Above are papers related to the following paper: [Paper_ID] Paper #1131267. 	[Title] Parallel Markov chain Monte Carlo sampling. [Abstract] Markov chain Monte Carlo (MCMC) samplers have proved remarkably popular as tools for Bayesian computation. However, problems can arise in their application when the density of interest is high dimensional and strongly correlated. In these circumstances the sampler may be slow to traverse the state space and mixing is poor. In this article we offer a partial solution to this problem. The state space of the Markov chain is augmented to accommodate multiple chains in parallel. Updates to individual chains are based around a genetic style crossover operator acting on `parent' states drawn from the population of chains. This process makes efficient use of gradient information implicitly encoded within the distribution of states across the population. Empirical studies support the claim that the crossover operator acting on a parallel population of chains improves mixing. This is illustrated with an example of sampling a high dimensional posterior probability density from a complex predictive model. By adopting a latent variable approach the methodology is extended to deal with variable selection and model averaging in high dimensions. This is illustrated with an example of knot selection for a spline interpolant. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #24966. 	[Title] A Practical Bayesian Framework for Backprop Networks [Abstract] A quantitative and practical Bayesian framework is described for learning of mappings in feedforward networks. The framework makes possible: (1) objective comparisons between solutions using alternative network architectures; (2) objective stopping rules for network pruning or growing procedures; (3) objective choice of magnitude and type of weight decay terms or additive regularisers (for penalising large weights, etc.); (4) a measure of the effective number of well-determined parameters in a model; (5) quantified estimates of the error bars on network parameters and on network output; (6) objective comparisons with alternative learning and interpolation models such as splines and radial basis functions. The Bayesian `evidence' automatically embodies `Occam's razor,' penalising over-flexible and over-complex models. The Bayesian approach helps detect poor underlying assumptions in learning models. For learning models well matched to a problem, a good correlation between generalisation ability and the Bayesian evidence is obtained. This paper makes use of the Bayesian framework for regularisation and model comparison described in the companion paper `Bayesian interpolation' (MacKay, 1991a). This framework is due to Gull and Skilling (Gull, 1989a). Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #16008. 	[Title] [12] J. Whittaker. Graphical Models in Applied Mathematical Multivariate Statis- [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #105865. 	[Title] Statistical Ideas for Selecting Network Architectures [Abstract] Choosing the architecture of a neural network is one of the most important problems in making neural networks practically useful, but accounts of applications usually sweep these details under the carpet. How many hidden units are needed? Should weight decay be used, and if so how much? What type of output units should be chosen? And so on. We address these issues within the framework of statistical theory for model This paper is principally concerned with architecture selection issues for feed-forward neural networks (also known as multi-layer perceptrons). Many of the same issues arise in selecting radial basis function networks, recurrent networks and more widely. These problems occur in a much wider context within statistics, and applied statisticians have been selecting and combining models for decades. Two recent discussions are [4, 5]. References [3, 20, 21, 22] discuss neural networks from a statistical perspective. choice, which provides a number of workable approximate answers. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #642827. 	[Title] Decomposable graphical Gaussian model determination [Abstract] We propose a methodology for Bayesian model determination in decomposable graphical Gaussian models. To achieve this aim we consider a hyper inverse Wishart prior distribution on the concentration matrix for each given graph. To ensure compatibility across models, such prior distributions are obtained by marginalisation from the prior conditional on the complete graph. We explore alternative structures for the hyperparameters of the latter, and their consequences for the model. Model determination is carried out by implementing a reversible jump MCMC sampler. In particular, the dimension-changing move we propose involves adding or dropping an edge from the graph. We characterise the set of moves which preserve the decomposability of the graph, giving a fast algorithm for maintaining the junction tree representation of the graph at each sweep. As state variable, we propose to use the incomplete variance-covariance matrix, containing only the elements for which the corresponding element of the inverse is nonzero. This allows all computations to be performed locally, at the clique level, which is a clear advantage for the analysis of large and complex data-sets. Finally, the statistical and computational performance of the procedure is illustrated by means of both artificial and real multidimensional data-sets. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1131149. 	[Title] Least Absolute Shrinkage is Equivalent to Quadratic Penalization [Abstract] Adaptive ridge is a special form of ridge regression, balancing the quadratic penalization on each parameter of the model. This paper shows the equivalence between adaptive ridge and lasso (least absolute shrinkage and selection operator). This equivalence states that both procedures produce the same estimate. Least absolute shrinkage can thus be viewed as a particular quadratic penalization. From this observation, we derive an EM algorithm to compute the lasso solution. We finally present a series of applications of this type of algorithm in regres sion problems: kernel regression, additive modeling and neural net training. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #202639. 	[Title] PREDICTING SUNSPOTS AND EXCHANGE RATES WITH CONNECTIONIST NETWORKS [Abstract] We investigate the effectiveness of connectionist networks for predicting the future continuation of temporal sequences. The problem of overfitting, particularly serious for short records of noisy data, is addressed by the method of weight-elimination: a term penalizing network complexity is added to the usual cost function in back-propagation. The ultimate goal is prediction accuracy. We analyze two time series. On the benchmark sunspot series, the networks outperform traditional statistical approaches. We show that the network performance does not deteriorate when there are more input units than needed. Weight-elimination also manages to extract some part of the dynamics of the notoriously noisy currency exchange rates and makes the network solution interpretable. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #137873. 	[Title] Stacked Density Estimation [Abstract] Technical Report No. 97-36, Information and Computer Science Department, University of California, Irvine Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #145176. 	[Title] Bayesian Non-linear Modelling for the Prediction Competition [Abstract] The 1993 energy prediction competition involved the prediction of a series of building energy loads from a series of environmental input variables. Non-linear regression using `neural networks' is a popular technique for such modeling tasks. Since it is not obvious how large a time-window of inputs is appropriate, or what preprocessing of inputs is best, this can be viewed as a regression problem in which there are many possible input variables, some of which may actually be irrelevant to the prediction of the output variable. Because a finite data set will show random correlations between the irrelevant inputs and the output, any conventional neural network (even with reg-ularisation or `weight decay') will not set the coefficients for these junk inputs to zero. Thus the irrelevant variables will hurt the model's performance. The Automatic Relevance Determination (ARD) model puts a prior over the regression parameters which embodies the concept of relevance. This is done in a simple and `soft' way by introducing multiple regularisation constants, one associated with each input. Using Bayesian methods, the regularisation constants for junk inputs are automatically inferred to be large, preventing those inputs from causing significant overfitting. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1105433. 	[Title] Priors for Infinite Networks [Abstract] Technical Report CRG-TR-94-1 Department of Computer Science University of Toronto 10 King's College Road Toronto, Canada M5S 1A4 Abstract Bayesian inference begins with a prior distribution for model parameters that is meant to capture prior beliefs about the relationship being modeled. For multilayer perceptron networks, where the parameters are the connection weights, the prior lacks any direct meaning | what matters is the prior over functions computed by the network that is implied by this prior over weights. In this paper, I show that priors over weights can be defined in such a way that the corresponding priors over functions reach reasonable limits as the number of hidden units in the network goes to infinity. When using such priors, there is thus no need to limit the size of the network in order to avoid "overfitting". The infinite network limit also provides insight into the properties of different priors. A Gaussian prior for hidden-to-output weights results in a Gaussian process prior for functions, which can be smooth, Brownian, or fractional Brownian, depending on the hidden unit activation function and the prior for input-to-hidden weights. Quite different effects can be obtained using priors based on non-Gaussian stable distributions. In networks with more than one hidden layer, a combination of Gaussian and non-Gaussian priors appears most interesting. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #124828. 	[Title] Ensemble Training: Some Recent Experiments with Postal Zip Data [Abstract] Recent findings suggest that a classification scheme based on an ensemble of networks is an effective way to address overfitting. We study optimal methods for training an ensemble of networks. Some recent experiments on Postal Zip-code character data suggest that weight decay may not be an optimal method for controlling the variance of a classifier. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #92065. 	[Title] ADAPTIVE REGULARIZATION [Abstract] Regularization, e.g., in the form of weight decay, is important for training and optimization of neural network architectures. In this work we provide a tool based on asymptotic sampling theory, for iterative estimation of weight decay parameters. The basic idea is to do a gradient descent in the estimated generalization error with respect to the regularization parameters. The scheme is implemented in our Designer Net framework for network training and pruning, i.e., is based on the diagonal Hessian approximation. The scheme does not require essential computational overhead in addition to what is needed for training and pruning. The viability of the approach is demonstrated in an experiment concerning prediction of the chaotic Mackey-Glass series. We find that the optimized weight decays are relatively large for densely connected networks in the initial pruning phase, while they decrease as pruning proceeds. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #423463. 	[Title] Model Selection for Generalized Linear Models via GLIB, with Application to Epidemiology 1 [Abstract] 1 This is the first draft of a chapter for Bayesian Biostatistics, edited by Donald A. Berry and Darlene K. Strangl. Adrian E. Raftery is Professor of Statistics and Sociology, Department of Statistics, GN-22, University of Washington, Seattle, WA 98195, USA. Sylvia Richardson is Directeur de Recherche, INSERM Unite 170, 16 avenue Paul Vaillant Couturier, 94807 Villejuif CEDEX, France. Raftery's research was supported by ONR contract no. N-00014-91-J-1074, by the Ministere de la Recherche et de l'Espace, Paris, by the Universite de Paris VI, and by INRIA, Rocquencourt, France. Raftery thanks the latter two institutions, Paul Deheuvels and Gilles Celeux for hearty hospitality during his Paris sabbatical in which part of this chapter was written. The authors are grateful to Christine Montfort for excellent research assistance and to Mariette Gerber, Michel Chavance and David Madigan for helpful discussions. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #308920. 	[Title] A Graphical Characterization of Lattice Conditional Independence Models [Abstract] Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of non-monotone missing data patterns and of nonnested dependent linear regression models ( seemingly unrelated regressions). It is shown here that the class of LCI models coincides with a subclass of the class of graphical Markov models determined by acyclic digraphs (ADGs), namely, the subclass of transitive ADG models. An explicit graph - theoretic characterization of those ADGs that are Markov equivalent to some transitive ADG is obtained. This characterization allows one to determine whether a specific ADG D is Markov equivalent to some transitive ADG, hence to some LCI model, in polynomial time, without an exhaustive search of the (exponentially large) equivalence class [D ]. These results do not require the existence or positivity of joint densities. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #641976. 	[Title] A comparison of some error estimates for neural network models  Summary [Abstract] We discuss a number of methods for estimating the standard error of predicted values from a multi-layer perceptron. These methods include the delta method based on the Hessian, bootstrap estimators, and the "sandwich" estimator. The methods are described and compared in a number of examples. We find that the bootstrap methods perform best, partly because they capture variability due to the choice of starting weights. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #239800. 	[Title] Ensemble Learning and Evidence Maximization [Abstract] Ensemble learning by variational free energy minimization is a tool introduced to neural networks by Hinton and van Camp in which learning is described in terms of the optimization of an ensemble of parameter vectors. The optimized ensemble is an approximation to the posterior probability distribution of the parameters. This tool has now been applied to a variety of statistical inference problems. In this paper I study a linear regression model with both parameters and hyper-parameters. I demonstrate that the evidence approximation for the optimization of regularization constants can be derived in detail from a free energy minimization viewpoint. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #65212. 	[Title] Bayesian Training of Backpropagation Networks by the Hybrid Monte Carlo Method [Abstract] It is shown that Bayesian training of backpropagation neural networks can feasibly be performed by the "Hybrid Monte Carlo" method. This approach allows the true predictive distribution for a test case given a set of training cases to be approximated arbitrarily closely, in contrast to previous approaches which approximate the posterior weight distribution by a Gaussian. In this work, the Hybrid Monte Carlo method is implemented in conjunction with simulated annealing, in order to speed relaxation to a good region of parameter space. The method has been applied to a test problem, demonstrating that it can produce good predictions, as well as an indication of the uncertainty of these predictions. Appropriate weight scaling factors are found automatically. By applying known techniques for calculation of "free energy" differences, it should also be possible to compare the merits of different network architectures. The work described here should also be applicable to a wide variety of statistical models other than neural networks. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #40124. 	[Title] Probabilistic Networks: New Models and New Methods [Abstract] In this paper I describe the implementation of a probabilistic regression model in BUGS. BUGS is a program that carries out Bayesian inference on statistical problems using a simulation technique known as Gibbs sampling. It is possible to implement surprisingly complex regression models in this environment. I demonstrate the simultaneous inference of an interpolant and an input-dependent noise level. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #63549. 	[Title] A Practical Monte Carlo Implementation of Bayesian Learning [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #671293. 	[Title] Selecting Input Variables Using Mutual Information and Nonparametric Density Estimation [Abstract] In learning problems where a connectionist network is trained with a finite sized training set, better generalization performance is often obtained when unneeded weights in the network are eliminated. One source of unneeded weights comes from the inclusion of input variables that provide little information about the output variables. We propose a method for identifying and eliminating these input variables. The method first determines the relationship between input and output variables using nonparametric density estimation and then measures the relevance of input variables using the information theoretic concept of mutual information. We present results from our method on a simple toy problem and a nonlinear time series. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1105344. 	[Title] In Advances in Neural Information Processing Systems 8  Gaussian Processes for Regression [Abstract] The Bayesian analysis of neural networks is difficult because a simple prior over weights implies a complex prior distribution over functions. In this paper we investigate the use of Gaussian process priors over functions, which permit the predictive Bayesian analysis for fixed values of hyperparameters to be carried out exactly using matrix operations. Two methods, using optimization and averaging (via Hybrid Monte Carlo) over hyperparameters have been tested on a number of challenging problems and have produced excellent results. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #79809. 	[Title] MDL and MML Similarities and Differences (Introduction to Minimum Encoding Inference Part III) [Abstract] Tech Report 207 Department of Computer Science, Monash University, Clayton, Vic. 3168, Australia Abstract: This paper continues the introduction to minimum encoding inductive inference given by Oliver and Hand. This series of papers was written with the objective of providing an introduction to this area for statisticians. We describe the message length estimates used in Wallace's Minimum Message Length (MML) inference and Rissanen's Minimum Description Length (MDL) inference. The differences in the message length estimates of the two approaches are explained. The implications of these differences for applications are discussed. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152194. 	[Title]  [Abstract] Draft A Brief Introduction to Neural Networks Richard D. De Veaux Lyle H. Ungar Williams College University of Pennsylvania Abstract Artificial neural networks are being used with increasing frequency for high dimensional problems of regression or classification. This article provides a tutorial overview of neural networks, focusing on back propagation networks as a method for approximating nonlinear multivariable functions. We explain, from a statistician's vantage point, why neural networks might be attractive and how they compare to other modern regression techniques. KEYWORDS: nonparametric regression; function approximation; backpropagation. 1 Introduction Networks that mimic the way the brain works; computer programs that actually LEARN patterns; forecasting without having to know statistics. These are just some of the many claims and attractions of artificial neural networks. Neural networks (we will henceforth drop the term artificial, unless we need to distinguish them from biological neural networks) seem to be everywhere these days, and at least in their advertising, are able to do all that statistics can do without all the fuss and bother of having to do anything except buy a piece of software. Neural networks have been successfully used for many different applications including robotics, chemical process control, speech recognition, optical character recognition, credit card fraud detection, interpretation of chemical spectra and vision for autonomous navigation of vehicles. (Pointers to the literature are given at the end of this article.) In this article we will attempt to explain how one particular type of neural network, feedforward networks with sigmoidal activation functions ("backpropagation networks") actually works, how it is "trained", and how it compares with some more well known statistical techniques. As an example of why someone would want to use a neural network, consider the problem of recognizing hand written ZIP codes on letters. This is a classification problem, where the 1 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1106370. 	[Title] The Effective Size of a Neural Network: A Principal Component Approach [Abstract] Often when learning from data, one attaches a penalty term to a standard error term in an attempt to prefer simple models and prevent overfitting. Current penalty terms for neural networks, however, often do not take into account weight interaction. This is a critical drawback since the effective number of parameters in a network usually differs dramatically from the total number of possible parameters. In this paper, we present a penalty term that uses Principal Component Analysis to help detect functional redundancy in a neural network. Results show that our new algorithm gives a much more accurate estimate of network complexity than do standard approaches. As a result, our new term should be able to improve techniques that make use of a penalty term, such as weight decay, weight pruning, feature selection, Bayesian, and prediction-risk tech niques. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1104449. 	[Title] Using Neural Networks for Descriptive Statistical Analysis of Educational Data [Abstract] In this paper we discuss the methodological issues of using a class of neural networks called Mixture Density Networks (MDN) for discriminant analysis. MDN models have the advantage of having a rigorous probabilistic interpretation, and they have proven to be a viable alternative as a classification procedure in discrete domains. We will address both the classification and interpretive aspects of discriminant analysis, and compare the approach to the traditional method of linear discrimin- ants as implemented in standard statistical packages. We show that the MDN approach adopted performs well in both aspects. Many of the observations made are not restricted to the particular case at hand, and are applicable to most applications of discriminant analysis in educational research. fl URL: http://www.cs.Helsinki.FI/research/cosco/ Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #3192. 	[Title] Bayesian Graphical Models for Discrete Data [Abstract] z York's research was supported by a NSF graduate fellowship. The authors are grateful to Julian Besag, David Bradshaw, Jeff Bradshaw, James Carlsen, David Draper, Ivar Heuch, Robert Kass, Augustine Kong, Steffen Lauritzen, Adrian Raftery, and James Zidek for helpful comments and discussions. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #197452. 	[Title] Best-First Model Merging for Dynamic Learning and Recognition [Abstract] Best-first model merging is a general technique for dynamically choosing the structure of a neural or related architecture while avoiding overfitting. It is applicable to both learning and recognition tasks and often generalizes significantly better than fixed structures. We demonstrate the approach applied to the tasks of choosing radial basis functions for function learning, choosing local affine models for curve and constraint surface modelling, and choosing the structure of a balltree or bumptree to maximize efficiency of access. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #23116. 	[Title] Bayesian Methods for Adaptive Models [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #39890. 	[Title] A THEORY OF INFERRED CAUSATION perceive causal relationships in uncon trolled observations. 2. the task [Abstract] This paper concerns the empirical basis of causation, and addresses the following issues: We propose a minimal-model semantics of causation, and show that, contrary to common folklore, genuine causal influences can be distinguished from spurious covariations following standard norms of inductive reasoning. We also establish a sound characterization of the conditions under which such a distinction is possible. We provide an effective algorithm for inferred causation and show that, for a large class of data the algorithm can uncover the direction of causal influences as defined above. Finally, we ad dress the issue of non-temporal causation. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #561364. 	[Title] Markov Chain Monte Carlo Model Determination for Hierarchical and Graphical Log-linear Models [Abstract] The Bayesian approach to comparing models involves calculating the posterior probability of each plausible model. For high-dimensional contingency tables, the set of plausible models is very large. We focus attention on reversible jump Markov chain Monte Carlo (Green, 1995) and develop strategies for calculating posterior probabilities of hierarchical, graphical or decomposable log-linear models. Even for tables of moderate size, these sets of models may be very large. The choice of suitable prior distributions for model parameters is also discussed in detail, and two examples are presented. For the first example, a 2 fi 3 fi 4 table, the model probabilities calculated using our reversible jump approach are compared with model probabilities calculated exactly or by using an alternative approximation. The second example is a 2 6 contingency table for which exact methods are infeasible, due to the large number of possible models. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #310742. 	[Title] Bayesian Graphical Modeling for Intelligent Tutoring Systems [Abstract] Conventional Intelligent Tutoring Systems (ITS) do not acknowledge uncertainty about the student's knowledge. Yet, both the outcome of any teaching intervention and the exact state of the student's knowledge are uncertain. In recent years, researchers have made startling progress in the management of uncertainty in knowledge-based systems. Building on these developments, we describe an ITS architecture that explicitly models uncertainty. This will facilitate more accurate student modeling and provide ITS's which can learn. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #5086. 	[Title] Approximate Bayes Factors and Accounting for Model Uncertainty in Generalized Linear Models [Abstract] Technical Report no. 255 Department of Statistics, University of Washington August 1993; Revised March 1994 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #131042. 	[Title] Efficient Implementation of Gaussian Processes [Abstract] Neural networks and Bayesian inference provide a useful framework within which to solve regression problems. However their parameterization means that the Bayesian analysis of neural networks can be difficult. In this paper, we investigate a method for regression using Gaussian process priors which allows exact Bayesian analysis using matrix manipulations. We discuss the workings of the method in detail. We will also detail a range of mathematical and numerical techniques that are useful in applying Gaussian processes to general problems including efficient approximate matrix inversion methods developed by Skilling. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1106789. 	[Title] Model Selection for Consumer Loan Application Data [Abstract] Loan applications at banks are often long, requiring the applicant to provide large amounts of data. Is all of it necessary? Can we save the applicant some frustration and the bank some expense by using only a subset of the relevant variables? To answer this question, I have attempted to model the current loan approval process at a particular bank. I have used several model selection techniques for logistic regression, including stepwise regression, Occam's Window, Markov Chain Monte Carlo Model Composition (Raftery, Madigan, and Hoeting, 1993), and Bayesian Random Searching. The resulting models largely agree upon a subset of only one-third of the original variables. fl This paper was completed in partial fulfillment of the Ph.D. data analysis requirement. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #95719. 	[Title] Choice of Basis for Laplace Approximation [Abstract] Maximum a posteriori optimization of parameters and the Laplace approximation for the marginal likelihood are both basis-dependent methods. This note compares two choices of basis for models parameterized by probabilities, showing that it is possible to improve on the traditional choice, the probability simplex, by transforming to the `softmax' basis. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #162664. 	[Title] An Algorithm for the Construction of Bayesian Network Structures from Data [Abstract] Previous algorithms for the construction of Bayesian belief network structures from data have been either highly dependent on conditional independence (CI) tests, or have required an ordering on the nodes to be supplied by the user. We present an algorithm that integrates these two approaches - CI tests are used to generate an ordering on the nodes from the database which is then used to recover the underlying Bayesian network structure using a non CI based method. Results of preliminary evaluation of the algorithm on two networks (ALARM and LED) are presented. We also discuss some algo rithm performance issues and open problems. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #129897. 	[Title] The out-of-bootstrap method for model averaging and selection [Abstract] We propose a bootstrap-based method for model averaging and selection that focuses on training points that are left out of individual bootstrap samples. This information can be used to estimate optimal weighting factors for combining estimates from different bootstrap samples, and also for finding the best subsets the linear model setting. These proposals provide alternatives to Bayesian approaches to model averaging and selection, requiring less computation and fewer subjective choices. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1127530. 	[Title] Bayesian inference for nondecomposable graphical Gaussian models [Abstract] In this paper we propose a method to calculate the posterior probability of a nondecomposable graphical Gaussian model. Our proposal is based on a new device to sample from Wishart distributions, conditional on the graphical constraints. As a result, our methodology allows Bayesian model selection within the whole class of graphical Gaussian models, including nondecomposable ones. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #129896. 	[Title] Accounting for Model Uncertainty in Survival Analysis Improves Predictive Performance [Abstract] Survival analysis is concerned with finding models to predict the survival of patients or to assess the efficacy of a clinical treatment. A key part of the model-building process is the selection of the predictor variables. It is standard to use a stepwise procedure guided by a series of significance tests to select a single model, and then to make inference conditionally on the selected model. However, this ignores model uncertainty, which can be substantial. We review the standard Bayesian model averaging solution to this problem and extend it to survival analysis, introducing partial Bayes factors to do so for the Cox proportional hazards model. In two examples, taking account of model uncertainty enhances predictive performance, to an extent that could be clinically useful. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #27627. 	[Title] Information-based objective functions for active data selection [Abstract] Learning can be made more efficient if we can actively select particularly salient data points. Within a Bayesian learning framework, objective functions are discussed which measure the expected informativeness of candidate measurements. Three alternative specifications of what we want to gain information about lead to three different criteria for data selection. All these criteria depend on the assumption that the hypothesis space is correct, which may prove to be their main weakness. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #145134. 	[Title] Brief Papers Computing Second Derivatives in Feed-Forward Networks: A Review [Abstract] The calculation of second derivatives is required by recent training and analysis techniques of connectionist networks, such as the elimination of superfluous weights, and the estimation of confidence intervals both for weights and network outputs. We here review and develop exact and approximate algorithms for calculating second derivatives. For networks with jwj weights, simply writing the full matrix of second derivatives requires O(jwj 2 ) operations. For networks of radial basis units or sigmoid units, exact calculation of the necessary intermediate terms requires of the order of 2h + 2 backward/forward-propagation passes where h is the number of hidden units in the network. We also review and compare three approximations (ignoring some components of the second derivative, numerical differentiation, and scoring). Our algorithms apply to arbitrary activation functions, networks, and error functions (for instance, with connections that skip layers, or radial basis functions, or cross-entropy error and Softmax units, etc.). Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1106671. 	[Title] Bayesian Learning in Feed Forward Neural Networks [Abstract] Bayesian methods are applicable to complex modeling tasks. In this review, the principles of Bayesian inference are presented and applied to neural network models. Several approximate implementations are discussed, and their advantages over conventional fre-quentist model training and selection are outlined. It is argued that Bayesian methods are preferable to traditional approaches, although empirical evidence for this is still sparse. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #19697. 	[Title] Keeping Neural Networks Simple by Minimizing the Description Length of the Weights [Abstract] Supervised neural networks generalize well if there is much less information in the weights than there is in the output vectors of the training cases. So during learning, it is important to keep the weights simple by penalizing the amount of information they contain. The amount of information in a weight can be controlled by adding Gaussian noise and the noise level can be adapted during learning to optimize the trade-off between the expected squared error of the network and the amount of information in the weights. We describe a method of computing the derivatives of the expected squared error and of the amount of information in the noisy weights in a network that contains a layer of non-linear hidden units. Provided the output units are linear, the exact derivatives can be computed efficiently without time-consuming Monte Carlo simulations. The idea of minimizing the amount of information that is required to communicate the weights of a neural network leads to a number of interesting schemes for encoding the weights. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #131318. 	[Title] FLAT MINIMA Neural Computation 9(1):1-42 (1997) [Abstract] We present a new algorithm for finding low complexity neural networks with high generalization capability. The algorithm searches for a "flat" minimum of the error function. A flat minimum is a large connected region in weight-space where the error remains approximately constant. An MDL-based, Bayesian argument suggests that flat minima correspond to "simple" networks and low expected overfitting. The argument is based on a Gibbs algorithm variant and a novel way of splitting generalization error into underfitting and overfitting error. Unlike many previous approaches, ours does not require Gaussian assumptions and does not depend on a "good" weight prior instead we have a prior over input/output functions, thus taking into account net architecture and training set. Although our algorithm requires the computation of second order derivatives, it has backprop's order of complexity. Automatically, it effectively prunes units, weights, and input lines. Various experiments with feedforward and recurrent nets are described. In an application to stock market prediction, flat minimum search outperforms (1) conventional backprop, (2) weight decay, (3) "optimal brain surgeon" / "optimal brain damage". We also provide pseudo code of the algorithm (omitted from the NC-version). Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #3191. 	[Title] Model Selection and Accounting for Model Uncertainty in Linear Regression Models [Abstract] 1 Adrian E. Raftery is Professor of Statistics and Sociology, David Madigan is Assistant Professor of Statistics, and Jennifer Hoeting is a Ph.D. Candidate, all at the Department of Statistics, GN-22, University of Washington, Seattle, WA 98195. The research of Raftery and Hoeting was supported by ONR Contract N-00014-91-J-1074. Madigan's research was partially supported by NSF grant no. DMS 92111627. The authors are grateful to Danika Lew for research assistance. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1123576. 	[Title] Density Networks and their Application to Protein Modelling [Abstract] I define a latent variable model in the form of a neural network for which only target outputs are specified; the inputs are unspecified. Although the inputs are missing, it is still possible to train this model by placing a simple probability distribution on the unknown inputs and maximizing the probability of the data given the parameters. The model can then discover for itself a description of the data in terms of an underlying latent variable space of lower dimensionality. I present preliminary results of the application of these models to protein data. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1154042. 	[Title] 5 Bayesian estimation 5.1 Introduction [Abstract] This chapter takes a different standpoint to address the problem of learning. We will here reason only in terms of probability, and make extensive use of the chain rule known as "Bayes' rule". A fast definition of the basics in probability is provided in appendix A for quick reference. Most of this chapter is a review of the methods of Bayesian learning applied to our modelling purposes. Some original analyses and comments are also provided in section 5.8, 5.11 and 5.12. There is a latent rivalry between "Bayesian" and "Orthodox" statistics. It is by no means our intention to enter this kind of controversy. We are perfectly willing to accept orthodox as well as unorthodox methods, as long as they are scientifically sound and provide good results when applied to learning tasks. The same disclaimer applies to the two frameworks presented here. They have been the object of heated controversy in the past 3 years in the neural networks community. We will not take side, but only present both frameworks, with their strong points and their weaknesses. In the context of this work, the "Bayesian frameworks" are especially interesting as the provide some continuous update rules that can be used during regularised cost minimisation to yield an automatic selection of the regularisation level. Unlike the methods presented in chapter 3, it is not necessary to try several regularisation levels and perform as many optimisations. The Bayesian framework is the only one in which training is achieved through a one-pass optimisation procedure. Please summarise all the given information.

Paper #1131267 has the title `Parallel Markov chain Monte Carlo sampling.  
` and the abstract `Markov chain Monte Carlo (MCMC) samplers have proved remarkably popular as tools for Bayesian computation. However, problems can arise in their application when the density of interest is high dimensional and strongly correlated. In these circumstances the sampler may be slow to traverse the state space and mixing is poor. In this article we offer a partial solution to this problem. The state space of the Markov chain is augmented to accommodate multiple chains in parallel. Updates to individual chains are based around a genetic style crossover operator acting on `parent' states drawn from the population of chains. This process makes efficient use of gradient information implicitly encoded within the distribution of states across the population. Empirical studies support the claim that the crossover operator acting on a parallel population of chains improves mixing. This is illustrated with an example of sampling a high dimensional posterior probability density from a complex predictive model. By adopting a latent variable approach the methodology is extended to deal with variable selection and model averaging in high dimensions. This is illustrated with an example of knot selection for a spline interpolant. 
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Question: Which category should Paper #1131267 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #1131267 has the title `Parallel Markov chain Monte Carlo sampling.  
` and the abstract `Markov chain Monte Carlo (MCMC) samplers have proved remarkably popular as tools for Bayesian computation. However, problems can arise in their application when the density of interest is high dimensional and strongly correlated. In these circumstances the sampler may be slow to traverse the state space and mixing is poor. In this article we offer a partial solution to this problem. The state space of the Markov chain is augmented to accommodate multiple chains in parallel. Updates to individual chains are based around a genetic style crossover operator acting on `parent' states drawn from the population of chains. This process makes efficient use of gradient information implicitly encoded within the distribution of states across the population. Empirical studies support the claim that the crossover operator acting on a parallel population of chains improves mixing. This is illustrated with an example of sampling a high dimensional posterior probability density from a complex predictive model. By adopting a latent variable approach the methodology is extended to deal with variable selection and model averaging in high dimensions. This is illustrated with an example of knot selection for a spline interpolant. 
`.
Question: Which category should Paper #1131267 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