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December 2021 Online inference with multi-modal likelihood functions
Mathieu Gerber, Kari Heine
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Ann. Statist. 49(6): 3103-3126 (December 2021). DOI: 10.1214/21-AOS2076


Let (Yt)t1 be a sequence of i.i.d. observations and {fθ,θRd} be a parametric model. We introduce a new online algorithm for computing a sequence (θˆt)t1, which is shown to converge almost surely to argmaxθRdE[logfθ(Y1)] at rate O(log(t)(1+ε)/2t1/2), with ε>0 a user specified parameter. This convergence result is obtained under standard conditions on the statistical model and, most notably, we allow the mapping θE[logfθ(Y1)] to be multi-modal. However, the computational cost to process each observation grows exponentially with the dimension of θ, which makes the proposed approach applicable to low or moderate dimensional problems only. We also derive a version of the estimator θˆt, which is well suited to student-t linear regression models that are popular tools for robust linear regression. As shown by experiments on simulated and real data, the corresponding estimator of the regression coefficients is, as expected, robust to the presence of outliers and thus, as a by-product, we obtain a new adaptive and robust online estimation procedure for linear regression models.


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Mathieu Gerber. Kari Heine. "Online inference with multi-modal likelihood functions." Ann. Statist. 49 (6) 3103 - 3126, December 2021.


Received: 1 October 2018; Revised: 1 March 2021; Published: December 2021
First available in Project Euclid: 14 December 2021

Digital Object Identifier: 10.1214/21-AOS2076

Primary: 62F10 , 62F12 , 62L12 , 68W27
Secondary: 90C26

Keywords: multi-modal likelihood functions , Online inference , scalable inference

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 6 • December 2021
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