R.D. Shachter, B. D'Ambrosio, and B.A. Del Favro, "Symbolic probabilistic inference in belief networks," Automated Reasoning (1990): 126-131. In Operations Research Vol. 36, No.4, 198b.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ON HYPERTREE ORGANIZATION A. Classification of loops The difficulty of coherent inference in multiply connected graphical models (those with loops) of probabilistic knowledge is well known and many inference algorithms have been proposed. Those based on message passing, e.g. 20] 14] 10] [23], 4] all convert a multiply connected network into a tree. However, no formal arguments can be found, e.g. in [20] 9] 18] 3] which demonstrate convincingly that message passing cannot be made coherent in multiply connected networks. This leaves the question whether it is impossible to ....

R.D. Shachter, B. D'Ambrosio, and B.A. Del Favero. Symbolic probabilistic inference in belief networks. In Proc. 8th Natl. Conf. on Artificial Intelligence, pages 126--131, 1990.

....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 ....

R.D.Shachter,B.D'Ambrosio,andB.D.Del Favero. Symbolic probabilistic inference in belief networks, Proc. 8th National Conference on Artificial Intelligence, MIT Press, Boston, pp. 126---131, 1990.

....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 , ....

....Figure 17. We conclude: Theorem 3 Algorithm elim bel computes the posterior belief P (x 1 je) for any given ordering of the variables which is initiated by X 1 . 2 The peeling algorithm for genetic trees [8] Zhang and Poole s algorithm [51] as well as the SPI algorithm by D Ambrosio et al. [39] are all variations of elim bel. Decimation algorithms in statistical physics are also related and were applied to Boltzmann trees [43] 21 B C D F A G B C D F A G D F B C A (a) b) c) Figure 18: Two orderings of the moral graph of our example problem 4.2 Complexity We see ....

[Article contains additional citation context not shown here]

B. D'Ambrosio R.D. Shachter and B.A. Del Favero. Symbolic probabilistic inference in belief networks. In National Conference on Artificial Intelligence (AAAI'90), pages 126--131, 1990. 50

....to determine the posterior probability of a variable or variables given some observations. In this section we outline a simple algorithm for belief net inference called VE [Zhang and Poole, 1996] or bucket elimination for belief assessment, BEBA [Dechter, 1996] that is based on the ideas of SPI [Shachter et al. 1990] . This is a query oriented algorithm that exploits network structure for efficient inference, similarly to clique tree propagation [Lauritzen and Spiegelhalter, 1988; Jensen et al. 1990] One difference is the factors represent conditional probabilities rather than the marginal probabilities ....

R. D. Shachter, B. D. D'Ambrosio, and B. D. Del Favero. Symbolic probabilistic inference in belief networks. In Proc. 8th National Conference on Artificial Intelligence, pages 126--131, Boston, 1990. MIT Press.

.... and an elimination order of width w, rc takes O(n exp(w log n) time under O(n) space, which is a new complexity result for linear space Bayesian network inference, and takes O(n exp(w) time under O(n exp(w) space, therefore, matching the complexity of clustering [7, 6] and elimination [10, 4, 11] algorithms. 1 rc is also equipped with a formula for computing its average running time under any amount of space. To introduce the key intuition underlying recursive conditioning, we note that the power of conditioning is in its ability to reduce network connectivity. In cutset conditioning, ....

R. Shachter, B.D. D'Ambrosio, and B. del Favero. Symbolic Probabilistic Inference in Belief Networks. In Proc. Conf. on Uncertainty in AI, pages 126-131, 1990.

.... is also known as the loop cutset method [25, 26, 28] The best known fact about this method is its linear space complexity, which is very attractive when compared to the exponential space complexity (in treewidth) of state of the art algorithms based on clustering [19, 30, 18, 17] and elimination [29, 20, 11, 32]. The worst known fact about cutset conditioning is its time complexity, which is exponential in the size of loop cutset. The loop cutset can be quite large, even for networks which can be solved in linear time and space using other methods. There have been improvements and variations on cutset ....

R. Shachter, B.D. D'Ambrosio, and B. del Favero. Symbolic Probabilistic Inference in Belief Networks. In Proc. Conf. on Uncertainty in AI, pages 126-131, 1990.

