HMMs (chains) Rab89] If the BN has (undirected) cycles or loops , as in the water sprinkler example, local message passing algorithms run the risk of double counting, and may not converge. For example, the ....

M. Peot and R. Shachter. Fusion and propogation with multiple observations in belief networks. Articial Intelligence, 48:299-318, 1991. 18

....computational issues involved in performing inference in hybrid DBNs in Section 6.3. 6 Inference We shall discuss how to perform inference in hybrid networks using the join tree algorithm [LS88, Jen96] which works on undirected Markov trees. Similar results have been derived for directed trees [Pea88, PS91, AA96, DM95]. We start by reviewing the discrete case, and then show how to generalize this to handle Gaussian networks, and nally hybrid networks [LS88, LW89, Lau92, Ole93, Lau96] 7 B C D E F A ABC BC BCD CD CDE DE DEF S3 B C D E F A B C D E F A (a) b) c) d) C1 C2 C3 C4 S1 S2 Figure 3: a) The ....

M. Peot and R. Shachter. Fusion and propogation with multiple observations in belief networks. Articial Intelligence, 48:299-318, 1991.

....in the short term. In particular, if the DBN is in fact an HMM, then a single FB iteration (2TN message computations) will result in the exact posteriors, whereas it requires T iterations of the decentralized protocol (each iteration computing 2TN messages in parallel) to reach the same result [16]. It is also easy to see that the fully factorized version of BK is equivalent to a single FB pass of LBP applied to a modi ed DBN, as shown in Figure 1. For each slice, we create two mega nodes that contains all the (hidden) nodes in that slice. The messages coming into the rst mega node are ....

M. Peot and R. Shachter. Fusion and propogation with multiple observations in belief networks. Articial Intelligence, 48:299-318, 1991.

.... Gamma U i X jU i = u) 1 U1 Y1 Y2 e (x) e (X) X U2 e (X Y1) e (X Y2) e (U1 X) e (U2 X) we can write in general that X (x) X X (x) Theta Y j Y j X (x) For leaves, we just write X U i (u 1 ) 1, since there is no evidence below X. To handle evidence, we follow Peot and Shachter [PS91], by letting a node X send a message to itself, X X (x) We compute X by introducing X s parents, to break the dependence on the upstream evidence, and then summing them out. We partition the evidence above X into the evidence in each subtree above each parent U i . Pr(X = xje X ) X ....

....not converge [MWJ99] 3 If the graph is a polytree, we can pick an arbitrary node as root. In the first pass, we send messages to it. If we go with an arrow, the messages are messages; if we go against an arrow, the messages are messages. On the second pass, we send messages from the root. See [PS91] for details. If the graph is a regular tree (not a polytree) there already is a single root. Note that a junction tree can be made into a regular tree, by picking an arbitrary clique as the root. Hence the first pass will only consist of sending messages, and the second pass will only consist ....

M. Peot and R. Shachter. Fusion and propogation with multiple observations in belief networks. Artificial Intelligence, 48:299--318, 1991.

....computational issues involved in performing inference in hybrid DBNs in Section 5.3. 5 Inference We shall discuss how to perform inference in hybrid networks using the join tree algorithm [LS88, Jen96] which works on undirected Markov trees. Similar results have been derived for directed trees [Pea88, PS91, AA96, DM95]. We start by reviewing the discrete case, and then show how to generalize this to handle Gaussian networks, and finally hybrid networks [LS88, LW89, Lau92, Ole93, Lau96] 5.1 Pure discrete case We will associate a potential function with each clique, which is the joint probability of its ....

M. Peot and R. Shachter. Fusion and propogation with multiple observations in belief networks. Artificial Intelligence, 48:299--318, 1991.

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M. Peot and R. Shachter. Fusion and propogation with multiple observations in belief networks. Artificial Intelligence, 48:299--318, 1991.

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