Above are papers related to the following paper: [Paper_ID] Paper #431206. 	[Title] Backfitting in Smoothing Spline ANOVA [Abstract] A scheme to compute smoothing spline ANOVA estimates for large data sets with a (near) tensor-product structure is proposed. Such data sets are common in spatial-temporal analysis and image analysis. This scheme combines backfitting algorithm with iterative imputation algorithm in order to save both computational space and time. The convergence of this algorithm and various ways to further speed it up, such as collapsing component functions and successive over-relaxation, are discussed. Issues related to its application in spatial-temporal analysis are discussed too. An application to a global analysis of historical surface temperature data is described. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #263553. 	[Title] A Generalized Approximate Cross Validation for Smoothing Splines with Non-Gaussian Data 1 [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #6913. 	[Title] On Convergence of the EM Algorithm and the Gibbs Sampler  SUMMARY [Abstract] In this article we investigate the relationship between the two popular algorithms, the EM algorithm and the Gibbs sampler. We show that the approximate rate of convergence of the Gibbs sampler by Gaussian approximation is equal to that of the corresponding EM type algorithm. This helps in implementing either of the algorithms as improvement strategies for one algorithm can be directly transported to the other. In particular, by running the EM algorithm we know approximately how many iterations are needed for convergence of the Gibbs sampler. We also obtain a result that under conditions, the EM algorithm used for finding the maximum likelihood estimates can be slower to converge than the corresponding Gibbs sampler for Bayesian inference which uses proper prior distributions. We illustrate our results in a number of realistic examples all based on the generalized linear mixed models. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1135108. 	[Title] A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian [Abstract] We describe the maximum-likelihood parameter estimation problem and how the Expectation-Maximization (EM) algorithm can be used for its solution. We first describe the abstract form of the EM algorithm as it is often given in the literature. We then develop the EM parameter estimation procedure for two applications: 1) finding the parameters of a mixture of Gaussian densities, and 2) finding the parameters of a hidden Markov model (HMM) (i.e., the Baum-Welch algorithm) for both discrete and Gaussian mixture observation models. We derive the update equations in fairly explicit detail but we do not prove any convergence properties. We try to emphasize intuition rather than mathematical rigor. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #646286. 	[Title] Rate of Convergence of the Gibbs Sampler by Gaussian Approximation  SUMMARY [Abstract] In this article we approximate the rate of convergence of the Gibbs sampler by a normal approximation of the target distribution. Based on this approximation, we consider many implementational issues for the Gibbs sampler, e.g., updating strategy, parameterization and blocking. We give theoretical results to justify our approximation and illustrate our methods in a number of realistic examples. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #662416. 	[Title] USING SMOOTHING SPLINE ANOVA TO EXAMINE THE RELATION OF RISK FACTORS TO THE INCIDENCE AND [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #646289. 	[Title] Bayesian Estimation and Model Choice in Item Response Models [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #78994. 	[Title] Adaptive tuning of numerical weather prediction models: Randomized GCV in three and four dimensional data assimilation [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152917. 	[Title]  [Abstract] Convergence Results for the EM Approach to Abstract The Expectation-Maximization (EM) algorithm is an iterative approach to maximum likelihood parameter estimation. Jordan and Jacobs (1993) recently proposed an EM algorithm for the mixture of experts architecture of Jacobs, Jordan, Nowlan and Hinton (1991) and the hierarchical mixture of experts architecture of Jordan and Jacobs (1992). They showed empirically that the EM algorithm for these architectures yields significantly faster convergence than gradient ascent. In the current paper we provide a theoretical analysis of this algorithm. We show that the algorithm can be regarded as a variable metric algorithm with its searching direction having a positive projection on the gradient of the log likelihood. We also analyze the convergence of the algorithm and provide an explicit expression for the convergence rate. In addition, we describe an acceleration technique that yields a significant speedup in simulation experiments. This report describes research done at the Dept. of Brain and Cognitive Sciences, the Center for Biological and Computational Learning, and the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for CBCL is provided in part by a grant from the NSF (ASC-9217041). Support for the laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Dept. of Defense. The authors were supported by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, by by grant IRI-9013991 from the National Science Foundation, by grant N00014-90-J-1942 from the Office of Naval Research, and by NSF grant ECS-9216531 to support an Initiative in Intelligent Control at MIT. Michael I. Jordan is a NSF Presidential Young Investigator. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152150. 	[Title]  [Abstract] TECHNICAL REPORT NO. 947 June 5, 1995 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #444191. 	[Title] Automatic Smoothing Spline Projection Pursuit  Automatic Smoothing Spline Projection Pursuit. [Abstract] kj = 1. The standard PPR algorithm of Friedman and Stuet-zle (1981) estimates the smooth functions f j using the supersmoother nonparametric scatterplot smoother. Friedman's algorithm constructs a model with M max linear combinations, then prunes back to a simpler model of size M M max , where M and M max are specified by the user. This paper discusses an alternative algorithm in which the smooth functions are estimated using smoothing splines. The direction coefficients ff j , the amount of smoothing in each direction, and the number of terms M and M max are determined to optimize a single generalized cross-validation measure. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1117219. 	[Title] Testing the Generalized Linear Model Null Hypothesis versus `Smooth' Alternatives 1 [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #662572. 	[Title] Spatial-Temporal Analysis of Temperature Using Smoothing Spline ANOVA [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1116336. 	[Title] On Computing the Largest Fraction of Missing Information for the EM Algorithm and the Worst [Abstract] We address the problem of computing the largest fraction of missing information for the EM algorithm and the worst linear function for data augmentation. These are the largest eigenvalue and its associated eigenvector for the Jacobian of the EM operator at a maximum likelihood estimate, which are important for assessing convergence in iterative simulation. An estimate of the largest fraction of missing information is available from the EM iterates; this is often adequate since only a few figures of accuracy are needed. In some instances the EM iteration also gives an estimate of the worst linear function. We show that the power method for eigencomputation can be used to compute efficient and accurate estimates of both quantities. Unlike eigenvalue decomposition, the power method computes only the largest eigenvalue and eigenvector of a matrix, it can take advantage of a good eigenvector estimate as an initial value and it can be terminated after only a few figures of accuracy are obtained. Moreover, the matrix products needed in the power method can be computed by extrapolation, obviating the need to form the Jacobian of the EM operator. We give results of simultation studies on multivariate normal data showing this approach becomes more efficient as the data dimension increases than methods that use a finite-difference approximation to the Jacobian, which is the only general-purpose alternative available. fl Funded by National Institutes of Health Small Business Innovation Reseach Grant 5R44CA65147-03, and by Office of Naval Research contracts N00014-96-1-0192 and N00014-96-1-0330. We are indebted to Tim Hesterberg, Jim Schimert, Doug Clarkson, Anne Greenbaum, and Adrian Raftery for comments and discussion that helped advance this research and improve this paper. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1132083. 	[Title] Spline Smoothing For Bivariate Data With Applications To Association Between Hormones [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1717. 	[Title] On Convergence Properties of the EM Algorithm for Gaussian Mixtures [Abstract] We build up the mathematical connection between the "Expectation-Maximization" (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite Gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix P , and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of P and provide new results analyzing the effect that P has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of Gaussian mixture models. This report describes research done at the Center for Biological and Computational Learning and the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the Center is provided in part by a grant from the National Science Foundation under contract ASC-9217041. Support for the laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N00000-00-A-0000. The authors were also supported by the HK RGC Earmarked Grant CUHK250/94E, by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, and by grant N00014-90-1-0777 from the Office of Naval Research. Michael I. Jordan is an NSF Presidential Young Investigator. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1154103. 	[Title]  [Abstract] TECHNICAL REPORT NO. 967 August 1996 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #169279. 	[Title] Smoothing Spline Models With Correlated Random Errors 1 [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #124224. 	[Title] Behavior Near Zero of the Distribution of GCV Smoothing Parameter Estimates 1 [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #428610. 	[Title] Non-linear Models for Time Series Using Mixtures of Experts [Abstract] We consider a novel non-linear model for time series analysis. The study of this model emphasizes both theoretical aspects as well as practical applicability. The architecture of the model is demonstrated to be sufficiently rich, in the sense of approximating unknown functional forms, yet it retains some of the simple and intuitive characteristics of linear models. A comparison to some more established non-linear models will be emphasized, and theoretical issues are backed by prediction results for benchmark time series, as well as computer generated data sets. Efficient estimation algorithms are seen to be applicable, made possible by the mixture based structure of the model. Large sample properties of the estimators are discussed as well, in both well specified as well as misspecified settings. We also demonstrate how inference pertaining to the data structure may be made from the parameterization of the model, resulting in a better, more intuitive, understanding of the structure and performance of the model. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #703953. 	[Title] Convergence in Norm for Alternating Expectation-Maximization (EM) Type Algorithms 1 [Abstract] We provide a sufficient condition for convergence of a general class of alternating estimation-maximization (EM) type continuous-parameter estimation algorithms with respect to a given norm. This class includes EM, penalized EM, Green's OSL-EM, and other approximate EM algorithms. The convergence analysis can be extended to include alternating coordinate-maximization EM algorithms such as Meng and Rubin's ECM and Fessler and Hero's SAGE. The condition for monotone convergence can be used to establish norms under which the distance between successive iterates and the limit point of the EM-type algorithm approaches zero monotonically. For illustration, we apply our results to estimation of Poisson rate parameters in emission tomography and establish that in the final iterations the logarithm of the EM iterates converge monotonically in a weighted Euclidean norm. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12631. 	[Title] Smoothing Spline ANOVA with Component-Wise Bayesian "Confidence Intervals" To Appear, J. Computational and Graphical Statistics [Abstract] We study a multivariate smoothing spline estimate of a function of several variables, based on an ANOVA decomposition as sums of main effect functions (of one variable), two-factor interaction functions (of two variables), etc. We derive the Bayesian "confidence intervals" for the components of this decomposition and demonstrate that, even with multiple smoothing parameters, they can be efficiently computed using the publicly available code RKPACK, which was originally designed just to compute the estimates. We carry out a small Monte Carlo study to see how closely the actual properties of these component-wise confidence intervals match their nominal confidence levels. Lastly, we analyze some lake acidity data as a function of calcium concentration, latitude, and longitude, using both polynomial and thin plate spline main effects in the same model. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12638. 	[Title] Soft Classification, a.k.a. Risk Estimation, via Penalized Log Likelihood and Smoothing Spline Analysis of Variance [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1131230. 	[Title] Identifiability, Improper Priors and Gibbs Sampling for Generalized Linear Models [Abstract] Alan E. Gelfand is a Professor in the Department of Statistics at the University of Connecti-cut, Storrs, CT 06269. Sujit K. Sahu is a Lecturer at the School of Mathematics, University of Wales, Cardiff, CF2 4YH, UK. The research of the first author was supported in part by NSF grant DMS 9301316 while the second author was supported in part by an EPSRC grant from UK. The authors thank Brad Carlin, Kate Cowles, Gareth Roberts and an anonymous referee for valuable comments. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #4330. 	[Title] Hierarchical Mixtures of Experts and the EM Algorithm [Abstract] We present a tree-structured architecture for supervised learning. The statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models (GLIM's). Learning is treated as a maximum likelihood problem; in particular, we present an Expectation-Maximization (EM) algorithm for adjusting the parameters of the architecture. We also develop an on-line learning algorithm in which the parameters are updated incrementally. Comparative simulation results are presented in the robot dynamics domain. This report describes research done at the Dept. of Brain and Cognitive Sciences, the Center for Biological and Computational Learning, and the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for CBCL is provided in part by a grant from the NSF (ASC-9217041). Support for the laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Dept. of Defense. The authors were supported by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, by by grant IRI-9013991 from the National Science Foundation, by grant N00014-90-J-1942 from the Office of Naval Research, and by NSF grant ECS-9216531 to support an Initiative in Intelligent Control at MIT. Michael I. Jordan is a NSF Presidential Young Investigator. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #6898. 	[Title] Sample Size Calculations for Smoothing Splines Based on Bayesian Confidence Intervals [Abstract] Bayesian confidence intervals of a smoothing spline are often used to distinguish two curves. In this paper, we provide an asymptotic formula for sample size calculations based on Bayesian confidence intervals. Approximations and simulations on special functions indicate that this asymptotic formula is reasonably accurate. Key Words: Bayesian confidence intervals; sample size; smoothing spline. fl Address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110. Tel.: (805)893-4870. Fax: (805)893-2334. E-mail: yuedong@pstat.ucsb.edu. Supported by the National Institute of Health under Grants R01 EY09946, P60 DK20572 and P30 HD18258. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1105011. 	[Title] On the Connection Between Stochastic Smoothing, Filtering and Estimation with Incomplete Data [Abstract] Connections between stochastic smoothing/filtering and estimation with incomplete data are investigated. It is shown, under the right censoring scheme, that the Kaplan-Meier estimator can be characterized as a moment estimate based on a stochastic filter/smoother(a pseudo-filter/smoother). Motivated by this result, a potentially useful martingale approach for estimation and convergence with incomplete data is proposed: estimators are characterized as pseudo-stochastic smoothers (which sometimes reduce to filters), which are described by a (system of) stochastic integral equation(s); recent results in convergence of stochastic integrals and stochastic differential equations are then applied to address convergence issues. As an illustration, the double censoring problem is revisited under this framework, a closed form estimator is proposed and convergence properties studied. Martingale theory plays a vital role in the entire analysis. This approach is in essence a self-consistency method. 1 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1123188. 	[Title] GRKPACK: FITTING SMOOTHING SPLINE ANOVA MODELS FOR EXPONENTIAL FAMILIES [Abstract] Wahba, Wang, Gu, Klein and Klein (1995) introduced Smoothing Spline ANalysis of VAriance (SS ANOVA) method for data from exponential families. Based on RKPACK, which fits SS ANOVA models to Gaussian data, we introduce GRKPACK: a collection of Fortran subroutines for binary, binomial, Poisson and Gamma data. We also show how to calculate Bayesian confidence intervals for SS ANOVA estimates. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #6910. 	[Title] Smoothing Spline ANOVA for Exponential Families, with Application to the Wisconsin Epidemiological Study of Diabetic [Abstract]  Please summarise all the given information.

Paper #431206 has the title `Backfitting in Smoothing Spline ANOVA  
` and the abstract `A scheme to compute smoothing spline ANOVA estimates for large data sets with a (near) tensor-product structure is proposed. Such data sets are common in spatial-temporal analysis and image analysis. This scheme combines backfitting algorithm with iterative imputation algorithm in order to save both computational space and time. The convergence of this algorithm and various ways to further speed it up, such as collapsing component functions and successive over-relaxation, are discussed. Issues related to its application in spatial-temporal analysis are discussed too. An application to a global analysis of historical surface temperature data is described. 
`.
Question: Which category should Paper #431206 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #431206 has the title `Backfitting in Smoothing Spline ANOVA  
` and the abstract `A scheme to compute smoothing spline ANOVA estimates for large data sets with a (near) tensor-product structure is proposed. Such data sets are common in spatial-temporal analysis and image analysis. This scheme combines backfitting algorithm with iterative imputation algorithm in order to save both computational space and time. The convergence of this algorithm and various ways to further speed it up, such as collapsing component functions and successive over-relaxation, are discussed. Issues related to its application in spatial-temporal analysis are discussed too. An application to a global analysis of historical surface temperature data is described. 
`.
Question: Which category should Paper #431206 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