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Powerlaw distributions in empirical data
 ISSN 00361445. doi: 10.1137/ 070710111. URL http://dx.doi.org/10.1137/070710111
, 2009
"... Powerlaw distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and manmade phenomena. Unfortunately, the empirical detection and characterization of power laws is made difficult by the large fluctuations that occur in the t ..."
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Cited by 199 (3 self)
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Powerlaw distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and manmade phenomena. Unfortunately, the empirical detection and characterization of power laws is made difficult by the large fluctuations that occur in the tail of the distribution. In particular, standard methods such as leastsquares fitting are known to produce systematically biased estimates of parameters for powerlaw distributions and should not be used in most circumstances. Here we describe statistical techniques for making accurate parameter estimates for powerlaw data, based on maximum likelihood methods and the KolmogorovSmirnov statistic. We also show how to tell whether the data follow a powerlaw distribution at all, defining quantitative measures that indicate when the power law is a reasonable fit to the data and when it is not. We demonstrate these methods by applying them to twentyfour realworld data sets from a range of different disciplines. Each of the data sets has been conjectured previously to follow a powerlaw distribution. In some cases we find these conjectures to be consistent with the data while in others the power law is ruled out.
Philosophy and the practice of Bayesian statistics
, 2010
"... A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually ..."
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Cited by 13 (5 self)
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A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypotheticodeductivism. We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science. Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework.
Risk of Bayesian Inference in Misspecified Models, and the Sandwich Covariance Matrix ∗
, 2009
"... It is well known that in misspecified parametric models, the maximum likelihood estimator (MLE) is consistent for the pseudotrue value and has an asymptotically normal sampling distribution with "sandwich " covariance matrix. Also, posteriors are asymptotically centered at the MLE, normal and of as ..."
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Cited by 2 (0 self)
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It is well known that in misspecified parametric models, the maximum likelihood estimator (MLE) is consistent for the pseudotrue value and has an asymptotically normal sampling distribution with "sandwich " covariance matrix. Also, posteriors are asymptotically centered at the MLE, normal and of asymptotic variance that is in general different than the sandwich matrix. It is shown that due to this discrepancy, Bayesian inference about the pseudotrue parameter value is in general of lower asymptotic frquentist risk when the original posterior is substituted by an artificial normal posterior centered at the MLE with sandwich covariance matrix. An algorithm is suggested that allows the implementation of this artificial posterior also in models with high dimensional nuisance parameters which cannot reasonably be estimated by maximizing the likelihood. JEL classification: C44, C11
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, 2012
"... doi: 10.3389/fncom.2012.00024 Selectionist and evolutionary approaches to brain function: a critical appraisal ..."
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doi: 10.3389/fncom.2012.00024 Selectionist and evolutionary approaches to brain function: a critical appraisal
The Safe Bayesian: learning the learning rate via the mixability gap
"... Abstract. Standard Bayesian inference can behave suboptimally if the model is wrong. We present a modification of Bayesian inference which continues to achieve good rates with wrong models. Our method adapts the Bayesian learning rate to the data, picking the rate minimizing the cumulative loss of s ..."
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Abstract. Standard Bayesian inference can behave suboptimally if the model is wrong. We present a modification of Bayesian inference which continues to achieve good rates with wrong models. Our method adapts the Bayesian learning rate to the data, picking the rate minimizing the cumulative loss of sequential prediction by posterior randomization. Our results can also be used to adapt the learning rate in a PACBayesian context. The results are based on an extension of an inequality due to T. Zhang and others to dependent random variables. 1
Language Acquisition and Probabilistic Models: keeping it simple
"... Hierarchical Bayesian Models (HBMs) have been used with some success to capture empirically observed patterns of under and overgeneralization in child language acquisition. However, as is well known, HBMs are “ideal ” learning systems, assuming access to unlimited computational resources that may n ..."
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Hierarchical Bayesian Models (HBMs) have been used with some success to capture empirically observed patterns of under and overgeneralization in child language acquisition. However, as is well known, HBMs are “ideal ” learning systems, assuming access to unlimited computational resources that may not be available to child language learners. Consequently, it remains crucial to carefully assess the use of HBMs along with alternative, possibly simpler, candidate models. This paper presents such an evaluation for a language acquisition domain where explicit HBMs have been proposed: the acquisition of English dative constructions. In particular, we present a detailed, empiricallygrounded modelselection comparison of HBMs vs. a simpler alternative based on clustering along with maximum likelihood estimation that we call linear competition learning (LCL). Our results demonstrate that LCL can match HBM model performance without incurring on the high computational costs associated with HBMs. 1
Address of the First and Second authors: CEREMADE
, 2010
"... Abstract: We study convergence rates of Bayesian density estimators based on finite locationscale mixtures of a kernel proportional to exp{−x  p}. We construct a finite mixture approximation of densities whose logarithm is locally βHölder, with squared integrable Hölder constant. Under additiona ..."
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Abstract: We study convergence rates of Bayesian density estimators based on finite locationscale mixtures of a kernel proportional to exp{−x  p}. We construct a finite mixture approximation of densities whose logarithm is locally βHölder, with squared integrable Hölder constant. Under additional tail and moment conditions, the approximation is minimax for both the KullbackLeibler divergence. We use this approximation to establish convergence rates for a Bayesian mixture model with priors on the weights, locations, and the number of components. Regarding these priors, we provide general conditions under which the posterior converges at a near optimal rate, and is rateadaptive with respect to the smoothness of the logarithm of the true density.