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140
Mixing Strategies for Density Estimation
 Ann. Statist
"... General results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under KullbackLeibler and square L 2 losses. It is shown that without knowing which strategy works best for the underlying density, a single strategy can be constructed by m ..."
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Cited by 59 (9 self)
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General results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under KullbackLeibler and square L 2 losses. It is shown that without knowing which strategy works best for the underlying density, a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks. A consequence is that under some mild conditions, an asymptotically minimaxrate adaptive estimator exists for a given countable collection of density classes, i.e., a single estimator can be constructed to be simultaneously minimaxrate optimal for all the function classes being considered. A demonstration is given for highdimensional density estimation on [0; 1] d where the constructed estimator adapts to smoothness and interactionorder over some piecewise Besov classes, and is consistent for all the densities with finite entropy. 1. Introduction. In Recent years, there has been an increasing interest in adaptive fu...
Combining Different Procedures for Adaptive Regression
 Journal of Multivariate Analysis
, 1998
"... Given any countable collection of regression procedures (e.g., kernel, spline, wavelet, local polynomial, neural nets, etc), we show that a single adaptive procedure can be constructed to share the advantages of them to a great extent in terms of global squared L 2 risk. The combined procedure basic ..."
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Cited by 58 (10 self)
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Given any countable collection of regression procedures (e.g., kernel, spline, wavelet, local polynomial, neural nets, etc), we show that a single adaptive procedure can be constructed to share the advantages of them to a great extent in terms of global squared L 2 risk. The combined procedure basically pays a price only of order 1=n for adaptation over the collection. An interesting consequence is that for a countable collection of classes of regression functions (possibly of completely different characteristics), a minimaxrate adaptive estimator can be constructed such that it automatically converges at the right rate for each of the classes being considered.
Mutual Information, Metric Entropy, and Cumulative Relative Entropy Risk
 Annals of Statistics
, 1996
"... Assume fP ` : ` 2 \Thetag is a set of probability distributions with a common dominating measure on a complete separable metric space Y . A state ` 2 \Theta is chosen by Nature. A statistician gets n independent observations Y 1 ; : : : ; Y n from Y distributed according to P ` . For each time ..."
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Cited by 52 (2 self)
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Assume fP ` : ` 2 \Thetag is a set of probability distributions with a common dominating measure on a complete separable metric space Y . A state ` 2 \Theta is chosen by Nature. A statistician gets n independent observations Y 1 ; : : : ; Y n from Y distributed according to P ` . For each time t between 1 and n, based on the observations Y 1 ; : : : ; Y t\Gamma1 , the statistician produces an estimated distribution P t for P ` , and suffers a loss L(P ` ; P t ). The cumulative risk for the statistician is the average total loss up to time n. Of special interest in information theory, data compression, mathematical finance, computational learning theory and statistical mechanics is the special case when the loss L(P ` ; P t ) is the relative entropy between the true distribution P ` and the estimated distribution P t . Here the cumulative Bayes risk from time 1 to n is the mutual information between the random parameter \Theta and the observations Y 1 ; : : : ;...
A Vector Quantization Approach to Universal Noiseless Coding and Quantization
 IEEE Trans. Inform. Theory
, 1996
"... AbstractA twostage code is a block code in which each block of data is coded in two stages: the first stage codes the identity of a block code among a collection of codes, and the second stage codes the data using the identified code. The collection of codes may he noiseless codes, fixedrate quan ..."
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Cited by 46 (11 self)
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AbstractA twostage code is a block code in which each block of data is coded in two stages: the first stage codes the identity of a block code among a collection of codes, and the second stage codes the data using the identified code. The collection of codes may he noiseless codes, fixedrate quantizers, or variablerate quantizers. We take a vector quantization approach to twostage coding, in which the first stage code can be regarded as a vector quantizer that “quantizes ” the input data of length n to one of a fixed collection of block codes. We apply the generalized Lloyd algorithm to the firststage quantizer, using induced measures of rate and distortion, to design locally optimal twostage, codes. On a source of medical images, twostage variahlerate vector quantizers designed in this way outperform standard (onestage) fixedrate vector quantizers by over 9 dB. The tail of the operational distortionrate function of the firststage quantizer determines the optimal rate of convergence of the redundancy of a universal sequence of twostage codes. We show that there exist twostage universal noiseless codes, fixedrate quantizers, and variablerate quantizers whose perletter rate and distortion redundancies converge to zero as (k/2)n ’ logn, when the universe of sources has finite dimension k. This extends the achievability part of Rissanen’s theorem from universal noiseless codes to universal quantizers. Further, we show that the redundancies converge as O(n’) when the universe of sources is countable, and as O(r~l+‘) when the universe of sources is infinitedimensional, under appropriate conditions. Index TermsTwostage, adaptive, compression, minimum description length, clustering. I.
Precise Minimax Redundancy and Regret
 IEEE TRANS. INFORMATION THEORY
, 2004
"... Recent years have seen a resurgence of interest in redundancy of lossless coding. The redundancy (regret) of universal xed{to{variable length coding for a class of sources determines by how much the actual code length exceeds the optimal (ideal over the class) code length. In a minimax scenario ..."
