III. Contributions to the mathematical theory of evolution

By K. Pearson

1. If a series of measurements, physical, biological, anthropological, or economical, not of the same object, but of a group of objects of the same type or family, be made, and a curve be constructed by plotting up the number of times the measurements fall within a given small unit of range to the range, this curve may be termed a frequency curve. As a rule this frequency curve takes the well known form of the curve of errors, and such a curve may be termed a normal frequency curve. The latter curve is symmetrical about its maximum ordinate. Occasionally, however, frequency curves do not take the normal form, and are then generally, but not necessarily, asymmetrical.

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Citations in the corpus

Learning Distributions from their Samples under Communication Constraints (2019, background): “INTRODUCTION Estimating a distribution from samples is a fundamental unsupervised learning problem that has been studied in statistics since the late nineteenth century [19].”
Noise Estimation Using Density Estimation for Self-Supervised Multimodal Learning (2020, methodology): “derlying the data, which is assumed to be i:i:d. Existing density estimation algorithms can be divided into two categories: (a) parametric or semiparametric approaches such as Gaussian Mixture models [57,1,16,40,90] and probabilistic graphical models [25,29,87]; and (b) non-parametric approaches suc”
Tensor decompositions for learning latent variable models (2012, neutral): “L G ]”
A Method of Moments for Mixture Models and Hidden Markov Models (2012, methodology): “For the discrete models such as the document topic model of Section 2.1 and discrete HMMs (Mos-sel and Roch, 2006; Hsu et al., 2009) Using standard techniques ( e.g. , Chaudhuri et al. (2009); Vershynin (2012)), the condition can also be shown to hold for mixtures of various continuous distributions”
Eliciting Risk and Time Preferences (2008, unknown): “…earlier finding that the discount rate was 10.1% in Section 2 masked two very different 30Mixture models have an astonishing pedigree in statistics: Pearson (1894) examined data on the ratio of forehead to body length of 1000 crabs to illustrate “the dissection of abnormal frequency curves into…”
Provable Meta-Learning of Linear Representations (2020, methodology): “An alternative to optimization-based approaches such as maximum likelihood estimation, the method-of-moments is among the oldest statistical techniques (Pearson, 1894) and has recently been used to estimate parameters in latent variable models (Anandkumar et al., 2012).”
Karl Pearson and the Origin of Kurtosis (2009, background): “The excess measures the excess of one third of the fourth moment over the square of the second moment; the defect measures the defect of the fourth moment from one-fifth of the ratio of the sixth moment to the second moment” (Pearson, 1894).”
Investigating population heterogeneity with factor mixture models. (2005, background): “Mixture modeling can serve not only to detect clusters of participants in a population but also to model distributions that are, for instance, skewed (Pearson, 1894; Pearson, 1895; see also McLachlan & Peel, 2000; Titterington, Smith, & Makov, 1985).”
Learning mixtures of spherical gaussians: moment methods and spectral decompositions (2012, methodology): “[25] K. Pearson.”
The MM, ME, ML, EL, EF and GMM approaches to estimation: a synthesis (2002, background): “He summarized (pp.219-220) the contribution of Pearson (1894) stating, " The paper is particularly noteworthy for its introduction of the method of moments as a means of fitting a theoretical curve to observed data.”
Latent Variable Modelling: A Survey * (2007, background): “An early reference to finite mixture models is Pearson (1894), even though earlier writings by Quetelet and other nineteenth century statisticians mention such approaches.”
Outlier-robust moment-estimation via sum-of-squares (2017, background): “…is that for many parameter estimation problems the best known algorithms in terms of provable guarantees are based on the method of moments , which means that they don’t require direct access to a sample from the distribution but instead only to its low-degree moments (e.g. [Pea94, KMV10, MV10]).”
Spectral Methods for Indian Buffet Process Inference (2014, background): “It has a rich history dating back over a century to mixture models for identifying crabs [27] and has served as a key tool for describing diverse sets of distributions ranging from text [10] to images [1] and user behavior [4].”