D. Heckerman and J. Breese. Causal independence for probability assessment and inference using Bayesian networks. Technical Report MSR-TR-94-08, Microsoft Research, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....simplicity we will assume decomposable combining rules . Such rules can be expressed using a set of separate, deterministic nodes in the support network, as shown in Figure 4. Most combining rule commonly employed in Bayesian networks such as noisy or or linear regression are decomposable (cp. [10]) Decomposable combining rules imply that for each node x 2 N there exist at most one clause c and a substitution s.t. body(c ) LH(B) and head(c ) x. Thus, while the same clause c can induce more than one node in N , all of these nodes have identical local structure: the associated pdfs ....

D. Heckerman and J. Breese. Causal Independence for Probability Assessment and Inference Using Bayesian Networks. Technical Report MSR-TR-94-08, Microsoft Research, 1994.

....Combining Rules We assumed decomposable combining rules. Such rules can be expressed using a set of separate, deterministic nodes in the support network, as shown in Figure 5. Most combining rules commonly employed in Bayesian networks such as noisy or or linear regression are decomposable (cp. HB94] Decomposable combining rules imply that for each node x 2 N there exist at most one clause c and a substitution s.t. body(c ) LH(B) and head(c ) x. Thus, while the same clause c can induce more than one node in N , all of these nodes have identical local structure: the associated ....

D. Heckerman and J. Breese. Causal Independence for Probability Assessment and Inference Using Bayesian Networks. Technical Report MSR-TR-94-08, Microsoft Research, 1994.

....the behavior of the gates that in a FT represent interactions between sub systems. Of particular attention for reliability aspects is one peculiar modeling feature often used in building BN models: noisy gates. The most common kind of noisy gate is the noisy or model [15] and its generalizations [7]; this kind of model can be profitably used in dependability, since it allows a simple probabilistic generalization of boolean gates of a FT. Consider, in the PLC case study, a situation in which more fault tolerance is added to the system by means of a third spare power supplier that is ....

D. Heckerman and J.S. Breese. Causal independence for probability assessment and inference using bayesian networks. IEEE Transactions on Systems, Man and Cybernetics, 26(6):826--831, 1996.

....we present results for approx mpe(i) on larger networks (Figure 16) Algorithm elim mpe was intractable on these problems. The most apparent phenomenon here is that the approximation improves with decreasing noise q, i.e. U=L 1 for q 0. In 4 Noisy OR is an example of causal independence [32] which implies that several causes (parent nodes) contribute independently to a common effect (child node) 29 0 20 40 60 80 100 0 0.10.20.30.40.50.60.70.80.9 1 Noise q i=8 i=14 i=20 of instances U L in [x , x ] 0.1 q q 0.5 X 74 16 48 27 4 10 15 7 1 2 3 ....

....small parent sets since the minibucket and the belief propagation approaches are, in general, time and space exponential in the parent set. This limitation can be removed by using the specific structure of deterministic CPTs in the coding networks, which is a special case of causal independence [32, 59]. Such networks can be transformed into networks having families of size three only. Indeed, in coding practice, the belief propagation algorithm exploits the special structure of the CPTs and is linear in the family size. 11 Conclusions The paper describes a new approximate inference scheme, ....

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D. Heckerman and J. Breese. Causal independence for probability assessment and inference using Bayesian networks. Technical Report MSR-TR-94-08, Microsoft Research, 1995.

....that one of the parents is permanently on, so that the child can turn on spontaneously . This leak node can be used as a catch all for all other, unmodelled causes. It is straightforward to generalize the noisy OR gate to nonbinary variables and other functions such as AND [Hen89, Sri93, HB94] It is also possible to loosen the assumption that all the causes are independent [MH97] Another popular compact representation for CPDs in the UAI community is a decision tree [BFGK96] This is a stochastic generalization of the concept of canalyzing function [Kau93] popular in the boolean ....

D. Heckerman and J.S. Breese. Causal independence for probability assessmentand inference using bayesian networks. IEEE Trans. on Systems, Man and Cybernetics, 26(6):826--831, 1994.

