Causal discovery from observational data remains a fundamental challenge in machine learning and statistics, particularly when variables represent inherently positive quantities such as gene expression levels, asset prices, company revenues, or population counts, which often follow multiplicative rather than additive dynamics. We propose the Hybrid Moment-Ratio Scoring (H-MRS) algorithm, a novel method for learning directed acyclic graphs (DAGs) from positive-valued data by combining moment-based scoring with log-scale regression. The key idea is that for positive-valued variables, the moment ratio $\frac{\mathbb{E}[X_j^2]}{\mathbb{E}[(\mathbb{E}[X_j \mid S])^2]}$ provides an effective criterion for causal ordering, where $S$ denotes candidate parent sets. H-MRS integrates log-scale Ridge regression for moment-ratio estimation with a greedy ordering procedure based on raw-scale moment ratios, followed by Elastic Net-based parent selection to recover the final DAG structure. Experiments on synthetic log-linear data demonstrate competitive precision and recall. The proposed method is computationally efficient and naturally respects positivity constraints, making it suitable for applications in genomics and economics. These results suggest that combining log-scale modeling with raw-scale moment ratios provides a practical framework for causal discovery in positive-valued domains.

Accurate classification requires not only high predictive accuracy but also well-calibrated confidence estimates. Yet, modern deep neural networks (DNNs) are often overconfident, primarily due to overfitting on the negative log-likelihood (NLL). While focal loss variants alleviate this issue, they typically reduce accuracy, revealing a persistent trade-off between calibration and predictive performance. Motivated by the complementary strengths of generative and discriminative classifiers, we propose Generative Cross-Entropy (GCE), which maximizes $p(x|y)$ and is equivalent to cross-entropy augmented with a class-level confidence regularizer. Under mild conditions, GCE is strictly proper. Across CIFAR-10/100, Tiny-ImageNet, and a medical imaging benchmark, GCE improves both accuracy and calibration over cross-entropy, especially in the long-tailed scenario. Combined with adaptive piecewise temperature scaling (ATS), GCE attains calibration competitive with focal-loss variants without sacrificing accuracy.

This study surveys the historical development of regularization, tracing its evolution from stepwise regression in the 1960s to recent advancements in formal error control, structured penalties for non-independent features, Bayesian methods, and l0-based regularization (among other techniques). We empirically evaluate the performance of four canonical frameworks -- Ridge, Lasso, ElasticNet, and Post-Lasso OLS -- across 134,400 simulations spanning a 7-dimensional manifold grounded in eight production-grade machine learning models. Our findings demonstrate that for prediction accuracy when the sample-to-feature ratio is sufficient (n/p >= 78), Ridge, Lasso, and ElasticNet are nearly interchangeable. However, we find that Lasso recall is highly fragile under multicollinearity; at high condition numbers (kappa) and low SNR, Lasso recall collapses to 0.18 while ElasticNet maintains 0.93. Consequently, we advise practitioners against using Lasso or Post-Lasso OLS at high kappa with small sample sizes. The analysis concludes with an objective-driven decision guide to assist machine learning engineers in selecting the optimal scikit-learn-supported framework based on observable feature space attributes.

High-dimensional statistical settings ($p \gg n$) pose fundamental challenges for classical inference, largely due to bias introduced by regularized estimators such as the LASSO. To address this, Javanmard and Montanari (2014) propose a debiased estimator that enables valid hypothesis testing and confidence interval construction. This report examines their debiased LASSO framework, which yields asymptotically normal estimators in high-dimensional settings. The key theoretical results underlying this approach are presented. Specifically, the construction of an optimized debiased estimator that restores asymptotic normality, which enables the computation of valid confidence intervals and $p$-values. To evaluate the claims of Javanmard and Montanari, a subset of the original simulation study and the real-data analysis is presented. The original empirical analysis is extended to the desparsified LASSO, which is referenced but not implemented in the original study. The results demonstrate that while the debiased LASSO achieves reliable coverage and controls Type I error, the LASSO projection estimator can offer improved power in idealized low-signal settings without compromising error rates. The results reveal a trade-off: the LASSO projection estimator performs well in low-signal settings, while Javanmard and Montanari's method is more robust to complex correlations, improving precision and signal detection in real data.

Understanding causal dependencies in observational data is critical for informing decision-making. These relationships are often modeled as Bayesian Networks (BNs) and Directed Acyclic Graphs (DAGs). Existing methods, such as NOTEARS and DAG-GNN, often face issues with scalability and stability in high-dimensional data, especially when there is a feature-sample imbalance. Here, we show that the denoising score matching objective of diffusion models could smooth the gradients for faster, more stable convergence. We also propose an adaptive k-hop acyclicity constraint that improves runtime over existing solutions that require matrix inversion. We name this framework Denoising Diffusion Causal Discovery (DDCD). Unlike generative diffusion models, DDCD utilizes the reverse denoising process to infer a parameterized causal structure rather than to generate data. We demonstrate the competitive performance of DDCDs on synthetic benchmarking data. We also show that our methods are practically useful by conducting qualitative analyses on two real-world examples. Code is available at this url: https://github.com/haozhu233/ddcd.

Can a safety gate permit unbounded beneficial self-modification while maintaining bounded cumulative risk? We formalize this question through dual conditions -- requiring sum delta_n < infinity (bounded risk) and sum TPR_n = infinity (unbounded utility) -- and establish a theory of their (in)compatibility. Classification impossibility (Theorem 1): For power-law risk schedules delta_n = O(n^{-p}) with p > 1, any classifier-based gate under overlapping safe/unsafe distributions satisfies TPR_n <= C_alpha * delta_n^beta via Holder's inequality, forcing sum TPR_n < infinity. This impossibility is exponent-optimal (Theorem 3). A second independent proof via the NP counting method (Theorem 4) yields a 13% tighter bound without Holder's inequality. Universal finite-horizon ceiling (Theorem 5): For any summable risk schedule, the exact maximum achievable classifier utility is U*(N, B) = N * TPR_NP(B/N), growing as exp(O(sqrt(log N))) -- subpolynomial. At N = 10^6 with budget B = 1.0, a classifier extracts at most U* ~ 87 versus a verifier's ~500,000. Verification escape (Theorem 2): A Lipschitz ball verifier achieves delta = 0 with TPR > 0, escaping the impossibility. Formal Lipschitz bounds for pre-LayerNorm transformers under LoRA enable LLM-scale verification. The separation is strict. We validate on GPT-2 (d_LoRA = 147,456): conditional delta = 0 with TPR = 0.352. Comprehensive empirical validation is in the companion paper [D2].

Vision transformers (ViT) rely on attention mechanism to weigh input features, and therefore attention scores have naturally been considered as explanations for its decision-making process. However, attention scores are almost always non-zero, resulting in noisy and diffused attention maps and limiting interpretability. Can we quantify uncertainty measures of attention scores and obtain regularized attention scores? To this end, we consider attention scores of ViT in a statistical framework where independent noise would lead to insignificant yet non-zero scores. Leveraging statistical learning techniques, we introduce the bootstrapping for attention scores which generates a baseline distribution of attention scores by resampling input features. Such a bootstrap distribution is then used to estimate significances and posterior probabilities of attention scores. In natural and medical images, the proposed \emph{Attention Regularization} approach demonstrates a straightforward removal of spurious attention arising from noise, drastically improving shrinkage and sparsity. Quantitative evaluations are conducted using both simulation and real-world datasets. Our study highlights bootstrapping as a practical regularization tool when using attention scores as explanations for ViT. Code available: https://github.com/ncchung/AttentionRegularization

Online health communities (OHCs) are vital for fostering peer support and improving health outcomes. Support groups within these platforms can provide more personalized and cohesive peer support, yet traditional support group formation methods face challenges related to scalability, static categorization, and insufficient personalization. To overcome these limitations, we propose two novel machine learning models for automated support group formation: the Group specific Dirichlet Multinomial Regression (gDMR) and the Group specific Structured Topic Model (gSTM). These models integrate user generated textual content, demographic profiles, and interaction data represented through node embeddings derived from user networks to systematically automate personalized, semantically coherent support group formation. We evaluate the models on a large scale dataset from MedHelp, comprising over 2 million user posts. Both models substantially outperform baseline methods including LDA, DMR, and STM in predictive accuracy (held out log likelihood), semantic coherence (UMass metric), and internal group consistency. The gDMR model yields group covariates that facilitate practical implementation by leveraging relational patterns from network structures and demographic data. In contrast, gSTM emphasizes sparsity constraints to generate more distinct and thematically specific groups. Qualitative analysis further validates the alignment between model generated groups and manually coded themes, showing the practical relevance of the models in informing groups that address diverse health concerns such as chronic illness management, diagnostic uncertainty, and mental health. By reducing reliance on manual curation, these frameworks provide scalable solutions that enhance peer interactions within OHCs, with implications for patient engagement, community resilience, and health outcomes.

Reinforcement learning algorithms assume that observations satisfy the Markov property, yet real-world sensors frequently violate this assumption through correlated noise, latency, or partial observability. Standard performance metrics conflate Markov breakdowns with other sources of suboptimality, leaving practitioners without diagnostic tools for such violations. This paper introduces a prediction-based scoring method that quantifies non-Markovian structure in observation trajectories. A random forest first removes nonlinear Markov-compliant dynamics; ridge regression then tests whether historical observations reduce prediction error on the residuals beyond what the current observation provides. The resulting score is bounded in [0, 1] and requires no causal graph construction. Evaluation spans six environments (CartPole, Pendulum, Acrobot, HalfCheetah, Hopper, Walker2d), three algorithms (PPO, A2C, SAC), controlled AR(1) noise at six intensity levels, and 10 seeds per condition. In post-hoc detection, 7 of 16 environment-algorithm pairs, primarily high-dimensional locomotion tasks, show significant positive monotonicity between noise intensity and the violation score (Spearman rho up to 0.78, confirmed under repeated-measures analysis); under training-time noise, 13 of 16 pairs exhibit statistically significant reward degradation. An inversion phenomenon is documented in low-dimensional environments where the random forest absorbs the noise signal, causing the score to decrease as true violations grow, a failure mode analyzed in detail. A practical utility experiment demonstrates that the proposed score correctly identifies partial observability and guides architecture selection, fully recovering performance lost to non-Markovian observations. Source code to reproduce all results is provided at https://github.com/NAVEENMN/Markovianes.

We introduce a novel class of graphical models, termed profile graphical models, that represent, within a single graph, how an external factor influences the dependence structure of a multivariate set of variables. This class is quite general and includes multiple graphs and chain graphs as special cases. Profile graphical models capture the conditional distributions of a multivariate random vector given different levels of a risk factor, and learn how the conditional independence structure among variables may vary across these risk profiles; we formally define this family of models and establish their corresponding Markov properties. We derive key structural and probabilistic properties that underpin a more powerful inferential framework than existing approaches, underscoring that our contribution extends beyond a novel graphical representation.Furthermore, we show that the resulting profile undirected graphical models are independence-compatible with two-block LWF chain graph models.We then develop a Bayesian approach for Gaussian undirected profile graphical models based on continuous spike-and-slab priors to learn shared sparsity structures across different levels of the risk factor. We also design a fast EM algorithm for efficient inference. Inferential properties are explored through simulation studies, including the comparison with competing methods. The practical utility of this class of models is demonstrated through the analysis of protein network data from various subtypes of acute myeloid leukemia. Our results show a more parsimonious network and greater patient heterogeneity than its competitors, highlighting its enhanced ability to capture subject-specific differences.