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31
Inference in Hybrid Bayesian Networks with Mixtures of Truncated Exponentials
, 2003
"... Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization for solving hybrid Bayesian networks. Any probability density function can be approximated with an MTE potential, which can always by marginalized in closed form. This allows propagation to be done exactly us ..."
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Cited by 16 (2 self)
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Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization for solving hybrid Bayesian networks. Any probability density function can be approximated with an MTE potential, which can always by marginalized in closed form. This allows propagation to be done exactly using the ShenoyShafer architecture for computing marginals, with no restrictions on the construction of a join tree.
Approximating probability density functions with mixtures of truncated exponentials
 Proceedings of the Tenth Conference on Information Processing and Management of Uncertainty in KnowledgeBased Systems (IPMU04), 2004
, 2006
"... Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization and Monte Carlo methods for solving hybrid Bayesian networks. Any probability density function (PDF) can be approximated by an MTE potential, which can always be marginalized in closed form. This allows propagat ..."
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Cited by 16 (9 self)
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Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization and Monte Carlo methods for solving hybrid Bayesian networks. Any probability density function (PDF) can be approximated by an MTE potential, which can always be marginalized in closed form. This allows propagation to be done exactly using the ShenoyShafer architecture for computing marginals, with no restrictions on the construction of a join tree. This paper presents MTE potentials that approximate standard PDF’s and applications of these potentials for solving inference problems in hybrid Bayesian networks. These approximations will extend the types of inference problems that can be modeled with Bayesian networks, as demonstrated using three examples.
Inference in hybrid Bayesian networks using dynamic discretization
 Statistics and Computing
, 2007
"... We consider approximate inference in hybrid Bayesian Networks (BNs) and present a new iterative algorithm that efficiently combines dynamic discretisation with robust propagation algorithms on junction trees structures. Our approach offers a significant extension to Bayesian Network theory and pract ..."
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Cited by 13 (7 self)
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We consider approximate inference in hybrid Bayesian Networks (BNs) and present a new iterative algorithm that efficiently combines dynamic discretisation with robust propagation algorithms on junction trees structures. Our approach offers a significant extension to Bayesian Network theory and practice by offering a flexible way of modelling continuous nodes in BNs conditioned on complex configurations of evidence and intermixed with discrete nodes as both parents and children of continuous nodes. Our algorithm is implemented in a commercial Bayesian Network software package, AgenaRisk, which allows model construction and testing to be carried out easily. The results from the empirical trials clearly show how our software can deal effectively with different type of hybrid models containing elements of expert judgement as well as statistical inference. In particular, we show how the rapid convergence of the algorithm towards zones of high probability density, make robust inference analysis possible even in situations where, due to the lack of information in both prior and data, robust sampling becomes unfeasible.
A Bayesian network reliability modeling and analysis framework. Phd dissertation
 IEEE Transactions on Reliability
, 2005
"... (CTBN) framework for dynamic systems reliability modeling and analysis. Dynamic systems exhibit complex behaviors and interactions between their components; where not only the combination of failure events matters, but so does the sequence ordering of the failures. Similar to dynamic fault trees, th ..."
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Cited by 10 (2 self)
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(CTBN) framework for dynamic systems reliability modeling and analysis. Dynamic systems exhibit complex behaviors and interactions between their components; where not only the combination of failure events matters, but so does the sequence ordering of the failures. Similar to dynamic fault trees, the CTBN framework defines a set of ‘basic ’ BN constructs that capture welldefined system components ’ behaviors and interactions. Combining, in a structured way, the various ‘basic ’ Bayesian network constructs enables the user to construct, in a modular and hierarchical fashion, the system model. Within the CTBN framework, one can perform various analyses, including reliability, sensitivity, and uncertainty analyses. All the analyses allow the user to obtain closedform solutions. Index Terms—Bayesian networks, dynamic systems, reliability modeling and analysis. ACRONYMS 1 BN Bayesian network CPD conditional probability distribution CSP cold spare CTBN continuoustime Bayesian network DBN dynamic Bayesian network DFT dynamic fault tree DTBN discretetime Bayesian network FDEP functional dependency FT fault tree HSP hot spare MPD marginal probability distribution MTTF mean time to failure PAND priority AND PDF probability density function RBD reliability block diagram RV random variable SEQ sequence enforcing WSP warm spare NOTATION dormancy factor impulse function, , , , failure rates marginal probability density function of variable
Bayesian Networks in Reliability
, 2005
"... Over the last decade, Bayesian Networks (BNs) have become a popular tool for modelling many kinds of statistical problems. We have also seen a growing interest for using BNs in the reliability analysis community. In this paper we will discuss the properties of the modelling framework that make BNs ..."
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Cited by 9 (1 self)
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Over the last decade, Bayesian Networks (BNs) have become a popular tool for modelling many kinds of statistical problems. We have also seen a growing interest for using BNs in the reliability analysis community. In this paper we will discuss the properties of the modelling framework that make BNs particularly well suited for reliability applications, and point to ongoing research that is relevant for practitioners in reliability.
Hearty P., Modelling Dependable Systems using Hybrid Bayesian Networks
 Proc. of First International Conference on Availability, Reliability and Security (ARES 2006
, 2006
"... A hybrid Bayesian Network (BN) is one that incorporates both discrete and continuous nodes. In our extensive applications of BNs for system dependability assessment the models are invariably hybrid and the need for efficient and accurate computation is paramount. We apply a new iterative algorithm t ..."
