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Learning to be Bayesian without supervision
 in Adv. Neural Information Processing Systems (NIPS*06
, 2007
"... Bayesian estimators are defined in terms of the posterior distribution. Typically, this is written as the product of the likelihood function and a prior probability density, both of which are assumed to be known. But in many situations, the prior density is not known, and is difficult to learn from ..."
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Cited by 24 (7 self)
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Bayesian estimators are defined in terms of the posterior distribution. Typically, this is written as the product of the likelihood function and a prior probability density, both of which are assumed to be known. But in many situations, the prior density is not known, and is difficult to learn from data since one does not have access to uncorrupted samples of the variable being estimated. We show that for a wide variety of observation models, the Bayes least squares (BLS) estimator may be formulated without explicit reference to the prior. Specifically, we derive a direct expression for the estimator, and a related expression for the mean squared estimation error, both in terms of the density of the observed measurements. Each of these priorfree formulations allows us to approximate the estimator given a sufficient amount of observed data. We use the first form to develop practical nonparametric approximations of BLS estimators for several different observation processes, and the second form to develop a parametric family of estimators for use in the additive Gaussian noise case. We examine the empirical performance of these estimators as a function of the amount of observed data. 1
Least squares estimation without priors or supervision
 Neural Computation
, 2011
"... Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values), or a prior probability model for the true values. Here, we consider the problem of obtaining a leastsquares estimator given a measurement process with ..."
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Cited by 4 (2 self)
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Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values), or a prior probability model for the true values. Here, we consider the problem of obtaining a leastsquares estimator given a measurement process with known statistics (i.e., a likelihood function), and a set of unsupervised measurements, each arising from a corresponding true value drawn randomly from an unknown distribution. We develop a general expression for a nonparametric empirical Bayes least squares (NEBLS) estimator, that expresses the optimal leastsquares estimator in terms of the measurement density, with no explicit reference to the unknown (prior) density. We study the conditions under which such estimators exist, and derive specific forms for a variety of different measurement processes. We further show that each of these NEBLS estimators may be used to express the mean squared estimation error as an expectation over the measurement density alone, thus generalizing Stein’s unbiased risk estimator (SURE) which provides such an expression for the additive Gaussian noise case. This error expression may then be optimized over noisy measurement samples, in the absence of supervised training data, yielding a generalized SUREoptimized parametric
Learning least squares estimators without assumed priors or supervision
, 2009
"... The two standard methods of obtaining a leastsquares optimal estimator are (1) Bayesian estimation, in which one assumes a prior distribution on the true values and combines this with a model of the measurement process to obtain an optimal estimator, and (2) supervised regression, in which one opti ..."
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Cited by 3 (2 self)
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The two standard methods of obtaining a leastsquares optimal estimator are (1) Bayesian estimation, in which one assumes a prior distribution on the true values and combines this with a model of the measurement process to obtain an optimal estimator, and (2) supervised regression, in which one optimizes a parametric estimator over a training set containing pairs of corrupted measurements and their associated true values. But many realworld systems do not have access to either supervised training examples or a prior model. Here, we study the problem of obtaining an optimal estimator given a measurement process with known statistics, and a set of corrupted measurements of random values drawn from an unknown prior. We develop a general form of nonparametric empirical Bayesian estimator that is written as a direct function of the measurement density, with no explicit reference to the prior. We study the observation conditions under which such “priorfree ” estimators may be obtained, and we derive specific forms for a variety of different corruption processes. Each of these priorfree estimators may also be used to express the mean squared estimation error as an expectation over the measurement density, thus generalizing Stein’s unbiased risk estimator (SURE) which provides such an expression for the additive Gaussian noise case. Minimizing this expression over measurement samples provides an “unsupervised
ARTICLE Communicated by Konrad Paul Kording Least Squares Estimation Without Priors or Supervision
"... Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values) or a prior probability model for the true values. Here, we consider the problem of obtaining a least squares estimator given a measurement process with kn ..."
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Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values) or a prior probability model for the true values. Here, we consider the problem of obtaining a least squares estimator given a measurement process with known statistics (i.e., a likelihood function) and a set of unsupervised measurements, each arising from a corresponding true value drawn randomly from an unknown distribution. We develop a general expression for a nonparametric empirical Bayes least squares (NEBLS) estimator, which expresses the optimal least squares estimator in terms of the measurement density, with no explicit reference to the unknown (prior) density. We study the conditions under which such estimators exist and derive specific forms for a variety of different measurement processes. We further show that each of these NEBLS estimators may be used to express the mean squared estimation error as an expectation over the measurement density alone, thus generalizing Stein’s unbiased