....some i s, additional factorizations are usually introduced. These include decision tree and decision graph models (Friedman and Goldszmidt 1996; Chickering, Meek, and Heckerman, 1997) noisy or gates, leaky noisy or gates, max gates and causal independence models (Pearl, 1988, Henrion, 1987, and Heckerman and Breese, 1996). These models share the following characteristic. For each variable x i in the Bayesian network, a subset of k i states of p i are designated as reference states. The components of B n;m : Theta ae R n R m are defined by x a i jp b i ;u c i = f i ( x a i jp 0 i ; ....

Heckerman, D. and Breese, J. (1996). Causal independence for probability assessment and inference using Bayesian networks. IEEE, Systems, Man, and Cybernetics, 26:826--831.

....problems Next, we ran a set of experiments on randomly generated networks whose CPTs have a noisy OR structure. Noisy OR CPTs assume a disjunctive interaction between boolean variables where several causes contribute independently to a common effect (a property known as causal independence [27]) A noisy OR CPT defines a logical OR gate disturbed by noise as follows: given a child node x, and its parents y 1 , y n , each y i is associated with a noise parameter q i , defined as q i = P (x = 0jy i = 1; y k = 0) for k 6= i. The conditional probabilities are defined as follows [43] P ....

....parent sets since the minibucket approach and the belief propagation approaches are, in general, time and space exponential in the parent set. This limitation can be eliminated by using the specific structure of deterministic CPTs in coding networks, which is a special case of causal independence [27, 55]. Such networks can be transformed into networks having families of size three only. Indeed, in coding practice, the belief propagation algorithm exploits the special structure of CPTs and is linear in the family size. 66 11 Conclusions This paper has described the general framework of ....

[Article contains additional citation context not shown here]

D. Heckerman and J. Breese. Causal independence for probability assessment and inference using Bayesian networks. Technical Report MSR-TR-94-08, Microsoft Research, 1995.

....chain. BeliefNetwork B f Variable z f parents running(dt=30s latest=t 1)f this variable g g g Belief network C, like A, makes use of distributions which must read their descriptions from the input stream. The NoisyOr conditional distribution (described, for example, by Heckerman and Breese [7]) is a discrete distribution, but instead of representing the distribution as a table, a few parameters are specified from which an internal representation is generated by the input function associated with the NoisyOr type. Network C also contains a variable which is not of the standard Variable ....

Heckerman, D., and J. Breese. (1995) "Causal Independence for Probability Assessment and Inference Using Bayesian Networks." Microsoft Research Technical Report MSR-TR-94-08.

....expensive. In this section, we describe an approximation wherein the repair probabilities can be computed without copying the original Bayesian network. This approximation relies on representing the behavior of the device using a form of causal independence (see, e.g. Srinivas [1993] and Heckerman and Breese [1996]) The notion of causal independence is illustrated in Figure 3. The effect of set of causes 3 In doing so, we assume that the direct effects of actions are deterministic. The more general case can be handled with influence diagrams in canonical form [Heckerman and Shachter, 1995] 6 e . ....

Heckerman, D. and Breese, J. (1996). Causal independence for probability assessment and inference using Bayesian networks. IEEE, Systems, Man, and Cybernetics, 26. to appear.

....for some i s, additional factorizations are usually introduced. These include decision tree and decision graph models (Friedman and Goldszmidt 1996; Chickering, Meek, and Heckerman, 1997) noisy or gates, leaky noisy or gates, max gates and causal independence models (Pearl, 1988; Henrion, 1987; Heckerman and Breese, 1996; Meek and Heckerman, 1997) These models share the following characteristic. For each variable x i in the DAG model, a subset of k i states of p i are designated as reference states. The components of B n;m : Theta ae R n R m are defined by x a i jp b i ;u c i = f i ( x a i jp 0 ....

Heckerman, D. and Breese, J. (1996). Causal independence for probability assessment and inference using Bayesian networks. IEEE, Systems, Man, and Cybernetics, 26:826--831.

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D. Heckerman and J. Breese. Causal independence for probability assessment and inference using Bayesian networks. Technical Report MSR-TR-94-08, Microsoft Research, 1995.

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: Heckerman D. and Breese J.S. Causal independence for probability assessment and inference using bayesian networks.

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