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Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
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Cited by 428 (62 self)
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We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being kcolorable or having a aeclique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edgequeries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...
Diagnostic Measures for Model Criticism
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 1996
"... ... In this article we present the general outlook and discuss general families of elaborations for use in practice; the exponential connection elaboration plays a key role. We then describe model elaborations for use in diagnosing: departures from normality, goodness of fit in generalized linear mo ..."
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Cited by 13 (1 self)
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... In this article we present the general outlook and discuss general families of elaborations for use in practice; the exponential connection elaboration plays a key role. We then describe model elaborations for use in diagnosing: departures from normality, goodness of fit in generalized linear models, and variable selection in regression and outlier detection. We illustrate our approach with two applications.
Can Finite Samples Detect Singularities of RealValued Functions?
 Proceedings of the 24th Annual ACM Symposium on the Theory of Computer Science
, 1992
"... Consider the following type of problem: There is an unknown function, f : R n ! R m , there is also a blackbox that on query x (2 R n ) returns f(x). Is there an algorithm that, using probes to the blackbox, can figure out analytic information about f? (For an example: "Is f a polynomia ..."
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Cited by 11 (1 self)
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Consider the following type of problem: There is an unknown function, f : R n ! R m , there is also a blackbox that on query x (2 R n ) returns f(x). Is there an algorithm that, using probes to the blackbox, can figure out analytic information about f? (For an example: "Is f a polynomial? ", "Is f a second order differentiable at x = (0; 0; : : : ; 0)?" etc.). Clearly, for examples as these, if we bound the number of probes an algorithm has to settle for, no algorithm can carry the task. On the other hand, if one allows an infinite iteration of a `probe compute and guess' process, then, (quite surprisingly) for many such questions, there are algorithms that are guaranteed to be correct in all but finitely many of their guesses. We call such questions Decidable In the Limit, (DIL). We analyze the class of DIL problems and provide a necessary and sufficient condition for the membership of a decision problem in this class. We offer an algorithm for any DIL problem, and apply it to several types of learning tasks. We introduce a an extension of the usual Inductive Inference learning model  Inductive Inference with a Cheating Teacher. In this model the teacher may choose to present to the learner, not only a language belonging to the agreed  upon family of languages, but also an arbitrary language outside this family. In such a case we require that the learner will be able to eventually detect the faulty choice made by the teacher. We show that such strong type of learning is possible, and there exist learning algorithms that will fail only on arbitrarily small sets of faulty languages. Furthermore, if an apriori probability distribution P , according to which f is being chosen, is available to the algorithm, then it can be strengthened into a finite A prelimi...
On the Testability of Identification in Some Nonparametric Models with Endogeneity
, 2013
"... This paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity. The first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogenei ..."
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Cited by 5 (0 self)
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This paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity. The first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogeneity involving mean independence restrictions. These conditions are typically referred to as completeness conditions. The second and third hypothesis testing problems we examine concern testing for identification directly in some nonparametric models with endogeneity involving quantile independence restrictions. For each of these hypothesis testing problems, we provide conditions under which any test will have power no greater than size against any alternative. In this sense, we conclude that no nontrivial tests for these hypothesis testing problems exist.
Some Bayesian perspectives on statistical modelling
, 1988
"... I would like to thank my supervisor, Professor A. F. M. Smith, for all his advice and encourage ..."
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Cited by 3 (2 self)
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I would like to thank my supervisor, Professor A. F. M. Smith, for all his advice and encourage
GoodnessofFit Tests Based on the Kernel Density Estimate
, 2002
"... Given an i.i.d. sample drawn from a density f on the real line, one has to test whether f is in a given class of densities. We investigate testing procedures constructed on the basis of minimizing the L 1 distance between the kernel density estimate and any density in the hypothesized class. Genera ..."
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Cited by 2 (0 self)
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Given an i.i.d. sample drawn from a density f on the real line, one has to test whether f is in a given class of densities. We investigate testing procedures constructed on the basis of minimizing the L 1 distance between the kernel density estimate and any density in the hypothesized class. General nonasymptotic bounds are derived for the power of the test. It is shown that the concentratedness of the datadependent smoothing factor plays a key role in the performance of the test, as well as the "size" of the hypothesized class of densities.
Bounds for the Loss in Probability of Correct Classification Under Model Based Approximation
"... In many pattern recognition/classification problem the true class conditional model and class probabilities are approximated for reasons of reducing complexity and/or of statistical estimation. The approximated classifier is expected to have worse performance, here measured by the probability of cor ..."
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In many pattern recognition/classification problem the true class conditional model and class probabilities are approximated for reasons of reducing complexity and/or of statistical estimation. The approximated classifier is expected to have worse performance, here measured by the probability of correct classification. We present an analysis valid in general, and easily computable formulas for estimating the degradation in probability of correct classification when compared to the optimal classifier. An example of an approximation is the Naïve Bayes classifier. We show that the performance of the Naïve Bayes depends on the degree of functional dependence between the features and labels. We provide a sufficient condition for zero loss of performance, too.
unknown title
"... Abstract – We study the question of determining whether an unknown function has a particular property or is ¤far from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may quer ..."
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Abstract – We study the question of determining whether an unknown function has a particular property or is ¤far from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being ¥colorable or having a ¦clique (clique of density ¦ w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only§©¨�����������¤� � edgequeries into the graph, where ¤ is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which correspond to the property being tested, if it holds for the input graph. 1.
. After every n, one must make a guess whether f
"... Let a class of densities be given. We draw an i.i.d. sample from a density f which may or may not be in . After every n, one must make a guess whether f or not. A class is almost surely discernible if there exists such a sequence of classification rules such that for any f , we make finitely m ..."
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Let a class of densities be given. We draw an i.i.d. sample from a density f which may or may not be in . After every n, one must make a guess whether f or not. A class is almost surely discernible if there exists such a sequence of classification rules such that for any f , we make finitely many errors almost surely. In this paper several results are given that allow one to decide whether a class is almost surely discernible. For example, continuity and square integrability are not discernible, but unimodality, logconcavity, and boundedness by a given constant are. Keywords and phrases. Density estimation, kernel estimate, convergence, discernibility, hypothesis testing, asymptotic optimality, minimax rate, minimum distance estimation, total boundedness. 1991 Mathematics Subject Classifications: Primary 62G05.