Consistent estimation of mixture complexity

James, Lancelot F.; Priebe, Carey E.; Marchette, David J.

doi:10.1214/aos/1013203454

Cited by 60 publications

(23 citation statements)

References 28 publications

(30 reference statements)

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“…, π m ). The order of a Gaussian mixture can be consistently estimated (James et al (2001)), and therefore 1(τ 2 1 = 0), which is equal to the indicator of a single mixture component, is identifiable from data. Testing a homogeneous Gaussian (τ 2 1 = 0) against a mixture with more components is a hard problem (see Chauveau et al (2019) for some recent work and more references) and proofs of unbiasedness as required for distinguishability from data do not seem to exist, but existing tests at least allow to potentially distinguish τ 2 1 = 0 from τ 2 1 > 0.…”

Section: Distinguishing Independence and Dependencementioning

confidence: 99%

Parameters not identifiable or distinguishable from data, including correlation between Gaussian observations

Hennig¹

2021

Preprint

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It is shown that some theoretically identifiable parameters cannot be identified from data, meaning that no consistent estimator of them can exist. An important example is a constant correlation between Gaussian observations (in presence of such correlation not even the mean can be identified from data). Identifiability and three versions of distinguishability from data are defined. Two different constant correlations between Gaussian observations cannot even be distinguished from data. A further example are cluster membership parameters in k-means clustering. Several existing results in the literature are connected to the new framework.

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Section: Distinguishing Independence and Dependencementioning

confidence: 99%

Parameters not identifiable or distinguishable from data, including correlation between Gaussian observations

Hennig¹

2021

Preprint

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“…The difficulty of order estimation in stationary HMMs (or equivalently order estimation for finite mixture models) mainly comes from the monotonic increase of the likelihood as the order increases. The methods developed by Chen and Kalbfleisch (1996) and James et al (2001) try to minimize the distance between a nonparametric curve based on the sample of data and the fitted model. Poskitt and Zhang (2005) proposed to use a penalized quasi-likelihood estimator and investigated its asymptotic properties.…”

Section: Mixture Modelsmentioning

confidence: 99%

“…To ensure the validity of the asymptotic results, normally a restriction is imposed on the values found by cross-validation approaches (James et al, 2001). In addition, these approaches normally require intensive computation and make it too difficult to be applied on the large number of genes from microarray data.…”

Section: Choices Of the Threshold Parametersmentioning

confidence: 99%

An Order Estimation Based Approach to Identify Response Genes for Microarray Time Course Data

Zk¹,

Ob²,

Af³

2012

Statistical Applications in Genetics and Molecular Biology

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Gene expression profiles from microarray time course experiments provide a unique opportunity to examine genome-wide signal processing and gene responses. A fundamental issue in microarray experiments is that the treatment condition can only be controlled at the cell level rather than at the gene level. The treatment condition does not affect all genes equally. Some genes depend on other genes to detect external changes. The dependency between genes is not fully deterministic and may vary with treatment condition. Thus the expression of each gene is potentially affected by two confounding effects: the treatment effect and the gene context effect arising from the regulatory interactions among genes. This gene context effect is hard to isolate. Neither can it be simply ignored. Instead, this gene context information which may be different under different treatment conditions is of primary biological interest. We introduce an approach which deals with the confounding effects and takes into account the uncontrollable gene context effect. Our method is based on the estimation of the number of hidden states, which, in our development, corresponds to the order of a hidden Markov model (HMM). For each gene, its observed expression is modeled by a gamma distribution determined by the corresponding hidden state at each time point. Those genes showing evidence for more than one hidden state can be categorized as the signalling genes, or in a wider sense, as the response genes which are coordinated by a cell system in reaction to a specific external condition. These response genes can be used in the comparison of different treatment conditions, to investigate the gene context effect under different treatments. Microarray time course data are also analyzed to demonstrate our method.

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“…Throughout the paper, the number of mixing components K of the unknown density has been assumed to be known-for example, the scientist will have some insight in its choice. Nevertheless, K could also be treated as an unknown parameter to be estimated, this being a standard problem in mixture estimation (see James, Priebe and Marchette [24], and references therein). In fact, more unknown parameters can be introduced, as long as the mixture remains identifiable.…”

Section: 2mentioning

confidence: 99%

On random tomography with unobservable projection angles

Panaretos¹

2009

Ann. Statist.

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We formulate and investigate a statistical inverse problem of a random tomographic nature, where a probability density function on $\mathbb{R}^3$ is to be recovered from observation of finitely many of its two-dimensional projections in random and unobservable directions. Such a problem is distinct from the classic problem of tomography where both the projections and the unit vectors normal to the projection plane are observable. The problem arises in single particle electron microscopy, a powerful method that biophysicists employ to learn the structure of biological macromolecules. Strictly speaking, the problem is unidentifiable and an appropriate reformulation is suggested hinging on ideas from Kendall's theory of shape. Within this setup, we demonstrate that a consistent solution to the problem may be derived, without attempting to estimate the unknown angles, if the density is assumed to admit a mixture representation.Comment: Published in at http://dx.doi.org/10.1214/08-AOS673 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Consistent estimation of mixture complexity

Cited by 60 publications

References 28 publications

Parameters not identifiable or distinguishable from data, including correlation between Gaussian observations

Parameters not identifiable or distinguishable from data, including correlation between Gaussian observations

An Order Estimation Based Approach to Identify Response Genes for Microarray Time Course Data

On random tomography with unobservable projection angles

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