....table it speci es. The main goal of algorithms for Bayesian networks is to answer such queries without having to construct the table explicitly, which size is exponential in the number of network variables. The main approaches for inference in Bayesian networks are based on variable elimination [24, 8, 25], jointrees [13, 14] or conditioning [22, 12, 9, 5, 6] We present in this paper a new, comprehensive approach to inference in Bayesian networks which rests on compiling the network into a factored polynomial. Given a piece of evidence, the polynomial is then evaluated and its partial derivatives ....

.... x and network parameters f . Given a variable elimination order of induced width w and length n, we show how to compile the polynomial in O(n exp(w) time using a simple variable elimination algorithm. 2. We show how to answer a large number of queries relating to classical inference [13, 15, 8, 25, 24], parameter estimation [23, 19] model validation [4] and sensitivity analysis [18, 2, 3] in constant time once the partial derivatives of the compiled polynomial are computed. 3. We show the following on the complexity of computing the derivatives of such polynomials: a) rst partial ....

[Article contains additional citation context not shown here]

R. Shachter, B.D. D'Ambrosio, and B. del Favero. Symbolic Probabilistic Inference in Belief Networks. In Proc. Conf. on Uncertainty in AI, pages 126-131, 1990.

....indicators x and network parameters f . Given a variable elimination order of with w and length n, we show how to compile the polynomial in O(n exp(w) time using a simple variable elimination algorithm. 2. We show how to answer a large number of queries relating to classical inference [10, 13, 7, 21, 20], parameter estimation [19, 16] model validation [4] and sensitivity analysis [15, 2, 3] in constant time once the partial derivatives of the compiled polynomial are computed. 3. We show the following on the complexity of computing the derivatives of such polynomials: a) rst partial ....

....In Figure 6, where e = a, Pr (b j e) F(e) b ) F(e) 03= 3 = 1 and Pr( b j e) F(e) b ) F(e) 27= 3 = 9. The ability to compute such posteriors eciently is probably the key celebrated property of jointree algorithms [10, 11] as compared to variableelimination algorithms [20, 7, 21]. The latter class of algorithms is much simpler except that they can only compute such posteriors by invoking themselves once for each network variable, leading to a complexity of O(n 2 exp(w) Jointree algorithms can do this in O(n exp(w) however, but at the expense of a more complicated ....

R. Shachter, B.D. D'Ambrosio, and B. del Favero. Symbolic Probabilistic Inference in Belief Networks. In UAI90.

....agent through communication. Then H is connected. 4 On hypertree organization The difficulty of coherent inference in multiply connected (with loops) graphical models of probabilistic knowledge is well known and many inference algorithms have been proposed. Those based on message passing, e.g. [13, 9, 5, 15], all convert a multiply connected network into a tree. However, no formal arguments can be found, e.g. in [13, 4, 11, 1] which demonstrate convincingly that message passing cannot be made coherent in multiply connected networks. This leaves the question whether it is impossible to construct ....

R.D. Shachter, B. D'Ambrosio, and B.A. Del Favero. Symbolic probabilistic inference in belief networks. In Proc. 8th Natl. Conf. on Artificial Intelligence, pages 126--131, 1990.

....followed by knowledge propagation based on a student s answer, an adaptive testing system is produced. 3.1 Knowledge Propagation There are many different algorithms for propagating belief through a Bayes net. The approach in this paper uses Shachter, D Ambrosio and DelFavero s SPI algorithm [5]. While issues of Bayesian belief propagation efficiency are not the primary concern in this paper, the complexity of specifying a Bayes net is. In order for it to be truly useful, a course instructor must be able to construct a Bayes net for a course with a minimal amount of effort. It is hoped ....

R. Shachter, B. D'Ambrosio, and B. DelFavero. Symbolic probabilistic inference in belief networks. In Proceedings Eighth National Conference on AI, pages 126--131. AAAI, August 1990.

....1 s and 0 s. Furthermore, extra links need to be added among the variables which are not mutually independent in order to preserve the dependency properties of the original network. By applying on the original network the efficient algorithms developed for computing the exact belief of a variable [8,9,25], the complete probabilistic information for 9 This condition, P unevaluated L j P r(L j jS e ) 1 Gamma P evaluated L i P r(L i jS e ) has been proved in [20] to be the sufficient condition to stop further evaluation. 24 the sub network can be derived. Subsequently RLCM can be applied ....

....applied to identify the most probable general composite hypotheses which are the same as the MPGE to a given S e in the original network. However, in the process of deriving the probabilistic information relevant to each variable in the sub network, existing efficient algorithms for exact beliefs [8,9,21 25] would still have to deal with exponential number of terms which are not independent from the variables in the sub network with respect to S e . From this it can be seen that the overall computational load remains the same, and the computational load merely shifts from one step to another. ....