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Cited by 6 (3 self)
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A hybrid Bayesian Network (BN) is one that incorporates both discrete and continuous nodes. In our extensive applications of BNs for system dependability assessment the models are invariably hybrid and the need for efficient and accurate computation is paramount. We apply a new iterative algorithm that efficiently combines dynamic discretisation with robust propagation algorithms on junction tree structures to perform inference in hybrid BNs. We illustrate its use on two example dependability problems: reliability estimation and diagnosis of a faulty sensor in a temporal system. Dynamic discretisation can be used as an alternative to analytical or Monte Carlo methods with high precision and can be applied to a wide range of dependability problems.
✦ Parameter Estimation and Model Selection for Mixtures of Truncated Exponentials
, 2009
"... Bayesian networks with mixtures of truncated exponentials (MTEs) support efficient inference algorithms and provide a flexible way of modeling hybrid domains (domains containing both discrete and continuous variables). On the other hand, estimating an MTE from data has turned out to be a difficult t ..."
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Cited by 3 (3 self)
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Bayesian networks with mixtures of truncated exponentials (MTEs) support efficient inference algorithms and provide a flexible way of modeling hybrid domains (domains containing both discrete and continuous variables). On the other hand, estimating an MTE from data has turned out to be a difficult task, and most prevalent learning methods treat parameter estimation as a regression problem. The drawback of this approach is that by not directly attempting to find the parameter estimates that maximize the likelihood, there is no principled way of performing subsequent model selection using those parameter estimates. In this paper we describe an estimation method that directly aims at learning the parameters of an MTE potential following a maximum likelihood approach. Empirical results demonstrate that the proposed method yields significantly better likelihood results than existing regressionbased methods. We also show how model selection, which in the case of univariate MTEs amounts to partitioning the domain and selecting the number of exponential terms, can be performed using the BICscore. 1
Hybrid Bayesian networks with linear deterministic variables, in
 F. Bacchus, T. Jaakkola (Eds.), Uncertainty in Artificial Intelligence
, 2005
"... When a hybrid Bayesian network has conditionally deterministic variables with continuous parents, the joint density function for the continuous variables does not exist. Conditional linear Gaussian distributions can handle such cases when the continuous variables have a multivariate normal distribu ..."
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Cited by 2 (1 self)
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When a hybrid Bayesian network has conditionally deterministic variables with continuous parents, the joint density function for the continuous variables does not exist. Conditional linear Gaussian distributions can handle such cases when the continuous variables have a multivariate normal distribution and the discrete variables do not have continuous parents. In this paper, operations required for performing inference with conditionally deterministic variables in hybrid Bayesian networks are developed. These methods allow inference in networks with deterministic variables where continuous variables may be nonGaussian, and their density functions can be approximated by mixtures of truncated exponentials. There are no constraints on the placement of continuous and discrete nodes in the network. 1
GAODE and HAODE: Two Proposals based on AODE to Deal with Continuous Variables
"... AODE (Aggregating OneDependence Estimators) is considered one of the most interesting representatives of the Bayesian classifiers, taking into account not only the low error rate it provides but also its efficiency. Until now, all the attributes in a dataset have had to be nominal to build an AODE ..."
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Cited by 2 (1 self)
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AODE (Aggregating OneDependence Estimators) is considered one of the most interesting representatives of the Bayesian classifiers, taking into account not only the low error rate it provides but also its efficiency. Until now, all the attributes in a dataset have had to be nominal to build an AODE classifier or they have had to be previously discretized. In this paper, we propose two different approaches in order to deal directly with numeric attributes. One of them uses conditional Gaussian networks to model a dataset exclusively with numeric attributes; and the other one keeps the superparent on each model discrete and uses univariate Gaussians to estimate the probabilities for the numeric attributes and multinomial distributions for the categorical ones, it also being able to model hybrid datasets. Both of them obtain competitive results compared to AODE, the latter in particular being a very attractive alternative to AODE in numeric datasets. 1.
✦ Mixtures of Truncated Basis Functions
, 2011
"... In this paper we propose a framework, called mixtures of truncated basis functions (MoTBFs), for representing general hybrid Bayesian networks. The proposed framework generalizes both the mixture of truncated exponentials (MTEs) framework and the mixture of polynomials (MoPs) framework. Similar to M ..."
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Cited by 2 (2 self)
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In this paper we propose a framework, called mixtures of truncated basis functions (MoTBFs), for representing general hybrid Bayesian networks. The proposed framework generalizes both the mixture of truncated exponentials (MTEs) framework and the mixture of polynomials (MoPs) framework. Similar to MTEs and MoPs, MoTBFs are defined so that the potentials are closed under combination and marginalization, which ensures that inference in MoTBF networks can be performed efficiently using the ShaferShenoy architecture. Based on a generalized Fourier series approximation, we devise a method for efficiently approximating an arbitrary density function using the MoTBF framework. The translation method is more flexible than existing MTE or MoPbased methods, and it supports an online/anytime tradeoff between the accuracy and the complexity of the approximation. Experimental results show that the approximations obtained are either comparable or significantly better than the approximations obtained using existing methods. 1