Z. Li and B. D'Ambrosio. Efficient inference in Bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11, 55--81.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of interest. From this a posteriori convex set we can obtain probability intervals for each case of this variable. The paper is organized as follows: in section 2 we describe the problem of propagation of probabilities on Bayesian networks and how solve it using the variable elimination algorithm [33, 61, 20]; section 3 is devoted to study the use of probability trees to represent potentials in a compact way and how they can represent context specific independences; also, in this section we study how to build and how to operate with probability trees; in section 4 we shows basic notions about convex ....

Z. Li and B. D'Ambrosio. Efficient inference in Bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11:55--81, 1994.

....elimination ordering is NP complete. Symbolic probabilistic inference (SPI) views probabilistic inference as a combinatorial optimization problem, the optimal factoring problem. Probabilistic inference is the problem of finding an optimal factoring given a set of probabilistic distributions [SD90, LD94]. SPI is symbolic and querydriven. Differential approach compiles a Bayesian network into a multivariate polynomial and then computes the partial derivatives of this polynomial with respect to each variable [Da00] Once such derivatives aremadeavailable,onecancomputeanswerstoavery large class of ....

Z. Li and B. D'Ambrosio. Efficient Inference in Bayes Networks as a Combinatorial Optimization Problem. International Journal of Approximate Reasoning, 11, 55--81.

....property of com14 piling a theory into a backtrack free (i.e. greedy) theory, and their complexity is dependent on the induced width graph parameter. The algorithms are variations on known algorithms, and, for the most part, are not new in the sense that the basic ideas have existed for some time [8, 34, 31, 50, 28, 39, 32, 3, 45, 46, 48, 47]. Definition 2 (graph concepts) A directed graph is a pair, G = fV; Eg, where V = fX 1 ; Xng is a set of elements and E = f(X i ; X j )jX i ; X j 2 V; i 6= jg is the set of edges. If (X i ; X j ) 2 E, we say that X i points to X j . For each variable X i , the set of parent nodes of X i , ....

Z Li and B. D'Ambrosio. Efficient inference in bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11:1--58, 1994.

....reflected in observational error then we need take no further actions. If a sensor s error model and observation are reported separately or in the case of a human observer, both the accuracy of the observation made and the credibility of the sensor are required. For a fuller discussion of credibility and how to measure it see Schum (1994). The observational error cannot be removed from the dispersion for the Node of Interest in a sound manner. However, by setting states for credibility nodes to the Perfect state, we can effectively remove the uncertainty attributable to the lack of credibility for the experts and model sources. ....

Li, Z. and D'Ambrosio, B., 1994, Efficient inference in Bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning 11:55-81.

....as a a model for uncertainy in decision making and statistical inference [6] Several algorithms manipulate Bayesian networks to produce posterior values [8, 11, 13] We can identify two types of algorithms. There are algorithms that focus on algebraic operations, such as the SPI algorithm [9], the variable elimination algorithm [14] and the bucket elimination algorithm [3] And there are algorithms that focus on graphical properties, mostly dealing with junction trees [6] One of the advantages of junction tree algorithms is that it is possible to efficiently compute marginal ....

Z. Li and B. D'Ambrosio. Efficient inference in Bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11, 1994.

....we exploit. For a fixed collection of evidence, the posterior probability for all nodes in the network can be computed once the evidence has been distributed to all the clusters in the cluster tree. A Bayesian network can be used to structure query based computations of posterior probability [27, 52, 53]. For any given query, the posterior probability can be computed by marginalization, i.e. summing out the random variables not mentioned in the query. The remaining nodes are structured by the Bayesian network s factorization of the joint probability distribution, and the query can be computed by ....

Zhaoyu Li and Bruce D'Ambrosio. Efficient inference in Bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11:55--81, 1994.

....does not work, and more complex algorithms have been proposed. Two of the most efficient exact inference algorithms are the Lauritzen Spiegelhalter (LS) algorithm [Lauritzen and Spiegelhalter, 1988] and an optimal factoring algorithm based on the symbolic probabilistic inference (SPI) approach [Li and D Ambrosio, 1994]. Both of the algorithms have a time complexity that is exponential to the size of a general network. Both of the above mentioned algorithms have to use In proceedings of the Uncertainty in Artificial Intelligence Conference (UAI 95) pp. 376 385 August 18 20, 1995, Montreal, Canada a lot of ....

Li, Z. and D'Ambrosio, B. (1994). Efficient inference in Bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11(1):55 -- 81.

.... ) w n w O( n exp( w n Same as worst case Elimination Conditioning Average time Space worst case better than exp( n ) O( Worst case time knowledge compilation one solution Output Figure 12: Comparing elimination and conditioning basic ideas have existed for some time [8, 35, 33, 49, 30, 39, 34, 3, 45, 46, 48, 47]. What we are presenting here is a syntactic and uniform exposition emphasizing these algorithms form as a straightforward elimination algorithm. The presentation allows ideas and techniques to flow across the boundaries between areas of research. In particular, having noted that elimination ....

