We prove a descriptive version of Brooks’s theorem for directed graphs. In particular, we show that, if D is a Borel directed graph on a standard Borel space X such that the maximum degree of each vertex is at most d≥3, then unless D contains the complete symmetric directed graph on d+1 vertices, D admits a μ-measurable d-dicoloring with respect to any Borel probability measure μ on X, and D admits a τ-Baire-measurable d-dicoloring with respect to any Polish topology τ compatible with the Borel structure on X. We also prove a definable version of Gallai’s theorem on list dicolorings by showing that any Borel directed graph of bounded degree whose connected components are not Gallai trees is Borel degree-list-dicolorable.

ObjectiveMother’s health risks should be identified early so that the outcome of the pregnancy can be enhanced and the complications experienced during pregnancy can be minimized. This paper will design and test a leakage-regulated hybrid machine learning model to predict maternal health risk using the optimized ensemble models.MethodsIt used the publicly available UCI Maternal Health Risk dataset (n = 1015). A fixed random seed was used to stratify the dataset to 80% training and 20% independent testing subsets (42). SMOTENN resampling was only done to the training data to avoid data leakage. Internal cross-validation was resorted to as a means of hyperparameter tuning. We came up with the optimized XGBoost, blending, and hybrid stacking models. The performance of a model was measured in terms of accuracy, precision, recall, F1 score, ROC-AUC, confusion analysis, and probability mean squared error (Brier score).ResultsThe hybrid stacking model had a ROC-AUC of 0.911 and general accuracy of 80 percent over the independent test set. The model proved to be very sensitive with high-risk cases (recall = 0.85). The probability mean squared error (Brier score) was 0.07, which is good probability calibration. The hybrid framework proposed performed better in terms of discriminative capability as compared to baseline models (logistic regression, random forest, and SVM).ConclusionsThe suggested leakage-sensitive hybrid ensemble framework offers strong and clinically significant working outcomes on maternal health risk forecasting. The results show the significance of effective validation techniques and probabilistic evaluation measures in healthcare machine learning systems.

Abstract We prove the continuity of asymptotic entropy as a function of the step distribution for non-degenerate probability measures with finite entropy on wreath products A ≀ B = ⨁ B A ⋊ B A\wr B=\bigoplus_{B}A\rtimes B , where 𝐴 is any countable group and 𝐵 is a countable hyper-FC-central group that contains a finitely generated subgroup of at least cubic growth. As one step in proving the above, we show that, on any countable group 𝐺, the probability that the 𝜇-random walk on 𝐺 never returns to the identity is continuous in 𝜇, for measures 𝜇 such that the semigroup generated by the support of 𝜇 contains a finitely generated subgroup of at least cubic growth. Finally, we show that, among random walks on a group 𝐺 that admit a separable completely metrizable space 𝑋 as a model for their Poisson boundary, the weak continuity of the associated harmonic measures on 𝑋 implies the continuity of the asymptotic entropy. This result recovers the continuity of asymptotic entropy on known cases, such as Gromov hyperbolic groups and acylindrically hyperbolic groups, and extends it to new classes of groups, including linear groups and groups acting on CAT ⁢ ( 0 ) \mathrm{CAT}(0) spaces.

This paper extends the concept of probability measures induced by graphs by developing a comprehensive structural and analytical framework for probable graphs. A graph-induced probability measure is defined via normalized vertex degrees. It is shown that a graph is probable if and only if its size equals half its order, establishing a direct connection between probabilistic normalization and graph structure. New results include a characterization of probable graphs via average degree, connections with perfect matchings, entropy-based interpretations, and the introduction of a probability generating polynomial. These results provide a deeper understanding of the interplay between graph theory and probability, and open new directions for applications in network analysis and stochastic modeling.

Abstract In this paper, we investigate multivariate generalized gamma convolutions, commonly referred to as the Thorin class, which forms a distinguished subclass of the class of infinitely divisible distributions. The Thorin class contains all elementary gamma distributions and is closed under convolution and weak convergence; moreover, it is the smallest class of probability measures on $$\mathbb {R}^{p}$$ R p enjoying these properties. By developing multivariate Markov–Krein transformations, we establish new and significant connections between multivariate generalized gamma convolutions and multivariate generalized spline distributions. In particular, using the Markov–Krein formula, we show that the asymptotic behavior of multivariate generalized splines naturally leads to Thorin measures. Conversely, we prove that Thorin measures can be characterized through the asymptotics of generalized spline distributions. Finally, within this framework, we derive a representation of the Fourier–Laplace transform of Thorin measures, obtained via the asymptotic properties of multivariate generalized splines and the Markov–Krein transform.

Bayesian updating of atomic probabilities

Precision measurement of photon emission probabilities in 177Lu.