R.D. Shachter, B. D'Ambrosio, B.A. Del Favero, "Symbolic Probabilistic Inference in Belief Networks," Proc. of the 8th National Conf. on AI, Boston, Massachusetts, Aug. 1990.

....variables in a Bayesian network are being considered, we refer to such a combination a composite hypothesis. III. Review of relevant research Probabilistic inference in a Bayesian network has been viewed as answering queries relevant to the propositional variables in a Bayesian network [9]; in particular, the likelihoods of the simple or (local) composite hypotheses in the presence of an evidence 2 . Various inference algorithms were developed elsewhere and the details were in [2 3,5,10 19] To date, the most efficient computational method to deal with a simple hypothesis has a ....

R.D. Shachter, B. D'Ambrosio, B.A. Del Favero, "Symbolic Probabilistic Inference in Belief Networks," Proc. of the 8th National Conf. on AI, pp. 126-131, Boston, Massachusetts, Aug. 1990.

....the unobserved variables, so, if XO = xO is the observed evidence, the goal is to obtain P (X i jx O ) for every X i 2 X U n XO . Trying to solve this problem by applying Bayes rule directly is intractable even for a little number of variables. In the last years many algorithms ( 20] 21] 1] [22], 23] have been proposed to solve this problem (in an exact way) by taking advantage of the conditional independences among the variables given by the structure of the graph. These algorithms are called propagation algorithms, because the computations are performed locally, and the information ....

R.D. Shachter, B.D. D'Ambrosio, and B.D. Del Favero, "Symbolic probabilistic inference in belief networks," in 8th National Conference on Artificial Intelligence, Boston, 1990, pp. 126--131, MIT Press.

.... ) 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 ....

....is described in Figure 17. Theorem 3 Algorithm elim bel compute the posterior belief P (x 1 je) for any given ordering of the variables which is initiated by X 1 . Both the peeling algorithm for genetic trees [8] Zhang and Poole s algorithm [50] and the SPI algorithm by D ambrosio et.al [39] are variations of elim bel. Decimation algorithms in statistical physics are also related and were applied to Boltzmann trees [43] 21 Algorithm elim bel Input: A belief network BN = fP 1 ; Png; an ordering of the variables, d = X 1 ; Xn ; evidence e. Output: The belief in X 1 = x ....

[Article contains additional citation context not shown here]

B. D'Ambrosio R.D. Shachter and B.A. Del Favero. Symbolic probabilistic inference in belief networks. In National Conference on Artificial Intelligence (AAAI90), pages 126--131, 1990.

....to determine the posterior probability of a variable or variables given some observations. In this section we outline a simple algorithm for belief net inference called VE [Zhang and Poole, 1996] or bucket elimination for belief assessment, BEBA [Dechter, 1996] that is based on the ideas of SPI [Shachter et al. 1990] . This is a query oriented algorithm that exploits network structure for efficient inference, similarly to clique tree propagation [Lauritzen and Spiegelhalter, 1988; Jensen et al. 1990] One difference is the factors represent conditional probabilities rather than the marginal probabilities ....

R. D. Shachter, B. D. D'Ambrosio, and B. D. Del Favero. Symbolic probabilistic inference in belief networks. In Proc. 8th National Conference on Artificial Intelligence, pages 126--131, Boston, 1990. MIT Press.

....did not really use the direction of the links in the network. By standard we mean the Lauritzen Spiegelhalter (Lauritzen Spiegelhalter 1988) the Shafer Shenoy (Shafer Shenoy 1990) and the Hugin (Jensen, Lauritzen Olesen 1990) algorithms and the various variations over these algorithms ( (Shachter 1990) and (Jensen 1995) These algorithms build a secondary structure (a junction tree or a join tree) by triangulating the (moralized) network. This structure can be used for propagation for all information scenaria. Therefore, the algorithms do not exploit independences induced by the evidence. ....

....A is instantiated and no evidence has been entered to DAG 4 , then it is only necessary to send messages down to DAG 4 . We may relax the requirement to the updating algorithm such that we are only interested in updated probabilities for a very small set of variables. In that case the SPI method (Shachter, D Ambrosio DelFavero 1990) and the bucket sort algorithm (Dechter 1996) can utilize specific independences, as they consist of a collect operation only, where the variables are successively eliminated by multiplying the functions involving A (say) and marginalizing A out of this product. These methods, however, are not ....