Z Li and B. D'Ambrosio. Efficient inference in bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11:1--58, 1994.

....a NP hard computation step over using one algorithm of each type. Furthermore, this technique re enforces a conjecture of Jensen and Jensen [JJ94] in that it still requires a single NP hard step to set up the structure on which inference is performed, as we show by confirming Li and D Ambrosio s [LD94] conjectured NP hardness of OFP. 1 Overview Bayesian Networks(BN) provide a standard way to represent a probability distribution on a series of discrete propositional variables. By taking advantage of independence information between the variables, BN s can reduce the amount of space necessary to ....

.... algorithm to recover the marginals of all the variables is known as the tree of cliques approach [LS88] Pea88] Nea90] Jen96] Another approach to the calculation of a marginal probability distribution on a set of target variables, called Symbolic Probabilistic Inference (SPI) is discussed in [LD94]. It involves solving the Optimal Factoring Problem (OFP defined in Section 4) for the target set of variables whose distribution you are interested in. The solution to the OFP is then used to combine the conditional probability tables that describe the Bayesian Network and extract the desired ....

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Zhaoyu Li and Bruce D'Ambrosio. Efficient Inference in Bayes Networks as a Combinatorial Optimization Problem. International Journal of Approximate Reasoning, 11:55--81, 1994.

....number of nodes and arcs in the graph. The posterior distribution of the target set is equal to the product of the distributions of the relevant nodes marginalized down to the target set. If we for a moment refer to the distributions as functions, then the SPI algorithm can be described as below, (Li D Ambrosio 1994): 1. Construct a function set A containing all the distributions of the network relevant for calculating the posterior distribution of the target set. 2. Add all pairs of functions of A to B, except pairs in which each function is a marginal probability distribution over one variable and the two ....

Li, Z. & D'Ambrosio, B. (1994), `Efficient inference in Bayes networks as a combinatorial optimization problem', International Journal of Approximate Reasoning.

....The computational work needed to perform this on line evaluation is so straightford that it lends itself to easy implementations on different software and hardware platforms. This approach shares some commonality with other methods that symbolically manipulate probability expressions, like SPI [3, 5]; it differs with SPI on the objective of such manipulations and, hence, on the results obtained. SPI explicates the notion of an arithmentic expression to state that belief network inference can be viewed as an expression factoring operation. This allows results from optimization theory to be ....

Z. Li and B.D. D'Ambrosio. Efficient Inference in Bayes Networks as a Combinatorial Optimization Problem. International Journal of Approximate Reasoning, 11:55--81, 1994.

....under uncertainty and have been used in a number of practical systems. Although there exists a number of efficient inference algorithms and implementations for probabilistic reasoning in Bayesian networks with discrete variables (for example, Pearl, 1988, Lauritzen and Spiegelhalter, 1988, Li and D Ambrosio, 1994, Dechter, 1996] few algorithms support efficient inference in hybrid Bayesian networks, Bayesian networks where continuous and discrete variables are intermixed. Exact probabilistic inference in hybrid networks can be reduced to taking multidimensional integrals in the same way that exact ....

....in hybrid Bayesian networks, Bayesian networks where continuous and discrete variables are intermixed. Exact probabilistic inference in hybrid networks can be reduced to taking multidimensional integrals in the same way that exact inference in discrete networks can be reduced to computing sums [Li and D Ambrosio, 1994, Dechter, 1996] However, computing integrals exactly is possible only for a restricted class of continuous functions. For example, one of the hybrid Bayesian network classes where exact probabilistic inference is possible are networks with Conditional Gaussian (CG) density functions [Lauritzen ....

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Li, Z. and D'Ambrosio, B. (1994). Efficient inference in Bayes networks as a combinatorial optimization problem.

....probabilities conditioned on an empty set of nodes. Submitted to the special issue of IEEE Computers (December 1996) Applications for Shared Memory Multiprocessors Probabilistic inference in a belief network can be most generally understood as an evaluation of joint probability sums [7]. Let X i be the set of query nodes, the nodes whose probabilities we want to obtain. The probability distribution p(X i ) the set of numbers, each being the probability of occurrence of a certain particular combination of states or values of the nodes in X i can be obtained from: p(X i ) ....

....The joint probability summation algorithm presented here represents the structure of computations in the modern algorithms for probabilistic inference. The two most popular algorithms are the Lauritzen Spiegelhalter (LS) algorithm [6] and the Symbolic Probabilistic Inference (SPI) algorithm [7] based on the optimal factoring approach described above. While the LS algorithm builds the whole join tree in the memory and evaluates the probabilities of all nodes in the network each time a query is made, the SPI algorithm is query oriented and computes only the conditional probabilities that ....

