Above are papers related to the following paper: [Paper_ID] Paper #50336. 	[Title] Computing upper and lower bounds on likelihoods in intractable networks [Abstract] We present deterministic techniques for computing upper and lower bounds on marginal probabilities in sigmoid and noisy-OR networks. These techniques become useful when the size of the network (or clique size) precludes exact computations. We illustrate the tightness of the bounds by numerical experi ments. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #300806. 	[Title] Learning in Boltzmann Trees [Abstract] We introduce a large family of Boltzmann machines that can be trained using standard gradient descent. The networks can have one or more layers of hidden units, with tree-like connectivity. We show how to implement the supervised learning algorithm for these Boltzmann machines exactly, without resort to simulated or mean-field annealing. The stochastic averages that yield the gradients in weight space are computed by the technique of decimation. We present results on the problems of N -bit parity and the detection of hidden symmetries. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152904. 	[Title] GIBBS-MARKOV MODELS [Abstract] In this paper we present a framework for building probabilistic automata parameterized by context-dependent probabilities. Gibbs distributions are used to model state transitions and output generation, and parameter estimation is carried out using an EM algorithm where the M-step uses a generalized iterative scaling procedure. We discuss relations with certain classes of stochastic feedforward neural networks, a geometric interpretation for parameter estimation, and a simple example of a statistical language model constructed using this methodology. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #390922. 	[Title] A MEAN FIELD LEARNING ALGORITHM FOR UNSUPERVISED NEURAL NETWORKS [Abstract] We introduce a learning algorithm for unsupervised neural networks based on ideas from statistical mechanics. The algorithm is derived from a mean field approximation for large, layered sigmoid belief networks. We show how to (approximately) infer the statistics of these networks without resort to sampling. This is done by solving the mean field equations, which relate the statistics of each unit to those of its Markov blanket. Using these statistics as target values, the weights in the network are adapted by a local delta rule. We evaluate the strengths and weaknesses of these networks for problems in statistical pattern recognition. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #683404. 	[Title] Incremental Tradeoff Resolution in Qualitative Probabilistic Networks [Abstract] Qualitative probabilistic reasoning in a Bayesian network often reveals tradeoffs: relationships that are ambiguous due to competing qualitative influences. We present two techniques that combine qualitative and numeric probabilistic reasoning to resolve such tradeoffs, inferring the qualitative relationship between nodes in a Bayesian network. The first approach incrementally marginalizes nodes that contribute to the ambiguous qualitative relationships. The second approach evaluates approximate Bayesian networks for bounds of probability distributions, and uses these bounds to determinate qualitative relationships in question. This approach is also incremental in that the algorithm refines the state spaces of random variables for tighter bounds until the qualitative relationships are resolved. Both approaches provide systematic methods for tradeoff resolution at potentially lower computational cost than application of purely numeric methods. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #37884. 	[Title] Hidden Markov decision trees [Abstract] We study a time series model that can be viewed as a decision tree with Markov temporal structure. The model is intractable for exact calculations, thus we utilize variational approximations. We consider three different distributions for the approximation: one in which the Markov calculations are performed exactly and the layers of the decision tree are decoupled, one in which the decision tree calculations are performed exactly and the time steps of the Markov chain are decoupled, and one in which a Viterbi-like assumption is made to pick out a single most likely state sequence. We present simulation results for artificial data and the Bach chorales. Accepted for oral presentation at NIPS*96. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152272. 	[Title] GIBBS-MARKOV MODELS [Abstract] In this paper we present a framework for building probabilistic automata parameterized by context-dependent probabilities. Gibbs distributions are used to model state transitions and output generation, and parameter estimation is carried out using an EM algorithm where the M-step uses a generalized iterative scaling procedure. We discuss relations with certain classes of stochastic feedforward neural networks, a geometric interpretation for parameter estimation, and a simple example of a statistical language model constructed using this methodology. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #159897. 	[Title] Robustness Analysis of Bayesian Networks with Finitely Generated Convex Sets of Distributions [Abstract] This paper presents exact solutions and convergent approximations for inferences in Bayesian networks associated with finitely generated convex sets of distributions. Robust Bayesian inference is the calculation of bounds on posterior values given perturbations in a probabilistic model. The paper presents exact inference algorithms and analyzes the circumstances where exact inference becomes intractable. Two classes of algorithms for numeric approximations are developed through transformations on the original model. The first transformation reduces the robust inference problem to the estimation of probabilistic parameters in a Bayesian network. The second transformation uses Lavine's bracketing algorithm to generate a sequence of maximization problems in a Bayesian network. The analysis is extended to the *-contaminated, the lower density bounded, the belief function, the sub-sigma, the density bounded, the total variation and the density ratio classes of distributions. c fl1996 Carnegie Mellon University Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #29723. 	