A short history of algebraic statistics

Riccomagno, Eva

doi:10.1007/s00184-008-0222-3

Cited by 14 publications

(10 citation statements)

References 56 publications

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“…The above definition is often encountered in the field of algebraic statistics, where properties of statistical models are studied using techniques from algebraic geometry and commutative computer algebra, among others [18,38]. We next follow [22] in extending some standard terminology.…”

Section: Definitionmentioning

confidence: 99%

Sensitivity analysis in multilinear probabilistic models

Leonelli

Görgen

Smith

2017

Information Sciences

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Sensitivity methods for the analysis of the outputs of discrete Bayesian networks have been extensively studied and implemented in different software packages. These methods usually focus on the study of sensitivity functions and on the impact of a parameter change to the Chan-Darwiche distance. Although not fully recognized, the majority of these results rely heavily on the multilinear structure of atomic probabilities in terms of the conditional probability parameters associated with this type of network. By defining a statistical model through the polynomial expression of its associated defining conditional probabilities, we develop here a unifying approach to sensitivity methods applicable to a large suite of models including extensions of Bayesian networks, for instance context-specific ones. Our algebraic approach enables us to prove that for models whose defining polynomial is multilinear both the Chan-Darwiche distance and any divergence in the family of φ-divergences are minimized for a certain class of multi-parameter contemporaneous variations when parameters are proportionally covaried.

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Section: Definitionmentioning

confidence: 99%

Sensitivity analysis in multilinear probabilistic models

Leonelli

Görgen

Smith

2017

Information Sciences

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“…As the entries of A are taken to be nonnegative integer numbers, the atomic probabilities are monomials in the θ i 's and as θ varies in R k >0 the model describes an algebraic variety in ∆ q−1 [16,30]. The assumption of strictly positive probabilities is often met in practice, for instance for models learnt with complete data [22].…”

Section: Monomial Discrete Parametric Modelsmentioning

confidence: 99%

A geometric characterisation of sensitivity analysis in monomial models

Leonelli,

Riccomagno

2019

Preprint

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Sensitivity analysis in probabilistic discrete graphical models is usually conducted by varying one probability value at a time and observing how this affects output probabilities of interest. When one probability is varied then others are proportionally covaried to respect the sum-to-one condition of probability laws. The choice of proportional covariation is justified by a variety of optimality conditions, under which the original and the varied distributions are as close as possible under different measures of closeness. For variations of more than one parameter at a time proportional covariation is justified only in some special cases only. In this work, for the large class of discrete statistical models entertaining a regular monomial parametrisation, we demonstrate the optimality of newly defined proportional multiway schemes with respect to an optimality criterion based on the notion of I-divergence. We demonstrate that there are varying parameters choices for which proportional covariation is not optimal and identify the sub-family of model distributions where the distance between the original distribution and the one where probabilities are covaried proportionally is minimum. This is shown by adopting a new formal, geometric characterization of sensitivity analysis in monomial models, which include a wide array of probabilistic graphical models. We also demonstrate the optimality of proportional covariation for multi-way analyses in Naive Bayes classifiers.

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“…The computation of Markov bases for specific log-linear models has been considered by Rapallo [24]. For a review on the concept of the Diaconis-Sturmfels algorithm along with a compact and smooth presentation of the mapping of statistical probabilities to polynomials, we refer to Riccomagno [27]. The log-linear representation is connected to parametric (and therefore binomial) toric models as discussed in Rapallo [26].…”

Section: Model Fitting Using Markov Basesmentioning

confidence: 99%

Inference for Ordinal Log-Linear Models Based on Algebraic Statistics

Pham¹,

Kateri²

2019

J Alg Stat

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Tools of algebraic statistics combined with MCMC algorithms have been used in contingency table analysis for model selection and model fit testing of log-linear models. However, this approach has not been considered so far for association models, which are special log-linear models for tables with ordinal classification variables. The simplest association model for two-way tables, the uniform (U) association model, has just one parameter more than the independence model and is applicable when both classification variables are ordinal. Less parsimonious are the row (R) and column (C) effect association models, appropriate when at least one of the classification variables is ordinal. Association models have been extended for multidimensional contingency tables as well. Here, we adjust algebraic methods for association models analysis and investigate their eligibility, focusing mainly on two-way tables. They are implemented in the statistical software R and illustrated on real data tables. Finally the algebraic model fit and selection procedure is assessed and compared to the asymptotic approach in terms of a simulation study.

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A short history of algebraic statistics

Cited by 14 publications

References 56 publications

Sensitivity analysis in multilinear probabilistic models

Sensitivity analysis in multilinear probabilistic models

A geometric characterisation of sensitivity analysis in monomial models

Inference for Ordinal Log-Linear Models Based on Algebraic Statistics

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