Shachter, R., D'Ambrosio, B. & DelFavero, B. (1990), Symbolic probabilistic inference in belief networks, in `Proceedings Eighth National Conference on AI', pp. 126--131.

....as the sum of x and all incoming numbers except that from V . The sum can now be retrieved from the root. Next, we call DistributeMessage at the same root (c) The sum can now be retrieved from any node. 3 Probability propagation in JTs Various methods for inference in BNs have been constructed [6, 1, 4, 8, 9, 2]. Several [4, 9, 2] use a junction tree (JT) as runtime structure. We review how to convert a BN into a JT and then consider two of them. 3.1 Conversion of a BN into a JT A BN S is a triplet (N; D;P ) where N is a set of variables, D is a DAG whose nodes are labeled by elements of N , and P is a ....

R.D. Shachter, B. D'Ambrosio, and B.A. Del Favero. Symbolic probabilistic inference in belief networks. In Proc. 8th Natl. Conf. on Artificial Intelligence, pages 126--131, 1990.

....did not really use the direction of the links in the network. By standard we mean the LauritzenSpiegelhalter [Lauritzen and Spiegelhalter, 1988] the Shafer Shenoy [Shafer and Shenoy, 1990] and the Hugin [Jensen et al. 1990] algorithms and the various variations over these algorithms ( [Shachter, 1990] and [Jensen, 1995] These algorithms build a secondary structure (a junction tree or a join tree) by triangulating the (moralized) network. This structure can be used for propagation for all information scenaria. Therefore, the algorithms do not exploit independences induced by the evidence. ....

....A is instantiated and no evidence has been entered to DAG 4 , then it is only necessary to sent messages down to DAG 4 . We may relax the requirement to the updating algorithm such that we are only interested in updated probabilities for a very small set of variables. In that case the SPI method [Shachter et al. 1990] and the bucket sort algorithm [Dechter, 1996] can utilize specific independences, as they consist of a collect operation only, where the variables are successively eliminated by multiplying the functions involving A (say) and marginalizing A out of this product. These methods, however, are not ....

Shachter, R., D'Ambrosio, B., and DelFavero, B. (1990). Symbolic probabilistic inference in belief networks. In Proceedings Eighth National Conference on AI, pages 126--131.

....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 ....

R. Shachter, B.D. D'Ambrosio, and B. del Favero. Symbolic Probabilistic Inference in Belief Networks. In Proc. Conf. on Uncertainty in AI, pages 126--131, 1990.

....monitored. Finally, we motivated the notion of an alert s urgency, and we captured this notion within our framework: We provided an action based definition of urgency, Methods for exact inference include conditioning [10, 37, 39] clustering [21, 22, 27, 48] and symbolic probabilistic inference [29, 46]. 15 and we developed, from our temporally extended decision model, a quantitative measure of this urgency. We propose that our alarm framework can guide the design of effective alarm systems in a variety of domains, and that such systems can be realized by invoking appropriate engineering ....

Ross Shachter, Bruce D'Ambrosio, and Brendan Del Favero. Symbolic probabilistic inference in belief networks. In Proceedings of the 8th National Conference on AI, pages 126--131, Boston, Mass., August 1990.

No context found.

R.D. Shachter, B. D'Ambrosio, and B.A. Del Favro, "Symbolic probabilistic inference in belief networks," Automated Reasoning (1990): 126-131. In Operations Research Vol. 36, No.4, 198b.

No context found.

R. D. Shachter, B. D'Ambrosio, and B. A. Del Favero (1990), Symbolic Probabilistic Inference in Belief Networks, in AAAI-90, pp. 126-131.

No context found.

R. D. Shachter, B. D'Ambrosio, and B. D. Del Favero. Symbolic probabilistic inference in belief networks, Proc. 8th National Conference on Artificial Intelligence, MIT Press, Boston, pp. 126---131, 1990.

No context found.

Ross D. Shachter, Brendan A. Del Favero, and Bruce D'Ambrosio. Symbolic probabilistic inference in belief networks. In AAAI, pages 126--131, 1990.

No context found.

Shachter, R., D'Ambrosio, B., and Del Favero, B., Symbolic probabilistic inference in belief networks, in Proceedings of the 8th National Conference on AI, Boston, Mass., 126#131, 1990.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC