Above are papers related to the following paper: [Paper_ID] Paper #154982. 	[Title] Bias, variance and prediction error for classification rules [Abstract] We study the notions of bias and variance for classification rules. Following Efron (1978) we develop a decomposition of prediction error into its natural components. Then we derive bootstrap estimates of these components and illustrate how they can be used to describe the error behaviour of a classifier in practice. In the process we also obtain a bootstrap estimate of the error of a "bagged" classifier. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #8079. 	[Title] Bias Plus Variance Decomposition for Zero-One Loss Functions [Abstract] We present a bias-variance decomposition of expected misclassification rate, the most commonly used loss function in supervised classification learning. The bias-variance decomposition for quadratic loss functions is well known and serves as an important tool for analyzing learning algorithms, yet no decomposition was offered for the more commonly used zero-one (misclassification) loss functions until the recent work of Kong & Dietterich (1995) and Breiman (1996). Their decomposition suffers from some major shortcomings though (e.g., potentially negative variance), which our decomposition avoids. We show that, in practice, the naive frequency-based estimation of the decomposition terms is by itself biased and show how to correct for this bias. We illustrate the decomposition on various algorithms and datasets from the UCI repository. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #49811. 	[Title] Estimating the Accuracy of Learned Concepts [Abstract] This paper investigates alternative estimators of the accuracy of concepts learned from examples. In particular, the cross-validation and 632 bootstrap estimators are studied, using synthetic training data and the foil learning algorithm. Our experimental results contradict previous papers in statistics, which advocate the 632 bootstrap method as superior to cross-validation. Nevertheless, our results also suggest that conclusions based on cross-validation in previous machine learning papers are unreliable. Specifically, our observations are that (i) the true error of the concept learned by foil from independently drawn sets of examples of the same concept varies widely, (ii) the estimate of true error provided by cross-validation has high variability but is approximately unbiased, and (iii) the 632 bootstrap estimator has lower variability than cross-validation, but is systematically biased. Please summarise all the given information.

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

Above are papers related to the following paper: [Paper_ID] Paper #3233. 	[Title] A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection [Abstract] We review accuracy estimation methods and compare the two most common methods: cross-validation and bootstrap. Recent experimental results on artificial data and theoretical results in restricted settings have shown that for selecting a good classifier from a set of classifiers (model selection), ten-fold cross-validation may be better than the more expensive leave-one-out cross-validation. We report on a large-scale experiment|over half a million runs of C4.5 and a Naive-Bayes algorithm|to estimate the effects of different parameters on these algorithms on real-world datasets. For cross-validation, we vary the number of folds and whether the folds are stratified or not; for bootstrap, we vary the number of bootstrap samples. Our results indicate that for real-word datasets similar to ours, the best method to use for model selection is ten-fold stratified cross validation, even if computation power allows using more folds. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #59045. 	[Title] PREDICTION GAMES AND ARCING ALGORITHMS [Abstract] Technical Report 504 December 19, 1997 Statistics Department University of California Berkeley, CA. (4720 Abstract The theory behind the success of adaptive reweighting and combining algorithms (arcing) such as Adaboost (Freund and Schapire [1995].[1996]) and others in reducing generalization error has not been well understood. By formulating prediction, both classification and regression, as a game where one player makes a selection from instances in the training set and the other a convex linear combination of predictors from a finite set, existing arcing algorithms are shown to be algorithms for finding good game strategies. An optimal game strategy finds a combined predictor that minimizes the maximum of the error over the training set. A bound on the generalization error for the combined predictors in terms of their maximum error is proven that is sharper than bounds to date. Arcing algorithms are described that converge to the optimal strategy. Schapire et.al. [1997] offered an explanation of why Adaboost works in terms of its ability to reduce the margin. Comparing Adaboost to our optimal arcing algorithm shows that their explanation is not valid and that the answer lies elsewhere. In this situation the VC-type bounds are misleading. Some empirical results are given to explore the situation. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1109542. 	[Title] The covariance inflation criterion for adaptive model selection [Abstract] We propose a new criterion for model selection in prediction problems. The covariance inflation criterion adjusts the training error by the average covariance of the predictions and responses, when the prediction rule is applied to permuted versions of the dataset. This criterion can be applied to general prediction problems (for example regression or classification), and to general prediction rules (for example stepwise regression, tree-based models and neural nets). As a byproduct we obtain a measure of the effective number of parameters used by an adaptive procedure. We relate the covariance inflation criterion to other model selection procedures and illustrate its use in some regression and classification problems. We also revisit the conditional bootstrap approach to model selection. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #25772. 	[Title] Flexible Metric Nearest Neighbor Classiflcation [Abstract] The K-nearest-neighbor decision rule assigns an object of unknown class to the plurality class among the K labeled "training" objects that are closest to it. Closeness is usually deflned in terms of a metric distance on the Euclidean space with the input measurement variables as axes. The metric chosen to deflne this distance can strongly efiect performance. An optimal choice depends on the problem at hand as characterized by the respective class distributions on the input measurement space, and within a given problem, on the location of the unknown object in that space. In this paper new types of K-nearest-neighbor procedures are described that estimate the local relevance of each input variable, or their linear combinations, for each individual point to be classifled. This information is then used to separately customize the metric used to deflne distance from that object in flnding its nearest neighbors. These procedures are a hybrid between regular K-nearest-neighbor methods and treestructured recursive partitioning techniques popular in statistics and machine learning. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1130927. 	[Title] Parametric regression 1.1 Learning problem model f bw b w in turn is an estimator [Abstract] Let us present briefly the learning problem we will address in this chapter and the following. The ultimate goal is the modelling of a mapping f : x 7! y from multidimensional input x to output y. The output can be multi-dimensional, but we will mostly address situations where it is a one dimensional real value. Furthermore, we should take into account the fact that we scarcely ever observe the actual true mapping y = f (x). This is due to perturbations such as e.g. observational noise. We will rather have a joint probability p (x; y). We expect this probability to be peaked for values of x and y corresponding to the mapping. We focus on automatic learning by example. A set D = of data sampled from the joint distribution p (x; y) = p (yjx) p (x) is collected. With the help of this set, we try to identify a model of the data, parameterised by a set of 1.2 Learning and optimisation The fit of the model to the system in a given point x is measured using a criterion representing the distance from the model prediction b y to the system, e (y; f w (x)). This is the local risk . The performance of the model is measured by the expected This quantity represents the ability to yield good performance for all the possible situations (i.e. (x; y) pairs) and is thus called generalisation error . The optimal set 1 parameters w: f w : x 7! b y. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1130931. 	[Title] TH presentee par STATISTICAL LEARNING AND REGULARISATION FOR REGRESSION Application to system identification and time [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1130934. 	[Title] 82 Lag-space estimation in time-series modelling keep track of cases where the estimation of P [Abstract] When m = 0 (no delays), we set A 0 (ffi) = f(j; k) ; j 6= kg, such that P m (*jffi) depends only on *. The estimated probabilities above become quite noisy when the number of elements in set A m and B m are small. For this reason, we estimate the standard deviation of P m (*jffi). Notice that this estimate is the empirical average of a binomial variable (either a given couple satisfied the conditions on ffi and *, or it does not). The standard deviation is then estimated easily by: Generally speaking, P m (*jffi) increases with * (laxer output test), and when ffi approaches 0 (stricter input condition). Let us now define by P m (*) the maximum over ffi of P m (*jffi): P m (*) = max ffi&gt;0 P m (*jffi). The dependability index is defined as: P 0 (*) represents how much data passes the continuity test when no input information is available. This dependability index measures how much of the remaining continuity information is associated with involving input i m . This index is then averaged over * with respect to the probability (1 P 0 (*)): m (*) (1 P 0 (*)) d* (4.8) It is clear that m (*), and therefore its average, should be positive quantities. Furthermore, if the system is deterministic, the dependability is zero after a certain number of inputs, so the sum of averages saturates. If the system is also noise-free, they sum up to 1. For any m greater than the embedding dimension: refers to results obtained using this method. 4.6 Statistical variable selection Statistical variable selection (or feature selection) encompasses a number of techniques aimed at choosing a relevant subset of input variables in a regression or a classification problem. As in the rest of this document, we will limit ourselves to considerations related to the regression problem, even though most methods discussed below apply to classification as well. Variable selection can be seen as a part of the data analysis problem: the selection (or discard) of a variable tells us about the relevance of the associated measurement to the modelled system. In a general setting, this is a purely combinatorial problem: given V possible variables, there is 2 V possible subsets (including the empty set and the full set) of these variables. Given a performance measure, such as prediction error, the only optimal scheme is to test all these subset and choose the one that gives the best performance. It is easy to see that such an extensive scheme is only viable when the number of variables is rather low. Identifying 2 V models when we have more than a few variables requires too much computation. A number of techniques have been devised to overcome this combinatorial limit. Some of them use an iterative, locally optimal technique to construct an estimate of the relevant subset in a number of steps. We will refer to them as stepwise selection methods, not to be con fused with stepwise regression, a subset of these methods that we will address below. In forward selection, we start with an empty set of variables. At each step, we select a candidate variable using a selection criteria, check whether this variable should be added to the set, and iterate until a given stop condition is reached. On the contrary, backward elimination methods start with the full set of all input variables. At each step, the least significant variable is selected according to a selection criteria. If this variable is irrelevant, it is removed and the process is iterated until a stop condition is reached. It is easy to devise examples where the inclusion of a variable causes a previously included variable to become irrelevant. It thus seems appropriate to consider running a backward elimination each time a new variable is added by forward selection. This combination of both ap proaches is known as stepwise regression in the linear regression con Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #5055. 	