[Title] Boltzmann Chains and Hidden Markov Models [Abstract] We propose a statistical mechanical framework for the modeling of discrete time series. Maximum likelihood estimation is done via Boltzmann learning in one-dimensional networks with tied weights. We call these networks Boltzmann chains and show that they contain hidden Markov models (HMMs) as a special case. Our framework also motivates new architectures that address particular shortcomings of HMMs. We look at two such architectures: parallel chains that model feature sets with disparate time scales, and looped networks that model long-term dependencies between hidden states. For these networks, we show how to implement the Boltzmann learning rule exactly, in polynomial time, without resort to simulated or mean-field annealing. The necessary computations are done by exact decimation procedures from statistical mechanics. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #3229. 	[Title] Book Review  Introduction to the Theory of Neural Computation Reviewed by: 2 [Abstract] Neural computation, also called connectionism, parallel distributed processing, neural network modeling or brain-style computation, has grown rapidly in the last decade. Despite this explosion, and ultimately because of impressive applications, there has been a dire need for a concise introduction from a theoretical perspective, analyzing the strengths and weaknesses of connectionist approaches and establishing links to other disciplines, such as statistics or control theory. The Introduction to the Theory of Neural Computation by Hertz, Krogh and Palmer (subsequently referred to as HKP) is written from the perspective of physics, the home discipline of the authors. The book fulfills its mission as an introduction for neural network novices, provided that they have some background in calculus, linear algebra, and statistics. It covers a number of models that are often viewed as disjoint. Critical analyses and fruitful comparisons between these models Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #28957. 	[Title] Operations for Learning with Graphical Models decomposition techniques and the demonstration that graphical models provide [Abstract] This paper is a multidisciplinary review of empirical, statistical learning from a graphical model perspective. Well-known examples of graphical models include Bayesian networks, directed graphs representing a Markov chain, and undirected networks representing a Markov field. These graphical models are extended to model data analysis and empirical learning using the notation of plates. Graphical operations for simplifying and manipulating a problem are provided including decomposition, differentiation, and the manipulation of probability models from the exponential family. Two standard algorithm schemas for learning are reviewed in a graphical framework: Gibbs sampling and the expectation maximization algorithm. Using these operations and schemas, some popular algorithms can be synthesized from their graphical specification. This includes versions of linear regression, techniques for feed-forward networks, and learning Gaussian and discrete Bayesian networks from data. The paper concludes by sketching some implications for data analysis and summarizing how some popular algorithms fall within the framework presented. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #50337. 	[Title] Recursive algorithms for approximating probabilities in graphical models [Abstract] MIT Computational Cognitive Science Technical Report 9604 Abstract We develop a recursive node-elimination formalism for efficiently approximating large probabilistic networks. No constraints are set on the network topologies. Yet the formalism can be straightforwardly integrated with exact methods whenever they are/become applicable. The approximations we use are controlled: they maintain consistently upper and lower bounds on the desired quantities at all times. We show that Boltzmann machines, sigmoid belief networks, or any combination (i.e., chain graphs) can be handled within the same framework. The accuracy of the methods is verified exper imentally. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #6818. 	[Title] Exploiting Tractable Substructures in Intractable Networks [Abstract] We develop a refined mean field approximation for inference and learning in probabilistic neural networks. Our mean field theory, unlike most, does not assume that the units behave as independent degrees of freedom; instead, it exploits in a principled way the existence of large substructures that are computationally tractable. To illustrate the advantages of this framework, we show how to incorporate weak higher order interactions into a first-order hidden Markov model, treating the corrections (but not the first order structure) within mean field theory. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152162. 	[Title] GIBBS-MARKOV MODELS [Abstract] In this paper we present a framework for building probabilistic automata parameterized by context-dependent probabilities. Gibbs distributions are used to model state transitions and output generation, and parameter estimation is carried out using an EM algorithm where the M-step uses a generalized iterative scaling procedure. We discuss relations with certain classes of stochastic feedforward neural networks, a geometric interpretation for parameter estimation, and a simple example of a statistical language model constructed using this methodology. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #189856. 	[Title] State-Space Abstraction for Anytime Evaluation of Probabilistic Networks [Abstract] One important factor determining the computa - tional complexity of evaluating a probabilistic network is the cardinality of the state spaces of the nodes. By varying the granularity of the state spaces, one can trade off accuracy in the result for computational efficiency. We present an any - time procedure for approximate evaluation of probabilistic networks based on this idea. On application to some simple networks, the proce - dure exhibits a smooth improvement in approxi - mation quality as computation time increases. This suggests that statespace abstraction is one more useful control parameter for designing real-time probabilistic reasoners. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #66982. 	[Title] Learning Generative Models with the Up-Propagation Algorithm [Abstract] Up-propagation is an algorithm for inverting and learning neural network generative models. Sensory input is processed by inverting a model that generates patterns from hidden variables using top-down connections. The inversion process is iterative, utilizing a negative feedback loop that depends on an error signal propagated by bottom-up connections. The error signal is also used to learn the generative model from examples. The algorithm is benchmarked against principal component analysis in In his doctrine of unconscious inference, Helmholtz argued that perceptions are formed by the interaction of bottom-up sensory data with top-down expectations. According to one interpretation of this doctrine, perception is a procedure of sequential hypothesis testing. We propose a new algorithm, called up-propagation, that realizes this interpretation in layered neural networks. It uses top-down connections to generate hypotheses, and bottom-up connections to revise them. It is important to understand the difference between up-propagation and its ancestor, the backpropagation algorithm[1]. Backpropagation is a learning algorithm for recognition models. As shown in Figure 1a, bottom-up connections recognize patterns, while top-down connections propagate an error signal that is used to learn the recognition model. In contrast, up-propagation is an algorithm for inverting and learning generative models, as shown in Figure 1b. Top-down connections generate patterns from a set of hidden variables. Sensory input is processed by inverting the generative model, recovering hidden variables that could have generated the sensory data. This operation is called either pattern recognition or pattern analysis, depending on the meaning of the hidden variables. Inversion of the generative model is done iteratively, through a negative feedback loop driven by an error signal from the bottom-up connections. The error signal is also used for learning the connections experiments on images of handwritten digits. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #194223. 	[Title] Bayesian Unsupervised Learning of Higher Order Structure [Abstract] Multilayer architectures such as those used in Bayesian belief networks and Helmholtz machines provide a powerful framework for representing and learning higher order statistical relations among inputs. Because exact probability calculations with these models are often intractable, there is much interest in finding approximate algorithms. We present an algorithm that efficiently discovers higher order structure using EM and Gibbs sampling. The model can be interpreted as a stochastic recurrent network in which ambiguity in lower-level states is resolved through feedback from higher levels. We demonstrate the performance of the algorithm on bench mark problems. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #189623. 	[Title] Localized Partial Evaluation of Belief Networks [Abstract] in the network. Often, however, an application will not need information about every node in the network nor will it need exact probabilities. We present the localized partial evaluation (LPE) propagation algorithm, which computes interval bounds on the marginal probability of a specified query node by examining a subset of the nodes in the entire network. Conceptually, LPE ignores parts of the network that are "too far away" from the queried node to have much impact on its value. LPE has the "anytime" property of being able to produce better solutions (tighter intervals) given more time to consider more of the network. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1140547. 	[Title] Localist Attractor Networks [Abstract] Attractor networks, which map a continuous input space to a discrete output space, are useful for pattern completion, cleaning up noisy or missing features in an input. However, designing a net to have a given set of attractors is notoriously tricky; training procedures are CPU intensive and often produce spurious attractors and ill-conditioned attractor basins. These difficulties occur because each connection in the network participates in the encoding of multiple attractors. We describe an alternative formulation of attractor networks in which the encoding of knowledge is local, not distributed. Although localist attractor nets have similar dynamics to their distributed counterparts, they are much easier to work with and interpret. We propose a statistical formulation of localist attractor net dynamics, which yields a convergence proof and a mathematical interpretation of model parameters. We present simulation experiments that explore the behavior of lo-calist attractor nets, showing that they produce a gang effectthe presence of an attractor enhances the attractor basins of neighboring attractorsand that spurious attractors occur only at points of symmetry in state space. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #17476. 	[Title] Markov games as a framework for multi-agent reinforcement learning [Abstract] In the Markov decision process (MDP) formalization of reinforcement learning, a single adaptive agent interacts with an environment defined by a probabilistic transition function. In this solipsistic view, secondary agents can only be part of the environment and are therefore fixed in their behavior. The framework of Markov games allows us to widen this view to include multiple adaptive agents with interacting or competing goals. This paper considers a step in this direction in which exactly two agents with diametrically opposed goals share an environment. It describes a Q-learning-like algorithm for finding optimal policies and demonstrates its application to a simple two-player game in which the optimal policy is probabilistic. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #35922. 	[Title] Mean Field Theory for Sigmoid Belief Networks [Abstract] We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. Our mean field theory provides a tractable approximation to the true probability distribution in these networks; it also yields a lower bound on the likelihood of evidence. We demonstrate the utility of this framework on a benchmark problem in statistical pattern recognition|the classification of handwritten digits. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #13269. 	[Title] database [Abstract] MIT Computational Cognitive Science Technical Report 9701 Abstract We describe variational approximation methods for efficient probabilistic reasoning, applying these methods to the problem of diagnostic inference in the QMR-DT database. The QMR-DT database is a large-scale belief network based on statistical and expert knowledge in internal medicine. The size and complexity of this network render exact probabilistic diagnosis infeasible for all but a small set of cases. This has hindered the development of the QMR- DT network as a practical diagnostic tool and has hindered researchers from exploring and critiquing the diagnostic behavior of QMR. In this paper we describe how variational approximation methods can be applied to the QMR network, resulting in fast diagnostic inference. We evaluate the accuracy of our methods on a set of standard diagnostic cases and compare to stochastic sampling methods. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #293974. 	[Title] Tractable Inference for Complex Stochastic Processes [Abstract] The monitoring and control of any dynamic system depends crucially on the ability to reason about its current status and its future trajectory. In the case of a stochastic system, these tasks typically involve the use of a belief statea probability distribution over the state of the process at a given point in time. Unfortunately, the state spaces of complex processes are very large, making an explicit representation of a belief state intractable. Even in dynamic Bayesian networks (DBNs), where the process itself can be represented compactly, the representation of the belief state is intractable. We investigate the idea of maintaining a compact approximation to the true belief state, and analyze the conditions under which the errors due to the approximations taken over the lifetime of the process do not accumulate to make our answers completely irrelevant. We show that the error in a belief state contracts exponentially as the process evolves. Thus, even with multiple approximations, the error in our process remains bounded indefinitely. We show how the additional structure of a DBN can be used to design our approximation scheme, improving its performance significantly. We demonstrate the applicability of our ideas in the context of a monitoring task, showing that orders of magnitude faster inference can be achieved with only a small degradation in accuracy. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #683355. 	[Title] Using Qualitative Relationships for Bounding Probability Distributions [Abstract] We exploit qualitative probabilistic relationships among variables for computing bounds of conditional probability distributions of interest in Bayesian networks. Using the signs of qualitative relationships, we can implement abstraction operations that are guaranteed to bound the distributions of interest in the desired direction. By evaluating incrementally improved approximate networks, our algorithm obtains monotonically tightening bounds that converge to exact distributions. For supermodular utility functions, the tightening bounds monotonically reduce the set of admissible decision alternatives as well. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1105116. 	[Title] Large Deviation Methods for Approximate Probabilistic Inference, with Rates of Convergence a free parameter. The [Abstract] We study layered belief networks of binary random variables in which the conditional probabilities Pr[childjparents] depend monotonically on weighted sums of the parents. For these networks, we give efficient algorithms for computing rigorous bounds on the marginal probabilities of evidence at the output layer. Our methods apply generally to the computation of both upper and lower bounds, as well as to generic transfer function parameterizations of the conditional probability tables (such as sigmoid and noisy-OR). We also prove rates of convergence of the accuracy of our bounds as a function of network size. Our results are derived by applying the theory of large deviations to the weighted sums of parents at each node in the network. Bounds on the marginal probabilities are computed from two contributions: one assuming that these weighted sums fall near their mean values, and the other assuming that they do not. This gives rise to an interesting trade-off between probable explanations of the evidence and improbable deviations from the mean. In networks where each child has N parents, the gap between our upper and lower bounds behaves as a sum of two terms, one of order p In addition to providing such rates of convergence for large networks, our methods also yield efficient algorithms for approximate inference in fixed networks. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1121254. 	[Title] On Structured Variational Approximations [Abstract] The problem of approximating a probability distribution occurs frequently in many areas of applied mathematics, including statistics, communication theory, machine learning, and the theoretical analysis of complex systems such as neural networks. Saul and Jordan (1996) have recently proposed a powerful method for efficiently approximating probability distributions known as structured variational approximations. In structured variational approximations, exact algorithms for probability computation on tractable substructures are combined with variational methods to handle the interactions between the substructures which make the system as a whole intractable. In this note, I present a mathematical result which can simplify the derivation of struc tured variational approximations in the exponential family of distributions. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1128425. 	