Training Neural Networks for Likelihood/Density Ratio Estimation

Moustakides, George V.; Basioti, Kalliopi

doi:10.48550/arxiv.1911.00405

Cited by 5 publications

(7 citation statements)

References 14 publications

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“…) Alternative strategies to learn suitable elementwise D3F for binary detection are available in the recent literature; see, e.g., [79], [80]. The choice of the loss function and decision statistic depend heavily on the specific problem.…”

Section: A Problem Formulation and Learning Mechanismmentioning

confidence: 99%

Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis

Braca¹,

Millefiori²,

Aubry³

et al. 2022

Preprint

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We study the performance-and specifically the rate at which the error probability converges to zero-of Machine Learning (ML) classification techniques. Leveraging the theory of large deviations, we provide the mathematical conditions for a ML classifier to exhibit error probabilities that vanish exponentially, say ∼ exp (−n I + o(n)), where n is the number of informative observations available for testing (or another relevant parameter, such as the size of the target in an image) and I is the error rate. Such conditions depend on the Fenchel-Legendre transform of the cumulant-generating function of the Data-Driven Decision Function (D3F, i.e., what is thresholded before the final binary decision is made) learned in the training phase. As such, the D3F and, consequently, the related error rate I, depend on the given training set, which is assumed of finite size. Interestingly, these conditions can be verified and tested numerically exploiting the available dataset, or a synthetic dataset, generated according to the available information on the underlying statistical model. In other words, the classification error probability convergence to zero and its rate can be computed on a portion of the dataset available for training. Coherently with the large deviations theory, we can also establish the convergence, for n large enough, of the normalized D3F statistic to a Gaussian distribution. This property is exploited to set a desired asymptotic false alarm probability, which empirically turns out to be accurate even for quite realistic values of n. Furthermore, approximate error probability curves ∼ ζn exp (−n I) are provided, thanks to the refined asymptotic derivation (often referred to as exact asymptotics), where ζn represents the most representative sub-exponential terms of the error probabilities. Leveraging the refined asymptotic, we are able to compute an analytically accurate approximation of the classification performance for both the regimes of small and large values of n. Theoretical findings are corroborated by extensive numerical simulations. Real-world data, acquired by an X-band maritime radar system for surveillance, are exploited to assess the validity of the proposed theory.

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Section: A Problem Formulation and Learning Mechanismmentioning

confidence: 99%

Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis

Braca¹,

Millefiori²,

Aubry³

et al. 2022

Preprint

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“…1) Density Ratio Estimation: Instead of estimating the postchange density f 1 as in the GLR procedure, we may estimate the density ratio f 1 /f 0 directly (referred to as density ratio estimation [192]), based on which we develop sequential change detection procedures. A data-driven framework using neural networks was developed in [134]. More specifically, given two sets of data sampled from the densities of interest, an optimization problem is defined so that the solution, specified through neural networks, will correspond to the desired likelihood ratio function or its transformations and can then be used for sequential change detection.…”

Section: A Machine Learning and Change Detectionmentioning

confidence: 99%

Sequential (Quickest) Change Detection: Classical Results and New Directions

Xie

Zou

Xie

et al. 2021

Preprint

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Online detection of changes in stochastic systems, referred to as sequential change detection or quickest change detection, is an important research topic in statistics, signal processing, and information theory, and has a wide range of applications. This survey starts with the basics of sequential change detection, and then moves on to generalizations and extensions of sequential change detection theory and methods. We also discuss some new dimensions that emerge at the intersection of sequential change detection with other areas, along with a selection of modern applications and remarks on open questions.

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“…In most applications there exist sufficient amount of prior data that can be used for training, consequently it would be interesting to attempt to develop detection and estimation methods that are data-driven, namely do not require exact (or partial) knowledge of probability densities and therefore rely solely on data. Such techniques were developed in [3] for several versions of the binary hypothesis testing problem based on the direct estimation of the likelihood ratio of the two unknown densities which, as we know, is a sufficient statistic for the detection problem. Similar developments for parameter estimation, to our knowledge, do not seem to exist in any systematic way.…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Parameter Estimation

Moustakides¹

2022

Preprint

Self Cite

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Optimum parameter estimation methods require knowledge of a parametric probability density that statistically describes the available observations. In this work we examine Bayesian and non-Bayesian parameter estimation problems under a data-driven formulation where the necessary parametric probability density is replaced by available data. We present various data-driven versions that either result in neural network approximations of the optimum estimators or in well defined optimization problems that can be solved numerically. In particular, for the data-driven equivalent of non-Bayesian estimation we end up with optimization problems similar to the ones encountered for the design of generative networks.

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Training Neural Networks for Likelihood/Density Ratio Estimation

Cited by 5 publications

References 14 publications

Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis

Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis

Sequential (Quickest) Change Detection: Classical Results and New Directions

Data-Driven Parameter Estimation

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