Simulation metamodeling with dynamic Bayesian networks

This paper presents a novel approach to simulation metamodeling using dynamic Bayesian networks (DBNs) in the context of discrete event simulation. A DBN is a probabilistic model that represents the joint distribution of a sequence of random variables and enables the efficient calculation of their marginal and conditional distributions. In this paper, the construction of a DBN based on simulation data and its utilization in simulation analyses are presented. The DBN metamodel allows the study of the time evolution of simulation by tracking the probability distribution of the simulation state over the duration of the simulation. This feature is unprecedented among existing simulation metamodels. The DBN metamodel also enables effective what-if analysis which reveals the conditional evolution of the simulation. In such an analysis, the simulation state at a given time is fixed and the probability distributions representing the state at other time instants are updated. Simulation parameters can be included in the DBN metamodel as external random variables. Then, the DBN offers a way to study the effects of parameter values and their uncertainty on the evolution of the simulation. The accuracy of the analyses allowed by DBNs is studied by constructing appropriate confidence intervals. These analyses could be conducted based on raw simulation data but the use of DBNs reduces the duration of repetitive analyses and is expedited by available Bayesian network software. The construction and analysis capabilities of DBN metamodels are illustrated with two example simulation studies.     

Bayesian networks for mathematical models: Techniques for automatic construction and efficient inference

Expert knowledge in the form of mathematical models can be considered sufficient statistics of all prior experimentation in the domain, embodying generic or abstract knowledge of it. When used in a probabilistic framework, such models provide a sound foundation for data mining, inference, and decision making under uncertainty.We describe a methodology for encapsulating knowledge in the form of ordinary differential equations (ODEs) in dynamic Bayesian networks (DBNs). The resulting DBN framework can handle both data and model uncertainty in a principled manner, can be used for temporal data mining with noisy and missing data, and can be used to re-estimate model parameters automatically using data streams. A standard assumption when performing inference in DBNs is that time steps are fixed. Generally, the time step chosen is small enough to capture the dynamics of the most rapidly changing variable. This can result in DBNs having a natural time step that is very short, leading to inefficient inference; this is particularly an issue for DBNs derived from ODEs and for systems where the dynamics are not uniform over time.We propose an alternative to the fixed time step inference used in standard DBNs. In our algorithm, the DBN automatically adapts the time step lengths to suit the dynamics in each step. The resulting system allows us to efficiently infer probable values of hidden variables using multiple time series of evidence, some of which may be sparse, noisy or incomplete.We evaluate our approach with a DBN based on a variant of the van der Pol oscillator, and demonstrate an example where it gives more accurate results than the standard approach, but using only one tenth the number of time steps.We also apply our approach to a real-world example in critical care medicine. By incorporating knowledge in the form of an existing ODE model, we have built a DBN framework for efficiently predicting individualised patient responses using the available bedside and lab data.     

Networks of probabilistic events in discrete time

The usual methods of applying Bayesian networks to the modeling of temporal processes, such as Dean and Kanazawa’s dynamic Bayesian networks (DBNs), consist in discretizing time and creating an instance of each random variable for each point in time. We present a new approach called network of probabilistic events in discrete time (NPEDT), for temporal reasoning with uncertainty in domains involving probabilistic events. Under this approach, time is discretized and each value of a variable represents the instant at which a certain event may occur. This is the main difference with respect to DBNs, in which the value of a variable V_i represents the state of a real-world property at time t_i. Therefore, our method is more appropriate for temporal fault diagnosis, because only one variable is necessary for representing the occurrence of a fault and, as a consequence, the networks involved are much simpler than those obtained by using DBNs. In contrast, DBNs are more appropriate for monitoring tasks, since they explicitly represent the state of the system at each moment. We also introduce in this paper several types of temporal noisy gates, which facilitate the acquisition and representation of uncertain temporal knowledge. They constitute a generalization of traditional canonical models of multicausal interactions, such as the noisy OR-gate, which have been usually applied to static domains. We illustrate the approach with the example domain of modeling the evolution of traffic jams produced on the outskirts of a city, after the occurrence of an event that obliges traffic to stop indefinitely.     

