Papers

Firms struggle to choose AI projects that pay off: two projects can look equally promising to smart, motivated stakeholders and yet deserve opposite decisions. At the residential real-estate brokerage Compass, one AI product (Likely-to-Sell recommendations) flagged sales outreach opportunities and went on to account for nine figures in annual gross commission revenue. Another championed AI product (a Time-on-Market pricing tool) was rightly shelved. A simple ROI estimate could not distinguish the two. We present expected ROI (eROI), a framework that decomposes each bet into three components and rates them separately: Value if Successful, Likelihood of Success, and Investment Required. Each maps to a question executives can answer before building: How valuable would it be if it worked? How likely is it to work? And what would it cost to implement? Separating the three breaks a common catch-22: teams cannot estimate ROI until they know whether a project will work, yet cannot know whether it will work without building it. Judging Value if Successful on its own dissolves the loop, letting a team argue that a product would be valuable if it worked while it weighs how likely that is. The framework also asks, before ranking anything, whether there are enough good ideas on the table. After ranking, it guides assembling a portfolio of bets rather than funding only the single top-ranked project. We illustrate eROI on Compass's candidate AI products. Precise ROI estimates are hard to make given the inherent uncertainty of AI projects. Coarse business-level ratings of the three components are enough to tell strong bets from weak ones.

We study stochastic gradient estimation in black-box environments where only noisy simulation observations of function values are available. Finite-difference (FD) methods are among the most widely used zeroth-order gradient estimators in such settings, by measuring the change in function values against a perturbation size. While the optimal order in choosing this perturbation size with respect to the simulation budget is well understood, the optimal constant factor relies on model characteristics that are typically unknown and viewed to be as difficult to estimate as the gradient itself. Consequently, FD estimators are often based on ad hoc tuning of the perturbation size, which may exhibit highly unstable performance across problem instances. In this paper, we challenge this conventional wisdom from both theoretical and practical perspectives. We show that, by pilot-estimating these model quantities using a negligible fraction of the simulation budget, substantial robustness is attained in the resulting FD estimators. Theoretically, we show that using a perturbation size governed by this pilot estimation can already achieve an MSE that is first-order identical to the “oracle" MSE as if the optimal perturbation size is known in advance. Moreover, we show how such an approach is competitive against any choices of prescribed perturbation size, even if they are designed to be minimax-optimal over reasonable classes of target functions and FD schemes. Our proposed pilot estimation is practically easy to run, and a variety of numerical experiments demonstrate both the robustness and near-oracle optimality of our estimator relative to conventional FD schemes based on ad hoc tuning.

We study the problem of recovering latent inner products from a random geometric graph with anisotropic Gaussian latent points. More precisely, for an i.i.d. sample $x_1, \dots, x_n \sim N(0,\Sigma)$ where $\Sigma \in \mathbb{R}^{d \times d}$, an edge $(i,j)$ is present in the graph if and only if $\langle x_i, x_j \rangle \ge \zeta$ for a threshold $\zeta$. We assume the threshold $\zeta$ to be chosen such that the average edge density of the graph is of constant order. To address the undesired degree fluctuations amplified by the anisotropy of the latent points, we consider the doubly centered adjacency matrix of the graph, and estimate the latent inner products using a rank-$d$ spectral approximation of the doubly centered matrix. The estimator obtains a mean squared error with a rate involving the stable rank of the covariance matrix $\Sigma$. Notably, the rate of estimation matches the state of the art for the isotropic case $\Sigma = I_d$, and permits an ill-conditioned covariance matrix with a diverging condition number. The analysis of the spectral method proceeds via the entrywise Hermite expansion of the doubly centered adjacency matrix with respect to the latent inner products. Instead of the standard trace method, it uses a decoupling argument recently introduced by Kaushik, Romberg, and Muthukumar (2025) to control nonlinear error terms.

Distributional random forests replace mean-based CART splitting with criteria that compare the full conditional response distribution in candidate children. We implement and systematically study a family of such criteria inside a single honest-forest implementation: isotropic random-Fourier-feature maximum mean discrepancy (MMD), an anisotropic diagonal-bandwidth variant, an adaptive per-split frequency-selection variant, and a non-kernel sliced-Wasserstein criterion, together with post-hoc kernel-mean shrinkage of the forest weights. Using paired-seed comparisons across synthetic quantile mechanisms, real univariate benchmarks, a California-housing subsample curve, and multivariate synthetic and real responses, we characterize where each extension pays. Three findings recur. First, among distributional criteria ordinary isotropic MMD is already close to best in class: the anisotropic, adaptive-frequency, and sliced-Wasserstein extensions, and post-hoc shrinkage, do not systematically improve on it. Second, on scalar tabular regression mean-based CART splitting remains the robust default and wins many cells. Third, multivariate responses are the regime where distributional splitting clearly earns its keep, most sharply on a pure-dependence copula where the energy score separates the criteria even though marginal CRPS does not. The evidence supports a simple allocation story: distributional splitting helps only when non-location structure is both present and estimable; otherwise it dilutes split-selection power away from the mean. All criteria, the honest forest, and the paired-comparison harness are implemented in the open-source drforest library, whose Rust-backed split search makes broad criterion sweeps inexpensive.

