arXiv subjects

Shinto Eguchi

Publications and source records attributed to Shinto Eguchi.

At least 19 recordsLinked to original sources

arXiv2026

An information-geometric framework for mapping maximum potential biodiversity

Biodiversity measures are often used descriptively: one computes a diversity index from an observed or estimated community composition and maps the resulting values across space. Conservation planning, however, also requires a site-specific benchmark against which the observed community can be compared. This chapter develops an information-geometric framework for such \emph{potential diversity} and the associated \emph{diversity gap}. The central object is a pair of probability vectors on the species simplex: an observed or realized composition $p^{\mathrm{obs}}$, and a potential composition $p^{\mathrm{pot}}$ obtained by a constrained variational principle. The gap is then defined by comparing a diversity functional at these two compositions. The framework is developed for both Hill-type diversity, which measures abundance and evenness, and Rao's quadratic entropy, which incorporates trait, phylogenetic, or ecological dissimilarities among species. A spatial point-process interpretation clarifies how local ecological capacities can be defined before passing to the simplex. Escort constraints, capacity constraints, and divergence projections then provide a unified way to define nontrivial benchmarks beyond the uniform distribution. The resulting formulation separates two distinct questions: how diverse a community is, and how far it is from a locally admissible potential benchmark. It also connects the ecological idea of dark diversity with a continuous, abundance-weighted comparison on the probability simplex. We also outline a dynamic extension in which capacities, species migration, and climate-driven shifts vary over time. Empirical implementation with large-scale citizen-science biodiversity data and trait databases is left for future work.

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arXiv2026

Zero-Inflated Logistic Regression Models with Shared Design: Identifiability, Existence of Estimates, and a Relabeling Rule

The zero-inflated logistic regression model accommodates binary responses with excess zeros, which often arise from a latent mixture of susceptible and insusceptible subpopulations or asymmetric misclassification of the response. The model has two components: regression for the binary response and a latent binary indicator for the zero-inflation state. In applied settings, it is common to use the same design matrix for both components if there is no prior knowledge. However, this shared-design specification lacks guaranteed identifiability of the regression parameters, as established in prior works. This paper investigates the theoretical properties of the zero-inflated logistic regression model under the shared-design setting and computational methods for applications. First, to motivate the use of the zero-inflated model, we prove that ignoring the zero-inflation mechanism can lead to a sign flip in the pseudo-true coefficient value relative to the true value. We then establish sufficient conditions for the existence of the maximum likelihood estimate. As a main result, we establish that the model under the shared-design setting is identifiable up to exchange symmetry of the parameters for two components and that the expected log-likelihood has a unique maximizer on the resulting quotient space. The posterior bimodality is examined using a P\'olya-Gamma Gibbs sampler with replica exchange. Finally, we propose a simple relabeling rule to select a single ordered parameter pair, and evaluate its performance through simulation studies and an application to self-reported diabetes data.

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arXiv2026

Statistical Inference via Generative Models: Flow Matching and Causal Inference

Generative AI has achieved remarkable empirical success, but from the perspective of statistics it often remains opaque: its predictions may be accurate, yet the underlying mechanism is difficult to interpret, analyze, and trust. This book reinterprets generative AI in the language of statistics, using flow matching as a central example. The key idea is that generative models should be understood not merely as devices for producing plausible data, but as methods for the nonparametric learning of high-dimensional probability distributions. From this viewpoint, missing-data imputation becomes principled sampling from learned conditional distributions, counterfactual analysis becomes the estimation of intervention distributions, and distributional dynamics become statistically analyzable objects. Mathematically, flow matching represents distributional deformation through the continuity equation and a time-dependent velocity field, thereby extending score matching from the learning of static score fields to the learning of transport paths themselves. Building on this foundation, the book develops a statistical framework in which generative models are used to estimate nuisance components while inferential validity is maintained through orthogonalization and cross-fitting in the spirit of double/debiased machine learning. Applications to survival analysis, censoring, missingness, and causal inference show how generative models can be integrated into statistical inference for structured high-dimensional problems.

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arXiv2025

Implicit geometric regularization in flow matching via density weighted Stein operators

Flow Matching (FM) has emerged as a powerful paradigm for continuous normalizing flows, yet standard FM implicitly performs an unweighted $L^2$ regression over the entire ambient space. In high dimensions, this leads to a fundamental inefficiency: the vast majority of the integration domain consists of low-density ``void'' regions where the target velocity fields are often chaotic or ill-defined. In this paper, we propose {$\gamma$-Flow Matching ($\gamma$-FM)}, a density-weighted variant that aligns the regression geometry with the underlying probability flow. While density weighting is desirable, naive implementations would require evaluating the intractable target density. We circumvent this by introducing a Dynamic Density-Weighting strategy that estimates the \emph{target} density directly from training particles. This approach allows us to dynamically downweight the regression loss in void regions without compromising the simulation-free nature of FM. Theoretically, we establish that $\gamma$-FM minimizes the transport cost on a statistical manifold endowed with the $\gamma$-Stein metric. Spectral analysis further suggests that this geometry induces an implicit Sobolev regularization, effectively damping high-frequency oscillations in void regions. Empirically, $\gamma$-FM significantly improves vector field smoothness and sampling efficiency on high-dimensional latent datasets, while demonstrating intrinsic robustness to outliers.