Sub-sample swapping for sequential Monte Carlo approximation of high-dimensional densities in the context of complex object tracking

In this paper, we address the problem of complex object tracking using the particle filter framework, which essentially amounts to estimate high-dimensional distributions by a sequential Monte Carlo algorithm. For this purpose, we first exploit Dynamic Bayesian Networks to determine conditionally independent subspaces of the object’s state space, which allows us to independently perform the particle filter’s propagations and corrections over small spaces. Second, we propose a swapping process to transform the weighted particle set provided by the update step of the particle filter into a “new particle set” better focusing on high peaks of the posterior distribution. This new methodology, called Swapping-Based Partitioned Sampling, is proved to be mathematically sound and is successfully tested and validated on synthetic video sequences for single or multiple articulated object tracking.     

Structural-EM for learning PDG models from incomplete data

Probabilistic Decision Graphs (PDGs) are a class of graphical models that can naturally encode some context specific independencies that cannot always be efficiently captured by other popular models, such as Bayesian Networks. Furthermore, inference can be carried out efficiently over a PDG, in time linear in the size of the model. The problem of learning PDGs from data has been studied in the literature, but only for the case of complete data. We propose an algorithm for learning PDGs in the presence of missing data. The proposed method is based on the Expectation-Maximisation principle for estimating the structure of the model as well as the parameters. We test our proposal on both artificially generated data with different rates of missing cells and real incomplete data. We also compare the PDG models learnt by our approach to the commonly used Bayesian Network (BN) model. The results indicate that the PDG model is less sensitive to the rate of missing data than BN model. Also, though the BN models usually attain higher likelihood, the PDGs are close to them also in size, which makes the learnt PDGs preferable for probabilistic inference purposes.     

Multiagent bayesian forecasting of structural time-invariant dynamic systems with graphical models

Time series are found widely in engineering and science. We study forecasting of stochastic, dynamic systems based on observations from multivariate time series. We model the domain as a dynamic multiply sectioned Bayesian network (DMSBN) and populate the domain by a set of proprietary, cooperative agents. We propose an algorithm suite that allows the agents to perform one-step forecasts with distributed probabilistic inference. We show that as long as the DMSBN is structural time-invariant (possibly parametric time-variant), the forecast is exact and its time complexity is exponentially more efficient than using dynamic Bayesian networks (DBNs). In comparison with independent DBN-based agents, multiagent DMSBNs produce more accurate forecasts. The effectiveness of the framework is demonstrated through experiments on a supply chain testbed.     

Affine-Invariant Ensemble Transform Methods for Logistic Regression

We investigate the application of ensemble transform approaches to Bayesian inference of logistic regression problems. Our approach relies on appropriate extensions of the popular ensemble Kalman filter and the feedback particle filter to the cross entropy loss function and is based on a well-established homotopy approach to Bayesian inference. The arising finite particle evolution equations as well as their mean-field limits are affine-invariant. Furthermore, the proposed methods can be implemented in a gradient-free manner in case of nonlinear logistic regression and the data can be randomly subsampled similar to mini-batching of stochastic gradient descent. We also propose a closely related SDE-based sampling method which again is affine-invariant and can easily be made gradient-free. Numerical examples demonstrate the appropriateness of the proposed methodologies.

     

Bayesian inference for the power law process

The power law process has been used to model reliability growth, software reliability and the failure times of repairable systems. This article reviews and further develops Bayesian inference for such a process. The Bayesian approach provides a unified methodology for dealing with both time and failure truncated data. As well as looking at the posterior densities of the parameters of the power law process, inference for the expected number of failures and the probability of no failures in some given time interval is discussed. Aspects of the prediction problem are examined. The results are illustrated with two data examples.     

A Bayesian inference approach to identify a Robin coefficient in one-dimensional parabolic problems

This paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a Robin coefficient from boundary temperature measurement. A Bayesian inference approach is presented for the solution of this problem. The prior modeling is achieved via the Markov random field (MRF). The use of a hierarchical Bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed. The Markov chain Monte Carlo (MCMC) algorithm is used to explore the posterior state space. Numerical results indicate that MRF provides an effective prior regularization, and the Bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem.     