[Title] Boosting the Margin: A New Explanation for the Effectiveness of Voting Methods [Abstract] One of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large, and often is observed to decrease even after the training error reaches zero. In this paper, we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule, where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label. We show that techniques used in the analysis of Vapnik's support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error. We also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples. Finally, we compare our explanation to those based on the bias-variance decomposition. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1109439. 	[Title] Classifying Seismic Signals by Integrating Ensembles of Neural Networks [Abstract] This paper proposes a classification scheme based on integration of multiple Ensembles of ANNs. It is demonstrated on a classification problem, in which seismic signals of Natural Earthquakes must be distinguished from seismic signals of Artificial Explosions. A Redundant Classification Environment consists of several Ensembles of Neural Networks is created and trained on Bootstrap Sample Sets, using various data representations and architectures. The ANNs within the Ensembles are aggregated (as in Bagging) while the Ensembles are integrated non-linearly, in a signal adaptive manner, using a posterior confidence measure based on the agreement (variance) within the Ensembles. The proposed Integrated Classification Machine achieved 92.1% correct classifications on the seismic test data. Cross Validation evaluations and comparisons indicate that such integration of a collection of ANN's Ensembles is a robust way for handling high dimensional problems with a complex non-stationary signal space as in the current Seismic Classification problem. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1108551. 	[Title] From: Computational Learning Theory and Natural Systems, Chapter 18, "Cross-validation and Modal Theories", Cross-Validation and [Abstract] Cross-validation is a frequently used, intuitively pleasing technique for estimating the accuracy of theories learned by machine learning algorithms. During testing of a machine learning algorithm (foil) on new databases of prokaryotic RNA transcription promoters which we have developed, cross-validation displayed an interesting phenomenon. One theory is found repeatedly and is responsible for very little of the cross-validation error, whereas other theories are found very infrequently which tend to be responsible for the majority of the cross-validation error. It is tempting to believe that the most frequently found theory (the "modal theory") may be more accurate as a classifier of unseen data than the other theories. However, experiments showed that modal theories are not more accurate on unseen data than the other theories found less frequently during cross-validation. Modal theories may be useful in predicting when cross-validation is a poor estimate of true accuracy. We offer explanations 1 For correspondence: Department of Computer Science and Engineering, University of California, San Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1121867. 	[Title] MAJORITY VOTE CLASSIFIERS: THEORY AND APPLICATIONS [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1997. 	[Title] Cross-Validation and the Bootstrap: Estimating the Error Rate of a Prediction Rule [Abstract] A training set of data has been used to construct a rule for predicting future responses. What is the error rate of this rule? The traditional answer to this question is given by cross-validation. The cross-validation estimate of prediction error is nearly unbiased, but can be highly variable. This article discusses bootstrap estimates of prediction error, which can be thought of as smoothed versions of cross-validation. A particular bootstrap method, the 632+ rule, is shown to substantially outperform cross-validation in a catalog of 24 simulation experiments. Besides providing point estimates, we also consider estimating the variability of an error rate estimate. All of the results here are nonparametric, and apply to any possible prediction rule: however we only study classification problems with 0-1 loss in detail. Our simulations include "smooth" prediction rules like Fisher's Linear Discriminant Function, and unsmooth ones like Nearest Neighbors. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #192850. 	[Title] Characterizing the generalization performance of model selection strategies [Abstract] We investigate the structure of model selection problems via the bias/variance decomposition. In particular, we characterize the essential aspects of a model selection task by the bias and variance profiles it generates over the sequence of hypothesis classes. With this view, we develop a new understanding of complexity-penalization methods: First, the penalty terms can be interpreted as postulating a particular profile for the variances as a function of model complexityif the postulated and true profiles do not match, then systematic under-fitting or over-fitting results, depending on whether the penalty terms are too large or too small. Second, we observe that it is generally best to penalize according to the true variances of the task, and therefore no fixed penalization strategy is optimal across all problems. We then use this characterization to introduce the notion of easy versus hard model selection problems. Here we show that if the variance profile grows too rapidly in relation to the biases, then standard model selection techniques become prone to significant errors. This can happen, for example, in regression problems where the independent variables are drawn from wide-tailed distributions. To counter this, we discuss a new model selection strategy that dramatically outperforms standard complexity-penalization and hold-out meth ods on these hard tasks. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #39124. 	[Title] The Utility of Feature Weighting in Nearest-Neighbor Algorithms [Abstract] Nearest-neighbor algorithms are known to depend heavily on their distance metric. In this paper, we investigate the use of a weighted Euclidean metric in which the weight for each feature comes from a small set of options. We describe Diet, an algorithm that directs search through a space of discrete weights using cross-validation error as its evaluation function. Although a large set of possible weights can reduce the learner's bias, it can also lead to increased variance and overfitting. Our empirical study shows that, for many data sets, there is an advantage to weighting features, but that increasing the number of possible weights beyond two (zero and one) has very little benefit and sometimes degrades performance. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #2654. 	[Title] Experiments with a New Boosting Algorithm [Abstract] In an earlier paper, we introduced a new boosting algorithm called AdaBoost which, theoretically, can be used to significantly reduce the error of any learning algorithm that consistently generates classifiers whose performance is a little better than random guessing. We also introduced the related notion of a pseudo-loss which is a method for forcing a learning algorithm of multi-label concepts to concentrate on the labels that are hardest to discriminate. In this paper, we describe experiments we carried out to assess how well AdaBoost with and without pseudo-loss, performs on real learning problems. We performed two sets of experiments. The first set compared boosting to Breiman's bagging method when used to aggregate various classifiers (including decision trees and single attribute-value tests). We compared the performance of the two methods on a collection of machine-learning benchmarks. In the second set of experiments, we studied in more detail the performance of boosting using a nearest-neighbor classifier on an OCR problem. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #105057. 	[Title] Error-Correcting Output Codes for Local Learners [Abstract] Error-correcting output codes (ECOCs) represent classes with a set of output bits, where each bit encodes a binary classification task corresponding to a unique partition of the classes. Algorithms that use ECOCs learn the function corresponding to each bit, and combine them to generate class predictions. ECOCs can reduce both variance and bias errors for multiclass classification tasks when the errors made at the output bits are not correlated. They work well with algorithms that eagerly induce global classifiers (e.g., C4.5) but do not assist simple local classifiers (e.g., nearest neighbor), which yield correlated predictions across the output bits. We show that the output bit predictions of local learners can be decorrelated by selecting different features for each bit. We present promising empirical results for this combination of ECOCs, near est neighbor, and feature selection. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1133028. 	[Title] An Efficient Method To Estimate Bagging's Generalization Error [Abstract] In bagging [Bre94a] one uses bootstrap replicates of the training set [Efr79, ET93] to try to improve a learning algorithm's performance. The computational requirements for estimating the resultant generalization error on a test set by means of cross-validation are often prohibitive; for leave-one-out cross-validation one needs to train the underlying algorithm on the order of m- times, where m is the size of the training set and is the number of replicates. This paper presents several techniques for exploiting the bias-variance decomposition [GBD92, Wol96] to estimate the generalization error of a bagged learning algorithm without invoking yet more training of the underlying learning algorithm. The best of our estimators exploits stacking [Wol92]. In a set of experiments reported here, it was found to be more accurate than both the alternative cross-validation-based estimator of the bagged algorithm's error and the cross-validation-based estimator of the underlying algorithm's error. This improvement was particularly pronounced for small test sets. This suggests a novel justification for using bagging| im proved estimation of generalization error. 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 #1102442. 	[Title] Unbiased Assessment of Learning Algorithms [Abstract] In order to rank the performance of machine learning algorithms, many researchers conduct experiments on benchmark data sets. Since most learning algorithms have domain-specific parameters, it is a popular custom to adapt these parameters to obtain a minimal error rate on the test set. The same rate is then used to rank the algorithm, which causes an optimistic bias. We quantify this bias, showing, in particular, that an algorithm with more parameters will probably be ranked higher than an equally good algorithm with fewer parameters. We demonstrate this result, showing the number of parameters and trials required in order to pretend to outperform C4.5 or FOIL, respectively, for various benchmark problems. We then describe out how unbiased ranking experiments should be conducted. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #27249. 	