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arXiv2025

Robust inference using density-powered Stein operators

We introduce a density-power weighted variant of the Stein operator, called the $\gamma$-Stein operator, for robust inference with unnormalized probability models. The operator is motivated by the first variation of the $\gamma$-divergence under infinitesimal escort transport and weights the usual Stein field by a positive power of the model density. This weighting down-weights observations in low model-density regions, providing a principled robustness mechanism while retaining the normalizing-constant-free structure of score matching. We develop the resulting $\gamma$-score matching estimating equations and discuss their non-integrable, generalized-method-of-moments character. We further study two extensions: a $\gamma$-kernelized Stein discrepancy, interpreted as a robust diagnostic or contaminated-null goodness-of-fit procedure, and $\gamma$-Stein variational gradient descent for robust posterior approximation. Numerical examples on directional, mixture, and quartic-potential models illustrate the robustness--efficiency trade-off: positive $\gamma$ can stabilize inference under targeted contamination, whereas $\gamma=0$ remains preferable under clean well-specified models.

stat.ML↗

arXiv2025

The $\phi$-PCA Framework: A Unified and Efficiency-Preserving Approach with Robust Variants

Principal component analysis (PCA) is a fundamental tool in multivariate statistics, yet its sensitivity to outliers and limitations in distributed environments restrict its effectiveness in modern large-scale applications. To address these challenges, we introduce the $\phi$-PCA framework which provides a unified formulation of robust and distributed PCA. The class of $\phi$-PCA methods retains the asymptotic efficiency of standard PCA, while aggregating multiple local estimates using a proper $\phi$ function enhances ordering-robustness, leading to more accurate eigensubspace estimation under contamination. Notably, the harmonic mean PCA (HM-PCA), corresponding to the choice $\phi(u)=u^{-1}$, achieves optimal ordering-robustness and is recommended for practical use. Theoretical results further show that robustness increases with the number of partitions, a phenomenon seldom explored in the literature on robust or distributed PCA. Altogether, the partition-aggregation principle underlying $\phi$-PCA offers a general strategy for developing robust and efficiency-preserving methodologies applicable to both robust and distributed data analysis.

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arXiv2025

Minimum Copula Divergence for Robust Estimation

This paper introduces a robust estimation framework based solely on the copula function. We begin by introducing a family of divergence measures tailored for copulas, including the $\alpha$-, $\beta$-, and $\gamma$-copula divergences, which quantify the discrepancy between a parametric copula model and an empirical copula derived from data independently of marginal specifications. Using these divergence measures, we propose the minimum copula divergence estimator (MCDE), an estimation method that minimizes the divergence between the model and the empirical copula. The framework proves particularly effective in addressing model misspecifications and analyzing heavy-tailed data, where traditional methods such as the maximum likelihood estimator (MLE) may fail. Theoretical results show that common copula families, including Archimedean and elliptical copulas, satisfy conditions ensuring the boundedness of divergence-based estimators, thereby guaranteeing the robustness of MCDE, especially in the presence of extreme observations. Numerical examples further underscore MCDE's ability to adapt to varying dependence structures, ensuring its utility in real-world scenarios.

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arXiv2025

On a class of binary regression models and their robust estimation

A robust estimation framework for binary regression models is studied, aiming to extend traditional approaches like logistic regression models. While previous studies largely focused on logistic models, we explore a broader class of models defined by general link functions. We incorporate various loss functions to improve estimation under model misspecification. Our investigation addresses robustness against outliers and model misspecifications, leveraging divergence-based techniques such as the $\beta$-divergence and $\gamma$-divergence, which generalize the maximum likelihood approach. These divergences introduce loss functions that mitigate the influence of atypical data points while retaining Fisher consistency. We establish a theoretical property of the estimators under both correctly specified and misspecified models, analyzing their robustness through quantifying the effect of outliers in linear predictor. Furthermore, we uncover novel relationships between existing estimators and robust loss functions, identifying previously unexplored classes of robust estimators. Numerical experiments illustrate the efficacy of the proposed methods across various contamination scenarios, demonstrating their potential to enhance reliability in binary classification tasks. By providing a unified framework, this study highlights the versatility and robustness of divergence-based methods, offering insights into their practical application and theoretical underpinnings.

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arXiv2024

Information Geometry for Maximum Diversity Distributions

In recent years, biodiversity measures have gained prominence as essential tools for ecological and environmental assessments, particularly in the context of increasingly complex and large-scale datasets. We provide a comprehensive review of diversity measures, including the Gini-Simpson index, Hill numbers, and Rao's quadratic entropy, examining their roles in capturing various aspects of biodiversity. Among these, Rao's quadratic entropy stands out for its ability to incorporate not only species abundance but also functional and genetic dissimilarities. The paper emphasizes the statistical and ecological significance of Rao's quadratic entropy under the information geometry framework. We explore the distribution maximizing such a diversity measure under linear constraints that reflect ecological realities, such as resource competition or habitat suitability. Furthermore, we discuss a unified approach of the Leinster-Cobbold index combining Hill numbers and Rao's entropy, allowing for an adaptable and similarity-sensitive measure of biodiversity. Finally, we discuss the information geometry associated with the maximum diversity distribution focusing on the cross diversity measures such as the cross-entropy.

q-bio.PE↗

arXiv2024

A Generalized Mean Approach for Distributed-PCA

Principal component analysis (PCA) is a widely used technique for dimension reduction. As datasets continue to grow in size, distributed-PCA (DPCA) has become an active research area. A key challenge in DPCA lies in efficiently aggregating results across multiple machines or computing nodes due to computational overhead. Fan et al. (2019) introduced a pioneering DPCA method to estimate the leading rank-$r$ eigenspace, aggregating local rank-$r$ projection matrices by averaging. However, their method does not utilize eigenvalue information. In this article, we propose a novel DPCA method that incorporates eigenvalue information to aggregate local results via the matrix $\beta$-mean, which we call $\beta$-DPCA. The matrix $\beta$-mean offers a flexible and robust aggregation method through the adjustable choice of $\beta$ values. Notably, for $\beta=1$, it corresponds to the arithmetic mean; for $\beta=-1$, the harmonic mean; and as $\beta \to 0$, the geometric mean. Moreover, the matrix $\beta$-mean is shown to associate with the matrix $\beta$-divergence, a subclass of the Bregman matrix divergence, to support the robustness of $\beta$-DPCA. We also study the stability of eigenvector ordering under eigenvalue perturbation for $\beta$-DPCA. The performance of our proposal is evaluated through numerical studies.

stat.ML↗

arXiv2024

Minimum Gamma Divergence for Regression and Classification Problems

The book is structured into four main chapters. Chapter 1 introduces the foundational concepts of divergence measures, including the well-known Kullback-Leibler divergence and its limitations. It then presents a detailed exploration of power divergences, such as the $\alpha$, $\beta$, and $\gamma$-divergences, highlighting their unique properties and advantages. Chapter 2 explores minimum divergence methods for regression models, demonstrating how these methods can improve robustness and efficiency in statistical estimation. Chapter 3 extends these methods to Poisson point processes, with a focus on ecological applications, providing a robust framework for modeling species distributions and other spatial phenomena. Finally, Chapter 4 explores the use of divergence measures in machine learning, including applications in Boltzmann machines, AdaBoost, and active learning. The chapter emphasizes the practical benefits of these measures in enhancing model robustness and performance.

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arXiv2024

Cumulant-based approximation for fast and efficient prediction for species distribution

Species distribution modeling plays an important role in estimating the habitat suitability of species using environmental variables. For this purpose, Maxent and the Poisson point process are popular and powerful methods extensively employed across various ecological and biological sciences. However, the computational speed becomes prohibitively slow when using huge background datasets, which is often the case with fine-resolution data or global-scale estimations. To address this problem, we propose a computationally efficient species distribution model using a cumulant-based approximation (CBA) applied to the loss function of $\gamma$-divergence. Additionally, we introduce a sequential estimating algorithm with an $L_1$ penalty to select important environmental variables closely associated with species distribution. The regularized geometric-mean method, derived from the CBA, demonstrates high computational efficiency and estimation accuracy. Moreover, by applying CBA to Maxent, we establish that Maxent and Fisher linear discriminant analysis are equivalent under a normality assumption. This equivalence leads to an highly efficient computational method for estimating species distribution. The effectiveness of our proposed methods is illustrated through simulation studies and by analyzing data on 226 species from the National Centre for Ecological Analysis and Synthesis and 709 Japanese vascular plant species. The computational efficiency of the proposed methods is significantly improved compared to Maxent, while maintaining comparable estimation accuracy. A R package {\tt CBA} is also prepared to provide all programming codes used in simulation studies and real data analysis.

