Ordinal categorical data are routinely encountered in a wide range of practical applications. When the primary goal is to construct a regression model for ordinal outcomes, cumulative link models represent one of the most popular choices to link the cumulative probabilities of the response with a set of covariates through a parsimonious linear predictor, shared across response categories. When the number of observations grows, standard sampling algorithms for Bayesian inference scale poorly, making posterior computation increasingly challenging in large datasets. In this article, we propose three scalable algorithms for approximating the posterior distribution of the regression coefficients in cumulative probit models relying on Variational Bayes and Expectation Propagation. We compare the proposed approaches with inference based on Markov Chain Monte Carlo, demonstrating superior computational performance and remarkable accuracy; finally, we illustrate the utility of the proposed algorithms on a challenging case study to investigate the structure of a criminal network.

Linear regression is frequently applied in a variety of domains. In order to improve the efficiency of these methods, various methods have been developed that compute summaries or \emph{sketches} of the datasets. Certain domains, however, contain sensitive data which necessitates that the application of these statistical methods does not reveal private information. Differentially private (DP) linear regression methods have been developed for mitigating this problem. These techniques typically involve estimating a noisy version of the parameter vector. Instead, we propose releasing private sketches of the datasets. We present differentially private sketches for the problems of least squares regression, as well as least absolute deviations regression. The availability of these private sketches facilitates the application of commonly available solvers for regression, without the risk of privacy leakage.

Tokenization is an important first step in Natural Language Processing (NLP) pipelines because it decides how models learn and represent linguistic information. However, current subword tokenizers like SentencePiece or HuggingFace BPE are mostly designed for Latin or multilingual corpora and do not perform well on languages with rich morphology such as Bengali. To address this limitation, we present BengaliBPE, a Byte Pair Encoding (BPE) tokenizer specifically developed for the Bengali script. BengaliBPE applies Unicode normalization, grapheme-level initialization, and morphology-aware merge rules to maintain linguistic consistency and preserve subword integrity. We use a large-scale Bengali news classification dataset to compare BengaliBPE with three baselines: Whitespace, SentencePiece BPE, and HuggingFace BPE. The evaluation considers tokenization granularity, encoding speed, and downstream classification accuracy. While all methods perform reasonably well, BengaliBPE provides the most detailed segmentation and the best morphological interpretability, albeit with slightly higher computational cost. These findings highlight the importance of language-aware tokenization for morphologically rich scripts and establish BengaliBPE as a strong foundation for future Bengali NLP systems, including large-scale pretraining of contextual language models.

The loss function used to train a neural network is strongly connected to its output layer from a statistical point of view. This technical report analyzes common activation functions for a neural network output layer, like linear, sigmoid, ReLU, and softmax, detailing their mathematical properties and their appropriate use cases. A strong statistical justification exists for the selection of the suitable loss function for training a deep learning model. This report connects common loss functions such as Mean Squared Error (MSE), Mean Absolute Error (MAE), and various Cross-Entropy losses to the statistical principle of Maximum Likelihood Estimation (MLE). Choosing a specific loss function is equivalent to assuming a specific probability distribution for the model output, highlighting the link between these functions and the Generalized Linear Models (GLMs) that underlie network output layers. Additional scenarios of practical interest are also considered, such as alternative output encodings, constrained outputs, and distributions with heavy tails.

This work investigates the structural properties, cycloid trajectories, and non-asymptotic convergence guarantees of the Expectation-Maximization (EM) algorithm for two-component Mixed Linear Regression (2MLR) with unknown mixing weights and regression parameters. Recent studies have established global convergence for 2MLR with known balanced weights and super-linear convergence in noiseless and high signal-to-noise ratio (SNR) regimes. However, the theoretical behavior of EM in the fully unknown setting remains unclear, with its trajectory and convergence order not yet fully characterized. We derive explicit EM update expressions for 2MLR with unknown mixing weights and regression parameters across all SNR regimes and analyze their structural properties and cycloid trajectories. In the noiseless case, we prove that the trajectory of the regression parameters in EM iterations traces a cycloid by establishing a recurrence relation for the sub-optimality angle, while in high SNR regimes we quantify its discrepancy from the cycloid trajectory. The trajectory-based analysis reveals the order of convergence: linear when the EM estimate is nearly orthogonal to the ground truth, and quadratic when the angle between the estimate and ground truth is small at the population level. Our analysis establishes non-asymptotic guarantees by sharpening bounds on statistical errors between finite-sample and population EM updates, relating EM's statistical accuracy to the sub-optimality angle, and proving convergence with arbitrary initialization at the finite-sample level. This work provides a novel trajectory-based framework for analyzing EM in Mixed Linear Regression.