Adaptive Bayesian Nonstationary Modeling for Large Spatial Datasets Using Covariance Approximations

Gaussian process models have been widely used in spatial statistics but face tremendous modeling and computational challenges for very large nonstationary spatial datasets. To address these challenges, we develop a Bayesian modeling approach using a nonstationary covariance function constructed based on adaptively selected partitions. The partitioned nonstationary class allows one to knit together local covariance parameters into a valid global nonstationary covariance for prediction, where the local covariance parameters are allowed to be estimated within each partition to reduce computational cost. To further facilitate the computations in local covariance estimation and global prediction, we use the full-scale covariance approximation (FSA) approach for the Bayesian inference of our model. One of our contributions is to model the partitions stochastically by embedding a modified treed partitioning process into the hierarchical models that leads to automated partitioning and substantial computational benefits. We illustrate the utility of our method with simulation studies and the global Total Ozone Matrix Spectrometer (TOMS) data. Supplementary materials for this article are available online.     

Imperfect debugging in software reliability: A Bayesian approach

The objective of studying software reliability is to assist software engineers in understanding more of the probabilistic nature of software failures during the debugging stages and to construct reliability models. In this paper, we consider modeling of a multiplicative failure rate whose components are evolving stochastically over testing stages and discuss its Bayesian estimation. In doing so, we focus on the modeling of parameters such as the fault detection rate per fault and the number of faults. We discuss how the proposed model can account for “imperfect debugging” under certain conditions. We use actual inter-failure data to carry out inference on model parameters via Markov chain Monte Carlo methods and present additional insights from Bayesian analysis.     

Particle filtering in the Dempster–Shafer theory

This paper derives a particle filter algorithm within the Dempster–Shafer framework. Particle filtering is a well-established Bayesian Monte Carlo technique for estimating the current state of a hidden Markov process using a fixed number of samples. When dealing with incomplete information or qualitative assessments of uncertainty, however, Dempster–Shafer models with their explicit representation of ignorance often turn out to be more appropriate than Bayesian models.The contribution of this paper is twofold. First, the Dempster–Shafer formalism is applied to the problem of maintaining a belief distribution over the state space of a hidden Markov process by deriving the corresponding recursive update equations, which turn out to be a strict generalization of Bayesian filtering. Second, it is shown how the solution of these equations can be efficiently approximated via particle filtering based on importance sampling, which makes the Dempster–Shafer approach tractable even for large state spaces. The performance of the resulting algorithm is compared to exact evidential as well as Bayesian inference.     

Nonparametric Bayesian Modeling for Stochastic Order

In comparing two populations, sometimes a model incorporating stochastic order is desired. Customarily, such modeling is done parametrically. The objective of this paper is to formulate nonparametric (possibly semiparametric) stochastic order specifications providing richer, more flexible modeling. We adopt a fully Bayesian approach using Dirichlet process mixing. An attractive feature of the Bayesian approach is that full inference is available regarding the population distributions. Prior information can conveniently be incorporated. Also, prior stochastic order is preserved to the posterior analysis. Apart from the two sample setting, the approach handles the matched pairs problem, the k-sample slippage problem, ordered ANOVA and ordered regression models. We illustrate by comparing two rather small samples, one of diabetic men, the other of diabetic women. Measurements are of androstenedione levels. Males are anticipated to produce levels which will tend to be higher than those of females.     

Learning failure-free PRISM programs

PRISM is a probabilistic logic programming formalism which allows defining a probability distribution over possible worlds. This paper investigates learning a class of generative PRISM programs known as failure-free. The aim is to learn recursive PRISM programs which can be used to model stochastic processes. These programs generalise dynamic Bayesian networks by defining a halting distribution over the generative process. Dynamic Bayesian networks model infinite stochastic processes. Sampling from infinite process can only be done by specifying the length of sequences that the process generates. In this case, only observations of a fixed length of sequences can be obtained. On the other hand, the recursive PRISM programs considered in this paper are self-terminating upon some halting conditions. Thus, they generate observations of different lengths of sequences. The direction taken by this paper is to combine ideas from inductive logic programming and learning Bayesian networks to learn PRISM programs. It builds upon the inductive logic programming approach of learning from entailment.     