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Cited by 46 (15 self)
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Recent years have seen a resurgence of interest in redundancy of lossless coding. The redundancy (regret) of universal xed{to{variable length coding for a class of sources determines by how much the actual code length exceeds the optimal (ideal over the class) code length. In a minimax scenario one nds the best code for the worst source either in the worst case (called also maximal minimax) or on average. We rst study the worst case minimax redundancy over a class of stationary ergodic sources and replace Shtarkov's bound by an exact formula. Among others, we prove that a generalized Shannon code minimizes the worst case redundancy, derive asymptotically its redundancy, and establish some general properties. This allows us to obtain precise redundancy rates for memoryless, Markov and renewal sources. For example, we derive the exact constant of the redundancy rate for memoryless and Markov sources by showing that an integer nature of coding contributes log(log m=(m 1))= log m+ o(1) where m is the size of the alphabet. Then we deal with the average minimax redundancy and regret. Our approach
Predictability, Complexity, and Learning
, 2001
"... We define predictive information Ipred(T) as the mutual information between the past and the future of a time series. Three qualitatively different behaviors are found in the limit of large observation times T: Ipred(T) can remain finite, grow logarithmically, or grow as a fractional power law. If t ..."
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Cited by 46 (2 self)
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We define predictive information Ipred(T) as the mutual information between the past and the future of a time series. Three qualitatively different behaviors are found in the limit of large observation times T: Ipred(T) can remain finite, grow logarithmically, or grow as a fractional power law. If the time series allows us to learn a model with a finite number of parameters, then Ipred(T) grows logarithmically with a coefficient that counts the dimensionality of the model space. In contrast, powerlaw growth is associated, for example, with the learning of infinite parameter (or nonparametric) models such as continuous functions with smoothness constraints. There are connections between the predictive information and measures of complexity that have been defined both in learning theory and the analysis of physical systems through statistical mechanics and dynamical systems theory. Furthermore, in the same way that entropy provides the unique measure of available information consistent with some simple and plausible conditions, we argue that the divergent part of Ipred(T) provides the unique measure for the complexity of dynamics underlying a time series. Finally, we discuss how these ideas may be useful in problems in physics, statistics, and biology.
A General Minimax Result for Relative Entropy
 IEEE Trans. Inform. Theory
, 1996
"... : Suppose Nature picks a probability measure P ` on a complete separable metric space X at random from a measurable set P \Theta = fP ` : ` 2 \Thetag. Then, without knowing `, a statistician picks a measure Q on X. Finally, the statistician suffers a loss D(P ` jjQ), the relative entropy between P ..."
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Cited by 44 (2 self)
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: Suppose Nature picks a probability measure P ` on a complete separable metric space X at random from a measurable set P \Theta = fP ` : ` 2 \Thetag. Then, without knowing `, a statistician picks a measure Q on X. Finally, the statistician suffers a loss D(P ` jjQ), the relative entropy between P ` and Q. We show that the minimax and maximin values of this game are always equal, and there is always a minimax strategy in the closure of the set of all Bayes strategies. This generalizes previous results of Gallager, and Davisson and LeonGarcia. Index terms: minimax theorem, minimax redundancy, minimax risk, Bayes risk, relative entropy, KullbackLeibler divergence, density estimation, source coding, channel capacity, computational learning theory 1 Introduction Consider a sequential estimation game in which a statistician is given n independent observations Y 1 ; : : : ; Yn distributed according to an unknown distribution ~ P ` chosen at random by Nature from the set f ~ P ` : ` 2 \...
On predictive distributions and Bayesian networks
 Statistics and Computing
, 2000
"... this paper we are interested in discrete prediction problems for a decisiontheoretic setting, where the ..."
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Cited by 39 (30 self)
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this paper we are interested in discrete prediction problems for a decisiontheoretic setting, where the
Modelbased decoding, information estimation, and changepoint detection in multineuron spike trains
 UNDER REVIEW, NEURAL COMPUTATION
, 2007
"... Understanding how stimulus information is encoded in spike trains is a central problem in computational neuroscience. Decoding methods provide an important tool for addressing this problem, by allowing us to explicitly read out the information contained in spike responses. Here we introduce several ..."
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Cited by 37 (17 self)
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Understanding how stimulus information is encoded in spike trains is a central problem in computational neuroscience. Decoding methods provide an important tool for addressing this problem, by allowing us to explicitly read out the information contained in spike responses. Here we introduce several decoding methods based on pointprocess neural encoding models (i.e. “forward ” models that predict spike responses to novel stimuli). These models have concave loglikelihood functions, allowing for efficient fitting via maximum likelihood. Moreover, we may use the likelihood of the observed spike trains under the model to perform optimal decoding. We present: (1) a tractable algorithm for computing the maximum a posteriori (MAP) estimate of the stimulus — the most probable stimulus to have generated the observed single or multiplespike train response, given some prior distribution over the stimulus; (2) a Gaussian approximation to the posterior distribution, which allows us to quantify the fidelity with which various stimulus features are encoded; (3) an efficient method for estimating the mutual information between the stimulus and the response; and (4) a framework for the detection of changepoint times (e.g. the time at which the stimulus undergoes a change in mean or variance), by marginalizing over the posterior distribution of stimuli. We show several examples illustrating the performance of these estimators with simulated data.
Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
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