ABSTRACT It involves the completely novel ways of integrating ML algorithms with traditional statistical modelling that has changed the way we analyze data, do predictive analytics or make decisions in the fields of the data. In this paper, we study some ML and statistical model connections to understand ways in which some modern ML algorithms help'enrich' conventional models; we demonstrate how new algorithms improve performance, scale, flexibility and robustness of the tradi tional models. It shows that the hybrid models are of great improvement in predictive accuracy, robustness, and interpretability. Keyword: Machine Learning, Statistical Modelling, Regression, Classification, Predictive Analytics, Hybrid Models, Dimensiona lity Reduction, Algorithmic Bias, Interpretability, Cross - Disciplinary Applications 1. INTRODUCTION Statistical modelling has very historically been the theoretical framework to understand relationships between variables and make inferences and test hypothes es. Its strength is that it is able to offer interpretations in terms of interpretable parameters and probabilistic assumptions [15].

We propose a new method for statistical inference in generalized linear models. In the overparameterized regime, Principal Component Regression (PCR) reduces variance by projecting high-dimensional data to a low-dimensional principal subspace before fitting. However, PCR incurs truncation bias whenever the true regression vector has mass outside the retained principal components (PC). To mitigate the bias, we propose Calibrated Principal Component Regression (CPCR), which first learns a low-variance prior in the PC subspace and then calibrates the model in the original feature space via a centered Tikhonov step. CPCR leverages cross-fitting and controls the truncation bias by softening PCR's hard cutoff. Theoretically, we calculate the out-of-sample risk in the random matrix regime, which shows that CPCR outperforms standard PCR when the regression signal has non-negligible components in low-variance directions. Empirically, CPCR consistently improves prediction across multiple overparameterized problems. The results highlight CPCR's stability and flexibility in modern overparameterized settings.

A formal link between regression and classification has been tenuous. Even though the margin maximization term $\|w\|$ is used in support vector regression, it has at best been justified as a regularizer. We show that a regression problem with $M$ samples lying on a hyperplane has a one-to-one equivalence with a linearly separable classification task with $2M$ samples. We show that margin maximization on the equivalent classification task leads to a different regression formulation than traditionally used. Using the equivalence, we demonstrate a ``regressability'' measure, that can be used to estimate the difficulty of regressing a dataset, without needing to first learn a model for it. We use the equivalence to train neural networks to learn a linearizing map, that transforms input variables into a space where a linear regressor is adequate.

Riesz regression has garnered attention as a tool in debiased machine learning for causal and structural parameter estimation (Chernozhukov et al., 2021). This study shows that Riesz regression is closely related to direct density-ratio estimation (DRE) in important cases, including average treat- ment effect (ATE) estimation. Specifically, the idea and objective in Riesz regression coincide with the one in least-squares importance fitting (LSIF, Kanamori et al., 2009) in direct density-ratio estimation. While Riesz regression is general in the sense that it can be applied to Riesz representer estimation in a wide class of problems, the equivalence with DRE allows us to directly import exist- ing results in specific cases, including convergence-rate analyses, the selection of loss functions via Bregman-divergence minimization, and regularization techniques for flexible models, such as neural networks. Conversely, insights about the Riesz representer in debiased machine learning broaden the applications of direct density-ratio estimation methods. This paper consolidates our prior results in Kato (2025a) and Kato (2025b).

Conformal prediction has emerged as a powerful framework for constructing distribution-free prediction sets with guaranteed coverage assuming only the exchangeability assumption. However, this assumption is often violated in online environments where data distributions evolve over time. Several recent approaches have been proposed to address this limitation, but, typically, they slowly adapt to distribution shifts because they update predictions only in a forward manner, that is, they generate a prediction for a newly observed data point while previously computed predictions are not updated. In this paper, we propose a novel online conformal inference method with retrospective adjustment, which is designed to achieve faster adaptation to distributional shifts. Our method leverages regression approaches with efficient leave-one-out update formulas to retroactively adjust past predictions when new data arrive, thereby aligning the entire set of predictions with the most recent data distribution. Through extensive numerical studies performed on both synthetic and real-world data sets, we show that the proposed approach achieves faster coverage recalibration and improved statistical efficiency compared to existing online conformal prediction methods.