Recently, Forré (arXiv:2104.11547, 2021) introduced transitional conditional independence, a notion of conditional independence that provides a unified framework for both random and non-stochastic variables. The original paper establishes a strong global Markov property connecting transitional conditional independencies with suitable graphical separation criteria for directed mixed graphs with input nodes (iDMGs), together with a version of causal calculus for iDMGs in a general measure-theoretic setting. These notes aim to further illustrate the motivations behind this framework and its connections to the literature, highlight certain subtlies in the general measure-theoretic causal calculus, and extend the "one-line" formulation of the ID algorithm of Richardson et al. (Ann. Statist. 51(1):334--361, 2023) to the general measure-theoretic setting.

Graph Neural Networks (GNNs) have achieved impressive performance in graph-related tasks. However, they suffer from poor generalization on out-of-distribution (OOD) data, as they tend to learn spurious correlations. Such correlations present a phenomenon that GNNs fail to stably learn the mutual information between prediction representations and ground-truth labels under OOD settings. To address these challenges, we formulate a causal graph starting from the essence of node classification, adopt backdoor adjustment to block non-causal paths, and theoretically derive a lower bound for improving OOD generalization of GNNs. To materialize these insights, we further propose a novel approach integrating causal representation learning and a loss replacement strategy. The former captures node-level causal invariance and reconstructs graph posterior distribution. The latter introduces asymptotic losses of the same order to replace the original losses. Extensive experiments demonstrate the superiority of our method in OOD generalization and effectively alleviating the phenomenon of unstable mutual information learning.

Sentiment signals derived from sparse news are commonly used in financial analysis and technology monitoring, yet transforming raw article-level observations into reliable temporal series remains a largely unsolved engineering problem. Rather than treating this as a classification challenge, we propose to frame it as a causal signal reconstruction problem: given probabilistic sentiment outputs from a fixed classifier, recover a stable latent sentiment series that is robust to the structural pathologies of news data such as sparsity, redundancy, and classifier uncertainty. We present a modular three-stage pipeline that (i) aggregates article-level scores onto a regular temporal grid with uncertainty-aware and redundancy-aware weights, (ii) fills coverage gaps through strictly causal projection rules, and (iii) applies causal smoothing to reduce residual noise. Because ground-truth longitudinal sentiment labels are typically unavailable, we introduce a label-free evaluation framework based on signal stability diagnostics, information preservation lag proxies, and counterfactual tests for causality compliance and redundancy robustness. As a secondary external check, we evaluate the consistency of reconstructed signals against stock-price data for a multi-firm dataset of AI-related news titles (November 2024 to February 2026). The key empirical finding is a three-week lead lag pattern between reconstructed sentiment and price that persists across all tested pipeline configurations and aggregation regimes, a structural regularity more informative than any single correlation coefficient. Overall, the results support the view that stable, deployable sentiment indicators require careful reconstruction, not only better classifiers.

While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]^D$ for target distributions supported on $d$-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in $\mathbb{R}^D$. By employing an easily implementable infinite series expansion of the transition densities, we develop analytic tools to bound the score function and its approximation by sparse ReLU networks. For target densities with Sobolev smoothness $α$, we establish a convergence rate in the $1$-Wasserstein distance of order $n^{-\frac{α+1-δ}{2α+d}}$ for arbitrarily small $δ> 0$, demonstrating that the generative algorithm fully adapts to the intrinsic dimension $d$. These results confirm that the presence of reflecting boundaries does not degrade the fundamental statistical efficiency of the diffusion paradigm, matching the almost optimal rates known for unconstrained settings.

In clustering, strong dominance in the size of a particular cluster is often undesirable, motivating a measure of cluster size uniformity that can be used to filter such partitions. A basic requirement of such a measure is stability: partitions that differ only slightly in their point assignments should receive similar uniformity scores. A difficulty arises because cluster labels are not fixed objects; algorithms may produce different numbers of labels even when the underlying point distribution changes very little. Measures defined directly over labels can therefore become unstable under label-count perturbations. I introduce the Mass Agreement Score (MAS), a point-centric metric bounded in [0, 1] that evaluates the consistency of expected cluster size as measured from the perspective of points in each cluster. Its construction yields fragment robustness by design, assigning similar scores to partitions with similar bulk structure while remaining sensitive to genuine redistribution of cluster mass.

