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The Spectral Approach to Linear Rational expectations Models

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arXiv 2020年

作者： Al-Sadoon, Majid M. Durham University Business School United Kingdom

This paper considers linear rational expectations models in the frequency domain. The paper characterizes existence and uniqueness of solutions to particular as well as generic systems. The set of all solutions to a given system is shown to be a finite dimensional affine space in the frequency domain. It is demonstrated that solutions can be discontinuous with respect to the parameters of the models in the context of non-uniqueness, invalidating mainstream frequentist and Bayesian methods. The ill-posedness of the problem motivates regularized solutions with theoretically guaranteed uniqueness, continuity, and even differentiability *** Codes 91B84 Copyright © 2020, The Authors. All rights reserved.

关键词： expectation maximization algorithm

Blind Channel Estimation and Joint Symbol Detection with Data-Driven Factor Graphs

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arXiv 2024年

作者： Schmid, Luca Raviv, Tomer Shlezinger, Nir Schmalen, Laurent Hertzstr. 16 Karlsruhe76187 Germany School of ECE Ben-Gurion University of the Negev Be’er-Sheva Israel

We investigate the application of the factor graph framework for blind joint channel estimation and symbol detection on time-variant linear inter-symbol interference channels. In particular, we consider the expectation maximization (EM) algorithm for maximum likelihood estimation, which typically suffers from high complexity as it requires the computation of the symbol-wise posterior distributions in every iteration. We address this issue by efficiently approximating the posteriors using the belief propagation (BP) algorithm on a suitable factor graph. By interweaving the iterations of BP and EM, the detection complexity can be further reduced to a single BP iteration per EM step. In addition, we propose a data-driven version of our algorithm that introduces momentum in the BP updates and learns a suitable EM parameter update schedule, thereby significantly improving the performance-complexity tradeoff with a few offline training samples. Our numerical experiments demonstrate the excellent performance of the proposed blind detector and show that it even outperforms coherent BP detection in high signal-to-noise scenarios. Copyright © 2024, The Authors. All rights reserved.

关键词： expectation maximization algorithm

On the maximal correlation coefficient for the bivariate Marshall Olkin distribution

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arXiv 2024年

作者： Bücher, Axel Staud, Torben Ruhr-Universität Bochum Fakultät für Mathematik Austria

We prove a formula for the maximal correlation coefficient of the bivariate Marshall Olkin distribution that was conjectured in Lin, Lai, and Govindaraju (2016, Stat. Methodol., 29:1–9). The formula is applied to obtain a new proof for a variance inequality in extreme value statistics that links the disjoint and the sliding block maxima method. Copyright © 2024, The Authors. All rights reserved.

关键词： expectation maximization algorithm

MaxMin-RLHF: Alignment with Diverse Human Preferences

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arXiv 2024年

作者： Chakraborty, Souradip Qiu, Jiahao Yuan, Hui Koppel, Alec Manocha, Dinesh Huang, Furong Bedi, Amrit Singh Wang, Mengdi Department of Computer Science University of Maryland College ParkMD United States Department of Electrical and Computer Engineering Princeton University NJ United States JP Morgan Chase AI Research New York United States Department of Computer Science University of Central Florida FL United States

Reinforcement Learning from Human Feedback (RLHF) aligns language models to human preferences by employing a singular reward model derived from preference data. However, the single reward model overlooks the rich diversity of human preferences inherent in data collected from multiple users. In this work, we first derive an impossibility result of alignment with single reward RLHF, thereby highlighting its insufficiency in representing diverse human preferences. Next, we propose to learn a mixture of reward models via an expectation-maximization algorithm and solve a MaxMin alignment objective inspired by the Egalitarian principle in social choice theory to better honor diverse human preferences. We present comprehensive experimental results on small-scale (GPT-2) and large-scale language (with Tulu2-7B)) and show the efficacy of the proposed approach in the presence of diversity among human preferences. We remark that our findings in this work are not only limited to language models but also extend to reinforcement learning in general. Copyright © 2024, The Authors. All rights reserved.

关键词： expectation maximization algorithm

Quantifying predictive uncertainty of aphasia severity in stroke patients with sparse heteroscedastic Bayesian high-dimensional regression

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arXiv 2023年

作者： Zgodic, Anja Bai, Ray Zhang, Jiajia Wang, Yuan Rorden, Chris McLain, Alexander C. Department of Epidemiology and Biostatistics University of South Carolina South Carolina United States Department of Statistics University of South Carolina South Carolina United States Department of Psychology University of South Carolina South Carolina United States

Sparse linear regression methods for high-dimensional data commonly assume that residuals have constant variance, which can be violated in practice. For example, Aphasia Quotient (AQ) is a critical measure of language impairment and informs treatment decisions, but it is challenging to measure in stroke patients. It is of interest to use high-resolution T2 neuroimages of brain damage to predict AQ. However, sparse regression models show marked evidence of heteroscedastic error even after transformations are applied. This violation of the homoscedasticity assumption can lead to bias in estimated coefficients, prediction intervals (PI) with improper length, and increased type I errors. Bayesian heteroscedastic linear regression models relax the homoscedastic error assumption but can enforce restrictive prior assumptions on parameters, and many are computationally infeasible in the high-dimensional setting. This paper proposes estimating high-dimensional heteroscedastic linear regression models using a heteroscedastic partitioned empirical Bayes expectation Conditional maximization (H-PROBE) algorithm. H-PROBE is a computationally efficient maximum a posteriori estimation approach that requires minimal prior assumptions and can incorporate covariates hypothesized to impact heterogeneity. We apply this method by using high-dimensional neuroimages to predict and provide PIs for AQ that accurately quantify predictive uncertainty. Our analysis demonstrates that H-PROBE can provide narrower PI widths than standard methods without sacrificing coverage. Narrower PIs are clinically important for determining the risk of moderate to severe aphasia. Additionally, through extensive simulation studies, we exhibit that H-PROBE results in superior prediction, variable selection, and predictive inference compared to alternative methods. Copyright © 2023, The Authors. All rights reserved.

关键词： expectation maximization algorithm

Maximum Ideal Likelihood Estimator: An New Estimation and Inference Framework for Latent Variable Models

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arXiv 2024年

作者： Cai, Jake Yizhou Ma, Ting Fung Department of Statistics University of South Carolina 1523 Greene Street ColumbiaSC29208 United States

In this paper, a new estimation framework, Maximum Ideal Likelihood Estimator (MILE), is proposed for general parametric models with latent variables and missing values. Instead of focusing on the marginal likelihood of the observed data as in many traditional approaches, the MILE directly considers the joint distribution of the complete dataset by treating the latent variables as parameters (the ideal likelihood). The MILE framework remains valid, even when traditional methods are not applicable, e.g., non-finite conditional expectation of the marginal likelihood function, via different optimization techniques and algorithms. The statistical properties of the MILE, such as the asymptotic equivalence to the Maximum Likelihood Estimation (MLE), are proved under some mild conditions, which facilitate statistical inference and prediction. Simulation studies illustrate that MILE outperforms traditional approaches with computational feasibility and scalability using existing and our proposed algorithms. Copyright © 2024, The Authors. All rights reserved.

关键词： expectation maximization algorithm

Hidden Markov Models for Multivariate Panel Data

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arXiv 2024年

作者： Neal, Mackenzie R. Sochaniwsky, Alexa A. McNicholas, Paul D. Department of Mathematics & Statistics McMaster University ON Canada

While advances continue to be made in model-based clustering, challenges persist in modeling various data types such as panel data. Multivariate panel data present difficulties for clustering algorithms because they are often plagued by missing data and dropouts, presenting issues for estimation algorithms. This research presents a family of hidden Markov models that compensate for the issues that arise in panel data. A modified expectation-maximization algorithm capable of handling missing not at random data and dropout is presented and used to perform model estimation. Copyright © 2024, The Authors. All rights reserved.

关键词： expectation maximization algorithm

Confirmation Bias in Gaussian Mixture Models

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arXiv 2024年

作者： Balanov, Amnon Bendory, Tamir Huleihel, Wasim

Confirmation bias, the tendency to interpret information in a way that aligns with one's preconceptions, can profoundly impact scientific research, leading to conclusions that reflect the researcher's hypotheses even when the observational data do not support them. This issue is especially critical in scientific fields involving highly noisy observations, such as cryo-electron microscopy. This study investigates confirmation bias in Gaussian mixture models. We consider the following experiment: A team of scientists assumes they are analyzing data drawn from a Gaussian mixture model with known signals (hypotheses) as centroids. However, in reality, the observations consist entirely of noise without any informative structure. The researchers use a single iteration of the K-means or expectation-maximization algorithms, two popular algorithms to estimate the centroids. Despite the observations being pure noise, we show that these algorithms yield biased estimates that resemble the initial hypotheses, contradicting the unbiased expectation that averaging these noise observations would converge to zero. Namely, the algorithms generate estimates that mirror the postulated model, although the hypotheses (the presumed centroids of the Gaussian mixture) are not evident in the observations. Specifically, among other results, we prove a positive correlation between the estimates produced by the algorithms and the corresponding hypotheses. We also derive explicit closed-form expressions of the estimates for a finite and infinite number of hypotheses. This study underscores the risks of confirmation bias in low signal-to-noise environments, provides insights into potential pitfalls in scientific methodologies, and highlights the importance of prudent data interpretation. Copyright © 2024, The Authors. All rights reserved.

关键词： expectation maximization algorithm

A FLEXIBLE MODEL FOR RECORD LINKAGE

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arXiv 2024年

作者： Robach, Kayané van der Pas, Stéphanie L. van de Wiel, Mark A. Hof, Michel H. Department of Epidemiology and Data Science Amsterdam UMC location Vrije Universiteit Amsterdam De Boelelaan 1117 Amsterdam1081 HV Netherlands Amsterdam Public Health Methodology Netherlands

Combining data from various sources empowers researchers to explore innovative questions, for example those raised by conducting healthcare monitoring studies. However, the lack of a unique identifier often poses challenges. Record linkage procedures determine whether pairs of observations collected on different occasions belong to the same individual using partially identifying variables (e.g. birth year, postal code). Existing methodologies typically involve a compromise between computational efficiency and accuracy. Traditional approaches simplify this task by condensing information, yet they neglect dependencies among linkage decisions and disregard the one-to-one relationship required to establish coherent links. Modern approaches offer a comprehensive representation of the data generation process, at the expense of computational overhead and reduced flexibility. We propose a flexible method, that adapts to varying data complexities, addressing registration errors and accommodating changes of the identifying information over time. Our approach balances accuracy and scalability, estimating the linkage using a Stochastic expectation Maximisation algorithm on a latent variable model. We illustrate the ability of our methodology to connect observations using large real data applications and demonstrate the robustness of our model to the linking variables quality in a simulation study. The proposed algorithm FlexRL is implemented and available in an open source R package. © 2024, CC BY.

关键词： expectation maximization algorithm