Application of interactive threat matrix induced system dynamics model to determine risk probability and resilient policy measures for CO2 pipelines

Moments of the Cramér transform of log-concave probability measures

The exponential ergodicity of partially dissipative McKean-Vlasov SDEs in the L 1 -Wasserstein distance has been extensively studied using asymptotic reflection coupling. However, the reflection coupling method is not applicable for the exponential ergodicity in L 2 -Wasserstein distance and relative entropy. In this paper, we first establish uniform log-Sobolev inequalities (in the frozen measure variable with bounded second moments) for the invariant probability measure of the corresponding SDEs with frozen distribution. Second, for the McKean-Vlasov SDEs, we combine the log-Harnack inequality and Talagrand's inequality to derive exponential ergodicity in both L 2 -Wasserstein distance and relative entropy. Furthermore, we extend these main results to the case of degenerate diffusion.

Abstract Free-energy-based adaptive biasing methods, such as metadynamics, the adaptive biasing force and their variants, are enhanced sampling algorithms widely used in molecular simulations. Although their efficiency has been empirically acknowledged for decades, providing theoretical insights via a quantitative convergence analysis is a difficult problem, in particular for the kinetic Langevin diffusion, which is non-reversible and hypocoercive. We obtain the first exponential convergence result for such a process, in an idealized setting where the dynamics can be associated with a mean-field non-linear flow on the space of probability measures. A key of the analysis is the interpretation of the (idealized) algorithm as the gradient descent of a suitable functional over the space of probability distributions.

A classical theorem of Szegő states that for any probability measure μ = w d θ 2 π + μ s \mu =w\frac {\mathrm {d}\theta }{2\pi }+\mu _s on the unit circle the polynomials are dense in L 2 ( T , μ ) L^2(\mathbb {T},\mu ) if and only if log ⁡ ( w ) ∉ L 1 ( T ) \log (w)\notin L^1(\mathbb {T}) . A related question asks whether the monomials with exponents in some subset Λ ⊆ N 0 \Lambda \subseteq \mathbb {N}_0 already span L 2 ( T , μ ) L^2(\mathbb {T},\mu ) if log ⁡ ( w ) ∉ L 1 ( T ) \log (w)\notin L^1(\mathbb {T}) . A result by Olevskii and Ulanovskii [Israel J. Math. 246 (2021), pp. 335–351] gives an answer if μ \mu belongs to a class of absolutely continuous measures. We investigate the same question for Markoff measures.

Abstract We prove that a Kakeya set in a vector space over a finite field of size always supports a probability measure, whose Fourier transform is bounded by for all non‐zero frequencies. We show that this bound is sharp in all dimensions at least 2. In particular, this provides a Fourier analytic proof that a Kakeya set in dimension 2 has size at least (which is asymptotically sharp). We also establish analogous results for sets containing ‐planes in a given set of orientations.

Computable phenotypes derived from electronic health records (EHRs) are central to clinical research and quality reporting. Although large language models (LLMs) can extract clinically rich information from unstructured notes, routine application to all patients is computationally expensive. We evaluated whether uncertainty-guided selective use of LLMs can improve phenotyping accuracy while preserving scalability. We developed a selective augmentation framework integrating structured and unstructured EHR data using uncertainty-guided triage. An ensemble of heterogeneous classifiers trained on structured data generated probabilistic phenotype predictions and uncertainty measures to identify patients at elevated risk of misclassification. Only flagged patients underwent LLM-based analysis of unstructured clinical notes using retrieval-augmented generation. LLM-derived outputs were incorporated as additional predictors in a final probabilistic model. Performance was evaluated for two registry-based phenotypes: diabetes mellitus and peripheral arterial disease (PAD), using internal cross-registry and external validation cohorts. For diabetes mellitus, selective augmentation improved sensitivity in the internal validation cohort from 0.81 to 0.90 without loss of specificity (0.92). More than 70% of triage-flagged patients represented misclassifications by structured data alone. For PAD, selective augmentation markedly increased sensitivity from 0.18 to 0.97 while maintaining high specificity (0.99), requiring LLM analysis for only 10% of patients. Uncertainty-guided triage efficiently concentrated LLM use on patients most likely to benefit, improving case identification-particularly for phenotypes poorly captured by structured data-while minimizing computational burden. Selective, uncertainty-guided integration of LLMs enables scalable, interpretable, and accurate EHR-based phenotyping, offering a practical alternative to universal LLM deployment in real-world informatics workflows.