Belief updating in multiply sectioned Bayesian networks without repeated local propagations

Multiply sectioned Bayesian networks (MSBNs) provide a coherent and flexible formalism for representing uncertain knowledge in large domains. Global consistency among subnets in an MSBN is achieved by communication. When a subnet updates its belief with respect to an adjacent subnet, existing inference operations require repeated belief propagations (proportional to the number of linkages between the two subnets) within the receiving subnet, making communication less efficient. We redefine these operations such that two such propagations are sufficient. We prove that the new operations, while improving the efficiency, do not compromise the coherence. An MSBN must be initialized before inference can take place. The initialization involves dedicated operations not shared by inference operations according to existing methods. We show that the new inference operations presented here unify inference and initialization. Hence the new operations are not only more efficient but also simpler. The new results are presented such that their connection with the common inference methods for single Bayesian networks is highlighted.     

A Combinatorial user optimal dynamic traffic assignment algorithm

This paper introduces a polynomial combinatorial optimization algorithm for the dynamic user optimal problem. The approach can efficiently solve single destination networks and can be potentially extended to heuristically solve multidestinational networks. In the model, traffic is propagated according to sound traffic flow theoretical models rather than link exit functions; thereby allowing link queue evolution to be modeled more precisely. The algorithm is designed, proven, implemented and computationally tested.     

混合动力公交车能量控制策略的优化模型

针对混合动力公交车在循环工况内功率需求的特点,建立了未来功率需求贝叶斯预测模型;利用2-阶段随机动态规划模型将大规模的随机动态规划问题简化为多个小规模的随机动态规划问题和一个确定型动态规划问题;对于随机动态规划模型的求解,给出了稀疏表示的降维方法,将复杂的泛函极值问题转化为常规的随机动态优化问题,并采用分布估计算法和计算资源最优配置算法的计算机仿真优化算法对随机动态优化问题进行求解;给出了基于查表的在线控制策略,为模型的实际应用进行了有益的探索。     

A Bayesian model for repeated measures zero-inflated count data with application to outpatient psychiatric service use

In applications involving count data, it is common to encounter an excess number of zeros. In the study of outpatient service utilization, for example, the number of utilization days will take on integer values, with many subjects having no utilization (zero values). Mixed-distribution models, such as the zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB), are often used to fit such data. A more general class of mixture models, called hurdle models, can be used to model zero-deflation as well as zero-inflation. Several authors have proposed frequentist approaches to fitting zero-inflated models for repeated measures. We describe a practical Bayesian approach which incorporates prior information, has optimal small-sample properties, and allows for tractable inference. The approach can be easily implemented using standard Bayesian software. A study of psychiatric outpatient service use illustrates the methods.     

A methodology for developing Bayesian networks: An application to information technology (IT) implementation

Bayesian Networks (BNs) are probabilistic inference engines that support reasoning under uncertainty. This article presents a methodology for building an information technology (IT) implementation BN from client–server survey data. The article also demonstrates how to use the BN to predict the attainment of IT benefits, given specific implementation characteristics (e.g., application complexity) and activities (e.g., reengineering). The BN is an outcome of a machine learning process that finds the network’s structure and its associated parameters, which best fit the data. The article will be of interest to academicians who want to learn more about building BNs from real data and practitioners who are interested in IT implementation models that make probabilistic statements about certain implementation decisions.     

Bayesian analysis of constant elasticity of variance models

This paper focuses on the estimation of some models in finance and in particular, in interest rates. We analyse discretized versions of the constant elasticity of variance (CEV) models where the normal law showing up in the usual discretization of the diffusion part is replaced by a range of heavy‐tailed distributions. A further extension of the model is to allow the elasticity of variance to be a parameter itself. This generalized model allows great flexibility in modelling and simplifies the model implementation considerably using the scale mixtures representation. The mixing parameters provide a means to identify possible outliers and protect inference by down‐weighting the distorting effects of these outliers. For parameter estimation, Bayesian approach is adopted and implemented using the software WinBUGS (Bayesian inference using Gibbs sampler). Results from a real data analysis show that an exponential power distribution with a random shape parameter, which is highly leptokurtic compared with the normal distribution, forms the best CEV model for the data. Copyright © 2006 John Wiley & Sons, Ltd.