Zhaoyu Li and Bruce D'Ambrosio. Efficient inference in Bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11(1):55 -- 81, 1994.

....[D A94] LS88, Comments by W. S. Kendall] CGH95] 2.3 Exact Optimization Finding the best (highest probability) configuration. 2.3.1 Junction Tree Propagation Max propagation: DDP90] Daw92] 2.3. 2 Linear Programming Formalizations as linear or non linear programming problems: San94] LD94] 2.4 Approximation 2.4.1 Stochastic Simulation Markov Chain Monte Carlo based techniques (mostly Gibbs Sampling) GG84] CC87] CC90] Pea87b] DC93] DH93] JKK93] DH92] Hry90] Nea93] DKL95] Kja95b] KKR95] MC96] Of these, JKK93, JKK95, MC96, DKL95, JKK93, JKK95, ....

Zhaoyu Li and B. D'Ambrosio. Efficient inference in bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11(1):55--81, July 1994.

....does not work, and more complex algorithms have been proposed. Two of the most efficient exact inference algorithms are the Lauritzen Spiegelhalter (LS) algorithm [Lauritzen and Spiegelhalter, 1988] and an optimal factoring algorithm based on the symbolic probabilistic inference (SPI) approach [Li and D Ambrosio, 1994]. Both of the algorithms have a time complexity that is exponential to the size of a general network. Both of the above mentioned algorithms have to use To appear in Philippe Besnard and Steve Hanks, editors Uncertainty in Artificial Intelligence: Proceedings of the Eleventh Conference ....

Li, Z. and D'Ambrosio, B. (1994). Efficient inference in Bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11(1):55 -- 81.

....It should be clear then that such algorithms cannot be optimal because they are bound to miss some independences that could be useful computationally. This also seems to be the practice in the probabilistic literature, except possibly for some recent work on utilizing non structural independence [1, 18]. Linear Converging Diverging i j k i j k j i k Figure 6: There are three types of intermediate nodes on a given path. The type of a node is determined by its relation to its neighbors. A node is diverging if both neighbors are children. A node is linear if one neighbor is a parent and the other ....

Z. Li and B.D. D'Ambrosio. Efficient Inference in Bayes Networks as a Combinatorial Optimization Problem. International Journal of Approximate Reasoning, 11:55--81, 1994.

....about the values of other nodes (e.g. findings) Two types of approaches are taken to find the desired conditional probabilities. The first is exact algorithms, among which the two dominant ones are the Lauritzen Spiegelhalter (LS) algorithm [1, 2, 3, 4] and Symbolic Probabilistic Inference (SPI) [5]. For a general query and network, the probabilistic inference problem is NP hard [6] and exact algorithms are computationally very expensive. The second approach is based on approximate search algorithms or MonteCarlo simulations. For the precision needed by practitioners in many applications, ....

....as possible. The structure of the join tree is by and large determined by the problem. In many cases load balance can be improved by constructing the join tree can be constructed with the view of parallelization ahead. In general the problem is analogous to construction an optimal evaluation tree [5] and is computationally expensive. In the current implementation, we have chosen a simple way to construct a tree which facilitates speedup. Each clique was prescribed a computational cost. In the current version computational cost has been calculated as a simple sum of the number of potentials, ....

Zhaoyu Li and Bruce D'Ambrosio. Efficient inference in bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 1994. to be published.

....computational work needed to perform this on line evaluation is so straightforward that it lends itself to easy implementations on different software and hardware platforms. This approach shares some commonality with other methods that symbolically manipulate probability expressions, like SPI (Li D Ambrosio, 1994; Shachter, D Ambrosio, del Favero, 1990) it differs from SPI on the objective of such manipulations and, hence, on the results obtained. SPI explicates the notion of an arithmetic expression to state that belief network inference can be viewed as an expression factoring operation. This allows ....

....was at Rockwell Science Center. Special thanks to Jack Breese, Bruce D Ambrosio and to the anonymous reviewers for their useful comments on earlier drafts of this paper. 11. We have shown how clustering and conditioning algorithms can be used for Q DAG generation, but other algorithms such as SPI (Li D Ambrosio, 1994; Shachter et al. 1990) can be used as well. A Practical Paradigm for Implementing Belief Network Inference Appendix A. Proof of Theorem 1 Without loss of generality, we assume in this proof that all variables are declared as evidence variables. To prove this soundness theorem, all we need to ....

Li, Z., & D'Ambrosio, B. (1994). Efficient Inference in Bayes Networks as a Combinatorial Optimization Problem. International Journal of Approximate Reasoning, 11, 55--81.

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Z. Li and B. D'Ambrosio. Efficient inference in Bayes networks as a combinatorial optimization problem. International Journal of Approximate Reasoning, 11, 55--81.

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