Diabetes confers increased risk for fracture independently from FRAX-estimated fracture probability. To compare the relative performance of the rheumatoid arthritis (RA) input and trabecular bone score (TBS) adjustment, alone or in combination, to capture FRAX-independent risk associated with diabetes. We analyzed data on 54,609 individuals from the Manitoba Bone Density Program aged ≥ 40years with FRAX-based probability and TBS measurements (mean age 63.8years, 89.9% female) including 5274 (9.7%) with diabetes. Incident major osteoporotic fracture (MOF, 5723, 10.5%) and hip fractures (1715, 3.1%) were ascertained during mean 9.6years observation from population-based healthcare data. The effect of diabetes on fracture outcomes was modeled without (Cox regression) and with competing mortality (Fine-Gray regression), adjusted for FRAX-based probability before and after RA input and TBS adjustment. For MOF prediction in those with diabetes duration less than 5years, no FRAX adjustment was required. For those with duration 5-10years, FRAX adjusted with TBS was slightly better than the unadjusted FRAX output. For those with diabetes duration greater than 10years, the larger effect from RA was beneficial, with or without TBS. In contrast, hip fracture risk was consistently greater regardless of diabetes duration and required the use of TBS, with or without RA. Diabetes was associated with incident MOF and hip fracture independent of baseline fracture probability, but this risk was partially offset by excess mortality. TBS adjustment and RA input showed complementary benefits for improving fracture prediction that differed according to diabetes duration and fracture outcome.

The structure of the point spectrum in the limit (in time) states of dynamic conflict systems is studied in terms of probability measures. It is shown that a necessary and sufficient condition for the appearance of measures with point spectra is a single priority strategy. In this case, we establish the exponential rate of concentration of the distributions with point spectrum and its density in the phase space. The possibility of using information about the structure of the point spectrum in a new mathematical model of the formation of the beliefs of individuals in an abstract society is proposed.

In this paper, we primarily introduce the concepts of random measure-theoretic N-expansivity and random measure-theoretic cw-expansivity for a Borel probability measure, as well as the notions of random N-expansivity and random cw-expansivity for a random dynamical system. Subsequently, we prove that a random dynamical system is randomly N-expansive (res., randomly cw-expansive) if and only if every Borel probability measure is randomly measure-theoretic N-expansive (res., randomly measure-theoretic cw-expansive) of f and all Borel probability measures share a common random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic), namely, there is a measure random variable δ : Ω → ( 0 , + ∞ ) such that δ is a random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic) of every Borel probability measure.

Off-site risk area delineation under severe nuclear accident conditions: A deterministic-probabilistic coupled consequence analysis.

Sparsity and scalability remain recurring problems in Memory-Based (MB) Collaborative Filtering (CF).Hybrid Memory-Based (HMB) approaches were originally introduced as a way to combine User-Based (UB) and Item-Based (IB) models, with the expectation that the weaknesses of one could be offset by the other.In this work, we revisit that assumption within the HMB framework by evaluating three probabilistic similarity measures: Mutual Information (MI), Adjusted Mutual Information (AMI), and Kullback-Leibler divergence (KL).Across experiments on the MovieLens 100K and 1M datasets, we find that the optimal fusion parameter in many cases falls toward boundary values (0.0 or 1.0).In practice, this means that the hybrid setup simplifies into a standard MB model rather than benefiting from a mixed UB-IB contribution.MI-based variants obtain the strongest results, with NDCG@20 scores of 0.0366 and 0.1356 on MovieLens 100K and 1M, respectively.These values are higher than those of the baseline methods, i.e., HMB_PCC (0.0209, 0.0609), HMB_BC (0.0191, 0.0535), and the UB_Siamese model (0.0167, 0.0371), with statistically significant differences at < 0.05.These findings suggest that when a stronger, non-linear similarity measure such as MI is used, the benefit of linear hybrid fusion becomes limited.Rather than relying on increasingly complex hybrid architectures, our results indicate that improving the expressiveness of the underlying similarity measure can already lead to a simpler model that is both more accurate and computationally more efficient.

Nested nonparametric processes are vectors of random probability measures widely used in the Bayesian literature to model the dependence across distinct, though related, groups of observations. These processes allow a two-level clustering, both at the observational and group levels. However, most available models are either not computationally efficient or mathematically tractable. In the present paper, we introduce a range of nested processes that overcome these issues. Our proposal builds upon Compound Random Measures, introduced early by Griffin and Leisen (2017). We provide a complete investigation of the theoretical properties of our model, along with the posterior characterization for vectors of Compound Random Measures, which is interesting per se and still not available in the current literature. We develop the first Ferguson & Klass algorithm for nested nonparametric processes. Finally, we test the model’s performance on different simulated scenarios and we exploit the construction to study air pollution in various provinces of an Italian region (Lombardy). We empirically show how nested processes based on Compound Random Measures outperform other Bayesian competitors.