Background: Breast cancer polygenic risk scores (PRS) and traditional risk models (e.g., the Gail model [Gail]) are known to contribute largely independent information, but it is unclear how the overlap varies by ancestry, age, disease type (invasive breast cancer, DCIS), and risk threshold. Methods: In a retrospective case–control study, we evaluated risk prediction performance in 180,398 women (161,849 of European ancestry; 18,549 of Asian ancestry). Odds ratios (ORs) from logistic regression models and the area under the receiver operating characteristic curve (AUC) were estimated. Results: PRS for invasive disease showed a stronger association in younger (<50 years) women (OR = 2.51, AUC = 0.622) than in women ≥ 50 years (OR = 2.06, AUC = 0.653) of European ancestry. PRS performance in Asians was lower (OR range = 1.62–1.64, AUC = 0.551–0.600). Gail performance was modest across groups and poor in younger Asian women (OR = 0.94–0.99, AUC = 0.523–0.533). Age interactions were observed for both PRS (p < 0.001) and Gail (p < 0.001) in Europeans, whereas in Asians, age interaction was observed only for Gail (invasive: p < 0.001; DCIS: p = 0.002). PRS identified more high-risk individuals than Gail in Asian populations, especially ≥50 years, while Gail identified more in Europeans. Overlap between PRS, Gail, and family history was limited at higher thresholds. Calibration analysis, comparing empirical and model-based ROC curves, showed divergence for both PRS and Gail (p < 0.001), which indicates miscalibration. In Europeans, family history and prior biopsies drove Gail discrimination. In younger Asians, age at first live birth was influential. Conclusions: PRS adds value to risk stratification beyond traditional tools, especially in younger women and Asian ancestry populations.

Background: Background breast features are frequently noted in pathology reports alongside invasive breast cancer but rarely factor into prognosis or treatment decisions. Their relationship to tumor characteristics and patient outcomes remains incompletely characterized. Methods: We conducted a retrospective cohort study of 7603 patients with Stage I–III invasive breast cancer (diagnosed 1991–2022, age < 80 years) from the Joint Breast Cancer Registry in Singapore. Natural language processing (NLP) was applied to 9754 free-text pathology reports to extract co-existing background breast features, with accuracy validated by dual-reviewer assessment of 200 reports. Because background features are most reliably assessed on excision specimens, the primary analytic cohort comprised 3988 patients with available excision pathology reports. Unsupervised hierarchical clustering grouped extracted features into three categories. Associations with tumor characteristics were assessed with multinomial logistic regression and ten-year overall survival by Cox proportional hazards models (median follow-up 9.6 years; 620 deaths). Results: Here, we show that NLP-based extraction of background breast features from routine pathology reports achieves an accuracy of over 90% across features. Lobular neoplasia and benign proliferative changes are associated with less aggressive tumor characteristics, whereas early neoplastic and papillary lesions are more prevalent in HER2-enriched and luminal B tumor subtypes. Benign proliferative changes are associated with better survival in age- and year-adjusted models (hazard ratio 0.91, 95% CI 0.86–0.97), but this association is attenuated after adjustment for stage and subtype. Conclusions: NLP-enabled extraction of background breast features from pathology text is feasible at scale. These features reflect tumor biology but do not independently add prognostic information beyond established clinical variables.

We study group-averaged Markov chains obtained by augmenting a $π$-stationary transition kernel $P$ with a group action on the state space via orbit kernels. Given a group $\mathcal{G}$ with orbits $(\mathcal{O}_i)_{i=1}^k$, we analyse three canonical orbit kernels: namely the Gibbs $(G)$, Metropolis-Hastings $(M)$, and Barker $(B)$ kernels, as well as their multiplicative sandwiches $QPQ$ and the additive mixtures $\frac{1}{2}(P+Q)$ where $Q\in\{G,M,B\}$. We show that $M^t, B^t \to G$ blockwise as $t \to \infty$ under suitable conditions, that the projection chains induced by $(\mathcal{O}_i)_{i=1}^k$ coincide for $GPG$ and $P$, and that orbit averaging never deteriorates the absolute spectral gap or asymptotic variance when $P$ is reversible. We give a direct and simple proof of Pythagorean identity under the Kullback-Leibler (KL) divergence, showing that $GPG$ arises naturally as an information projection of $P$ onto the set of $G$-invariant transition matrices. For a given $P$, we characterise the optimal choice of $G$ with a fixed number of orbits that minimises the one-step KL divergence to stationarity. Analogously, for a given $G$, we characterise the optimal choice of $P$ and give sufficient conditions under which $GPG = Π$. We further show that alternating projections over multiple group actions converge at a rate governed by the singular values of an overlap matrix, and that in structured cases, this yields exact sampling where the number of group actions grows logarithmically with the size of the state space. Based on the theory, we propose two heuristics to tune $G$ in practice. We also illustrate the results on discrete uniform and multimodal examples, including the Curie-Weiss model where $GPG$ achieves polynomial (in inverse temperature and dimension) mixing while Glauber dynamics remains exponentially slow.

We study additive mixtures of Markov kernels of the form $$A_\alpha = \alpha P + (1-\alpha )G$$ A α = α P + ( 1 - α ) G , where $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] , P is a baseline sampler and G is a Gibbs kernel induced by a partition of the state space. We first motivate the study of $$A_\alpha $$ A α , which can be interpreted as the projection of a lifted Markov chain. We then consider the minimisation of distance to stationarity under two objectives: the squared Frobenius norm and the Kullback–Leibler (KL) divergence. For the Frobenius objective, we derive explicit trace formulae and identify a Cheeger-type functional that characterises optimal two-block partitions. This yields a structured combinatorial optimisation problem admitting a difference-of-submodular decomposition, enabling efficient approximation via majorisation–minimisation. We also obtain geometric decay rates governed by the absolute spectral gap of P . For the KL divergence, we establish convexity-based bounds showing that the divergence of $$A_\alpha $$ A α is controlled by those of both P and G , thereby reducing partition selection to the Gibbs component. Numerical experiments on the Curie–Weiss model demonstrate that suitable choice of both the partition and the parameter $$\alpha $$ α can significantly accelerate convergence in total variation distance. We observe a consistent trade-off between local exploration and global averaging, with intermediate values of $$\alpha $$ α achieving the best performance across regimes.