[Title] Implicit learning in 3D object recognition: The importance of temporal context [Abstract] A novel architecture and set of learning rules for cortical self-organization is proposed. The model is based on the idea that multiple information channels can modulate one another's plasticity. Features learned from bottom-up information sources can thus be influenced by those learned from contextual pathways, and vice versa. A maximum likelihood cost function allows this scheme to be implemented in a biologically feasible, hierarchical neural circuit. In simulations of the model, we first demonstrate the utility of temporal context in modulating plasticity. The model learns a representation that categorizes people's faces according to identity, independent of viewpoint, by taking advantage of the temporal continuity in image sequences. In a second set of simulations, we add plasticity to the contextual stream and explore variations in the architecture. In this case, the model learns a two-tiered representation, starting with a coarse view-based clustering and proceeding to a finer clustering of more specific stimulus features. This model provides a tenable account of how people may perform 3D object recognition in a hierarchical, bottom-up fashion. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #390894. 	[Title] A variational approach to Bayesian logistic regression models and their extensions [Abstract] We consider a logistic regression model with a Gaussian prior distribution over the parameters. We show that accurate variational techniques can be used to obtain a closed form posterior distribution over the parameters given the data thereby yielding a posterior predictive model. The results are readily extended to (binary) belief networks. For belief networks we also derive closed form posteriors in the presence of missing values. Finally, we show that the dual of the regression problem gives a latent variable density model, the variational formulation of which leads to exactly solvable EM updates. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #4335. 	[Title] A VIEW OF THE EM ALGORITHM THAT JUSTIFIES INCREMENTAL, SPARSE, AND OTHER VARIANTS [Abstract] The EM algorithm performs maximum likelihood estimation for data in which some variables are unobserved. We present a function that resembles negative free energy and show that the M step maximizes this function with respect to the model parameters and the E step maximizes it with respect to the distribution over the unobserved variables. From this perspective, it is easy to justify an incremental variant of the EM algorithm in which the distribution for only one of the unobserved variables is recalculated in each E step. This variant is shown empirically to give faster convergence in a mixture estimation problem. A variant of the algorithm that exploits sparse conditional distributions is also described, and a wide range of other variant algorithms are also seen to be possible. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #390896. 	[Title] IMPROVING THE MEAN FIELD APPROXIMATION VIA THE USE OF MIXTURE DISTRIBUTIONS [Abstract] Mean field methods provide computationally efficient approximations to posterior probability distributions for graphical models. Simple mean field methods make a completely factorized approximation to the posterior, which is unlikely to be accurate when the posterior is multimodal. Indeed, if the posterior is multi-modal, only one of the modes can be captured. To improve the mean field approximation in such cases, we employ mixture models as posterior approximations, where each mixture component is a factorized distribution. We describe efficient methods for optimizing the parameters in these models. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1105531. 	[Title] Belief Networks, Hidden Markov Models, and Markov Random Fields: a Unifying View [Abstract] The use of graphs to represent independence structure in multivariate probability models has been pursued in a relatively independent fashion across a wide variety of research disciplines since the beginning of this century. This paper provides a brief overview of the current status of such research with particular attention to recent developments which have served to unify such seemingly disparate topics as probabilistic expert systems, statistical physics, image analysis, genetics, decoding of error-correcting codes, Kalman filters, and speech recognition with Markov models. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #396412. 	[Title] Boltzmann Machine learning using mean field theory and linear response correction [Abstract] We present a new approximate learning algorithm for Boltzmann Machines, using a systematic expansion of the Gibbs free energy to second order in the weights. The linear response correction to the correlations is given by the Hessian of the Gibbs free energy. The computational complexity of the algorithm is cubic in the number of neurons. We compare the performance of the exact BM learning algorithm with first order (Weiss) mean field theory and second order (TAP) mean field theory. The learning task consists of a fully connected Ising spin glass model on 10 neurons. We conclude that 1) the method works well for paramagnetic problems 2) the TAP correction gives a significant improvement over the Weiss mean field theory, both for paramagnetic and spin glass problems and 3) that the inclusion of diagonal weights improves the Weiss approximation for paramagnetic problems, but not for spin glass problems. Please summarise all the given information.

Paper #50336 has the title `Computing upper and lower bounds on likelihoods in intractable networks  
` and the abstract `We present deterministic techniques for computing upper and lower bounds on marginal probabilities in sigmoid and noisy-OR networks. These techniques become useful when the size of the network (or clique size) precludes exact computations. We illustrate the tightness of the bounds by numerical experi ments.
`.
Question: Which category should Paper #50336 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #50336 has the title `Computing upper and lower bounds on likelihoods in intractable networks  
` and the abstract `We present deterministic techniques for computing upper and lower bounds on marginal probabilities in sigmoid and noisy-OR networks. These techniques become useful when the size of the network (or clique size) precludes exact computations. We illustrate the tightness of the bounds by numerical experi ments.
`.
Question: Which category should Paper #50336 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Probabilistic_Methods