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arXiv2023

Robust minimum divergence estimation in a spatial Poisson point process

Species distribution modeling (SDM) plays a crucial role in investigating habitat suitability and addressing various ecological issues. While likelihood analysis is commonly used to draw ecological conclusions, it has been observed that its statistical performance is not robust when faced with slight deviations due to misspecification in SDM. We propose a new robust estimation method based on a novel divergence for the Poisson point process model. The proposed method is characterized by weighting the log-likelihood equation to mitigate the impact of heterogeneous observations in the presence-only data, which can result from model misspecification. We demonstrate that the proposed method improves the predictive performance of the maximum likelihood estimation in our simulation studies and in the analysis of vascular plant data in Japan.

stat.ME↗

arXiv2023

Statistical learning for species distribution models in ecological studies

We discuss species distribution models (SDM) for biodiversity studies in ecology. SDM plays an important role to estimate abundance of a species based on environmental variables that are closely related with the habitat of the species. The resultant habitat map indicates areas where the species is likely to live, hence it is essential for conservation planning and reserve selection. We especially focus on a Poisson point process and clarify relations with other statistical methods. Then we discuss a Poisson point process from a view point of information divergence, showing the Kullback-Leibler divergence of density functions reduces to the extended Kullback-Leibler divergence of intensity functions. This property enables us to extend the Poisson point process to that derived from other divergence such as $β$ and $γ$ divergences. Finally, we discuss integrated SDM and evaluate the estimating performance based on the Fisher information matrices.

stat.AP↗

arXiv2022

Active Learning by Query by Committee with Robust Divergences

Active learning is a widely used methodology for various problems with high measurement costs. In active learning, the next object to be measured is selected by an acquisition function, and measurements are performed sequentially. The query by committee is a well-known acquisition function. In conventional methods, committee disagreement is quantified by the Kullback--Leibler divergence. In this paper, the measure of disagreement is defined by the Bregman divergence, which includes the Kullback--Leibler divergence as an instance, and the dual $γ$-power divergence. As a particular class of the Bregman divergence, the $β$-divergence is considered. By deriving the influence function, we show that the proposed method using $β$-divergence and dual $γ$-power divergence are more robust than the conventional method in which the measure of disagreement is defined by the Kullback--Leibler divergence. Experimental results show that the proposed method performs as well as or better than the conventional method.

stat.ML↗

arXiv2022

Minimum information divergence of Q-functions for dynamic treatment resumes

This paper aims at presenting a new application of information geometry to reinforcement learning focusing on dynamic treatment resumes. In a standard framework of reinforcement learning, a Q-function is defined as the conditional expectation of a reward given a state and an action for a single-stage situation. We introduce an equivalence relation, called the policy equivalence, in the space of all the Q-functions. A class of information divergence is defined in the Q-function space for every stage. The main objective is to propose an estimator of the optimal policy function by a method of minimum information divergence based on a dataset of trajectories. In particular, we discuss the $γ$-power divergence that is shown to have an advantageous property such that the $γ$-power divergence between policy-equivalent Q-functions vanishes. This property essentially works to seek the optimal policy, which is discussed in a framework of a semiparametric model for the Q-function. The specific choices of power index $γ$ give interesting relationships of the value function, and the geometric and harmonic means of the Q-function. A numerical experiment demonstrates the performance of the minimum $γ$-power divergence method in the context of dynamic treatment regimes.

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arXiv2022

Robust self-tuning semiparametric PCA for contaminated elliptical distribution

Principal component analysis (PCA) is one of the most popular dimension reduction methods. The usual PCA is known to be sensitive to the presence of outliers, and thus many robust PCA methods have been developed. Among them, the Tyler's M-estimator is shown to be the most robust scatter estimator under the elliptical distribution. However, when the underlying distribution is contaminated and deviates from ellipticity, Tyler's M-estimator might not work well. In this article, we apply the semiparametric theory to propose a robust semiparametric PCA. The merits of our proposal are twofold. First, it is robust to heavy-tailed elliptical distributions as well as robust to non-elliptical outliers. Second, it pairs well with a data-driven tuning procedure, which is based on active ratio and can adapt to different degrees of data outlyingness. Theoretical properties are derived, including the influence functions for various statistical functionals and asymptotic normality. Simulation studies and a data analysis demonstrate the superiority of our method.

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arXiv2020

Copula-based measures of asymmetry between the lower and upper tail probabilities

We propose a copula-based measure of asymmetry between the lower and upper tail probabilities of bivariate distributions. The proposed measure has a simple form and possesses some desirable properties as a measure of asymmetry. The limit of the proposed measure as the index goes to the boundary of its domain can be expressed in a simple form under certain conditions on copulas. A sample analogue of the proposed measure for a sample from a copula is presented and its weak convergence to a Gaussian process is shown. Another sample analogue of the presented measure, which is based on a sample from a distribution on $\mathbb{R}^2$, is given. Simple methods for interval estimation and nonparametric testing based on the two sample analogues are presented. As an example, the presented measure is applied to daily returns of S&P500 and Nikkei225.

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