We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all remaining edges follow the Erdős--Rényi model $\mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $\mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies that $\mathcal{G}(n, p, d, k)$ and $\mathcal{G}(n, p)$ are indistinguishable at the level of the marginals, and the signal lies entirely in the edge dependencies induced by the local geometry. We investigate both the information-theoretic and computational limits of detection. On the information-theoretic side, our upper bounds follow from three tests based on signed triangle counts: a global test, a scan test, and a constrained scan test; our lower bounds follow from two complementary methods: truncated second moment via Wishart--GOE comparison, and tensorization of KL divergence. These results together settle the detection threshold at $d = \widetildeΘ(k^2 \vee k^6/n^3)$ for fixed $p$, and extend the state-of-the-art bounds from the full model (i.e., $k = n$) for vanishing $p$. On the computational side, we identify a computational--statistical gap and provide evidence via the low-degree polynomial framework, as well as the suboptimality of signed cycle counts of length $\ell \geq 4$.

Standard masked discrete diffusion models face limitations in reasoning tasks due to their inability to correct their own mistakes on the masking path. Since they rely on a fixed number of denoising steps, they are unable to adjust their computation to the complexity of a given problem. To address these limitations, we introduce a method based on learning a Markov transition kernel that is trained on its own outputs. This design enables tokens to be remasked, allowing the model to correct its previous mistakes. Furthermore, we do not need a fixed time schedule but use a trained stopping criterion. This allows for adaptation of the number of function evaluations to the difficulty of the reasoning problem. Our adaptation adds two lightweight prediction heads, enabling reuse and fine-tuning of existing pretrained models. On the Sudoku-Extreme dataset we clearly outperform other flow based methods with a validity of 95%. For the Countdown-4 we only need in average of 10 steps to solve almost 96% of them correctly, while many problems can be solved already in 2 steps.

Loss functions play a central role in supervised classification. Cross-entropy (CE) is widely used, whereas the mean absolute error (MAE) loss can offer robustness but is difficult to optimize. Interpolating between the CE and MAE losses, generalized cross-entropy (GCE) has recently been introduced to provide a trade-off between optimization difficulty and robustness. Existing formulations of GCE result in a non-convex optimization over classification margins that is prone to underfitting, leading to poor performances with complex datasets. In this paper, we propose a minimax formulation of generalized cross-entropy (MGCE) that results in a convex optimization over classification margins. Moreover, we show that MGCEs can provide an upper bound on the classification error. The proposed bilevel convex optimization can be efficiently implemented using stochastic gradient computed via implicit differentiation. Using benchmark datasets, we show that MGCE achieves strong accuracy, faster convergence, and better calibration, especially in the presence of label noise.

Population-based learning paradigms, including evolutionary strategies, Population-Based Training (PBT), and recent model-merging methods, combine fast within-model optimisation with slower population-level adaptation. Despite their empirical success, a general mathematical description of the resulting collective training dynamics remains incomplete. We introduce a theoretical framework for neural network training based on two-time-scale population dynamics. We model a population of neural networks as an interacting agent system in which network parameters evolve through fast noisy gradient updates of SGD/Langevin type, while hyperparameters evolve through slower selection--mutation dynamics. We prove the large-population limit for the joint distribution of parameters and hyperparameters and, under strong time-scale separation, derive a selection--mutation equation for the hyperparameter density. For each fixed hyperparameter, the fast parameter dynamics relaxes to a Boltzmann--Gibbs measure, inducing an effective fitness for the slow evolution. The averaged dynamics connects population-based learning with bilevel optimisation and classical replicator--mutator models, yields conditions under which the population mean moves toward the fittest hyperparameter, and clarifies the role of noise and diversity in balancing optimisation and exploration. Numerical experiments illustrate both the large-population regime and the reduced two-time-scale dynamics, and indicate that access to the effective fitness, either in closed form or through population-level estimation, can improve population-level updates.

Understanding how biomarker distributions evolve over time is a central challenge in digital health and chronic disease monitoring. In diabetes, changes in the distribution of glucose measurements can reveal patterns of disease progression and treatment response that conventional summary measures miss. Motivated by a 26-week clinical trial comparing the closed-loop insulin delivery system t:slim X2 with standard therapy in children with type 1 diabetes, we propose a probabilistic framework to model the continuous-time evolution of time-indexed distributions using continuous glucose monitoring data (CGM) collected every five minutes. We represent the glucose distribution as a Gaussian mixture, with time-varying mixture weights governed by a neural ODE. We estimate the model parameter using a distribution-matching criterion based on the maximum mean discrepancy. The resulting framework is interpretable, computationally efficient, and sensitive to subtle temporal distributional changes. Applied to CGM trial data, the method detects treatment-related improvements in glucose dynamics that are difficult to capture with traditional analytical approaches.