[Title] Machine Learning Bias, Statistical Bias, and Statistical Variance of Decision Tree Algorithms [Abstract] The term "bias" is widely used|and with different meanings|in the fields of machine learning and statistics. This paper clarifies the uses of this term and shows how to measure and visualize the statistical bias and variance of learning algorithms. Statistical bias and variance can be applied to diagnose problems with machine learning bias, and the paper shows four examples of this. Finally, the paper discusses methods of reducing bias and variance. Methods based on voting can reduce variance, and the paper compares Breiman's bagging method and our own tree randomization method for voting decision trees. Both methods uniformly improve performance on data sets from the Irvine repository. Tree randomization yields perfect performance on the Letter Recognition task. A weighted nearest neighbor algorithm based on the infinite bootstrap is also introduced. In general, decision tree algorithms have moderate-to-high variance, so an important implication of this work is that variance|rather than appropriate or inappropriate machine learning bias|is an important cause of poor performance for decision tree algorithms. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #27250. 	[Title] Why Does Bagging Work? A Bayesian Account and its Implications bagging's success, both in a [Abstract] The error rate of decision-tree and other classification learners can often be much reduced by bagging: learning multiple models from bootstrap samples of the database, and combining them by uniform voting. In this paper we empirically test two alternative explanations for this, both based on Bayesian learning theory: (1) bagging works because it is an approximation to the optimal procedure of Bayesian model averaging, with an appropriate implicit prior; (2) bagging works because it effectively shifts the prior to a more appropriate region of model space. All the experimental evidence contradicts the first hypothesis, and confirms the second. Bagging (Breiman 1996a) is a simple and effective way to reduce the error rate of many classification learning algorithms. For example, in the empirical study described below, it reduces the error of a decision-tree learner in 19 of 26 databases, by 4% on average. In the bagging procedure, given a training set of size s, a "bootstrap" replicate of it is constructed by taking s samples with replacement from the training set. Thus a new training set of the same size is produced, where each of the original examples may appear once, more than once, or not. On average, 63% of the original examples will appear in the bootstrap sample. The learning algorithm is then applied to this training set. This procedure is repeated m times, and the resulting m models are aggregated by uniform voting. Bagging is one of several "multiple model" approaches that have recently received much attention (see, for example, (Chan, Stolfo, & Wolpert 1996)). Other procedures of this type include boosting (Freund & Schapire 1996) and stacking (Wolpert 1992). Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #3187. 	[Title] A Reference Bayesian Test for Nested Hypotheses And its Relationship to the Schwarz Criterion [Abstract]  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 #7032. 	[Title] Combining estimates in regression and classification [Abstract] We consider the problem of how to combine a collection of general regression fit vectors in order to obtain a better predictive model. The individual fits may be from subset linear regression, ridge regression, or something more complex like a neural network. We develop a general framework for this problem and examine a recent cross-validation-based proposal called "stacking" in this context. Combination methods based on the bootstrap and analytic methods are also derived and compared in a number of examples, including best subsets regression and regression trees. Finally, we apply these ideas to classification problems where the estimated combination weights can yield insight into the structure of the problem. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1103737. 	[Title] On learning hierarchical classifications [Abstract] Many significant real-world classification tasks involve a large number of categories which are arranged in a hierarchical structure; for example, classifying documents into subject categories under the library of congress scheme, or classifying world-wide-web documents into topic hierarchies. We investigate the potential benefits of using a given hierarchy over base classes to learn accurate multi-category classifiers for these domains. First, we consider the possibility of exploiting a class hierarchy as prior knowledge that can help one learn a more accurate classifier. We explore the benefits of learning category-discriminants in a hard top-down fashion and compare this to a soft approach which shares training data among sibling categories. In doing so, we verify that hierarchies have the potential to improve prediction accuracy. But we argue that the reasons for this can be subtle. Sometimes, the improvement is only because using a hierarchy happens to constrain the expressiveness of a hypothesis class in an appropriate manner. However, various controlled experiments show that in other cases the performance advantage associated with using a hierarchy really does seem to be due to the prior knowledge it encodes. Please summarise all the given information.

Paper #154982 has the title `Bias, variance and prediction error for classification rules  
` and the abstract `We study the notions of bias and variance for classification rules. Following Efron (1978) we develop a decomposition of prediction error into its natural components. Then we derive bootstrap estimates of these components and illustrate how they can be used to describe the error behaviour of a classifier in practice. In the process we also obtain a bootstrap estimate of the error of a "bagged" classifier. 
`.
Question: Which category should Paper #154982 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #154982 has the title `Bias, variance and prediction error for classification rules  
` and the abstract `We study the notions of bias and variance for classification rules. Following Efron (1978) we develop a decomposition of prediction error into its natural components. Then we derive bootstrap estimates of these components and illustrate how they can be used to describe the error behaviour of a classifier in practice. In the process we also obtain a bootstrap estimate of the error of a "bagged" classifier. 
`.
Question: Which category should Paper #154982 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