For the canonical problem of estimating a multivariate normal mean under squared error loss, we demonstrate, for the first time, the existence of proper Bayes minimax multiple shrinkage estimators by introducing a general approach for their explicit construction. As opposed to minimax shrinkage estimators that shrink towards a single prespecified target, minimax multiple shrinkage estimators adaptively shrink towards the more promising of a set of prespecified targets, substantially increasing the region of potential risk reduction while maintaining the protection of always being at least as good as the maximum likelihood estimator. These estimators are particularly useful in practice as they address the challenge of selecting a minimax shrinkage estimator when prior information suggests more than one viable shrinkage target to choose from. In contrast to previous formal Bayes minimax multiple shrinkage estimators, which were built on mixtures of superharmonic marginals, these proper Bayes minimax multiple shrinkage estimators are obtained via mixtures of square-root superharmonic marginals. Examples of such proper Bayes minimax multiple shrinkage estimators include an adaptive convex combination of the rescaled Strawderman shrinkage estimators.

LoRA fine-tuning can create intruder dimensions: new leading singular vectors of the updated weight matrix $W+BA$ that are nearly orthogonal to all pretrained singular vectors and that drive catastrophic forgetting. Since their discovery, no theory has predicted, layer by layer on measured spectra, when they appear. We derive a per-layer critical update strength $s^\ast=\bar\theta/(\gamma\sigma_1(BA))$, computed from the measured spectrum of $W$ alone through the rectangular spiked-deformation transform, together with an exact secular-equation characterization of the updated spectrum, with no fitted parameters. In a pre-specified study spanning four dense Transformer families, a state-space model, a mixture-of-experts model, and an encoder-decoder (18 adapters, 9{,}840 layer scans), the law localizes the empirical threshold within a factor of two on $82\%$ of layers, separates intruder-bearing from intruder-free layers at deployment with a mean AUC of $0.89$, holds unchanged on six third-party adapters, and predicts where WikiText-2 perplexity begins to degrade; a combination of the two pre-specified edge evaluations reaches $98\%$ and is confirmed out-of-bag on the external adapters ($0.997$). Full fine-tuning disperses its update far below the threshold of every layer, which resolves the asymmetry between LoRA and full fine-tuning. Norm-matched interventions confirm that threshold-crossing layers, rather than update magnitude, carry the forgetting, and a spike-budget rule derived from the thresholds, requiring one SVD and no validation sweeps, reduces forgetting by $62\%$ on the most fragile model at no task cost.

Recent years have witnessed increasing interests in tackling heteroscedastic noise in bandits and reinforcement learning. In these works, the cumulative variance of the noise $\Lambda = \sum_{t=1}^T \sigma_t^2$, where $\sigma_t^2$ is the variance of the noise at round $t$, is used to characterize the statistical complexity of the problem, yielding simple regret bounds of order $\tilde{\cal{O}}(d \sqrt{\Lambda / T^2})$ for $d$-dimensional linear bandits with heteroscedastic noise. However, with a closer look, $\Lambda$ remains the same order even if the noise is close to zero at half of the rounds, which indicates that the $\Lambda$-dependence is not optimal. In this paper, we revisit the stochastic linear bandit problem with heteroscedastic noise, where the action set is prefixed throughout the learning process. We propose a novel variance-adaptive algorithm VAEE (Variance-Aware Exploration with Elimination) for large action set, which actively explores actions that maximizes the information gain among a candidate set of actions that are not eliminated. With the active-exploration strategy, we show that VAEE achieves a simple regret with a nearly harmonic-mean dependent rate. For finitely many actions, we propose a variance-aware variant of G-optimal design based exploration, which achieves a simple regret with sharper dependence on $d$. We also establish a nearly matching lower bound for the fixed action set setting indicating that harmonic-mean dependent rate is unavoidable. To the best of our knowledge, this is the first work that breaks the $\sqrt{\Lambda}$ barrier for stochastic linear bandits with heteroscedastic noise.

Kernel ridge regression is a standard method for functional data analysis, but its exact behavior is less understood. We study tensor-product kernel ridge regression for estimating the $r$-th moment function of a random function based on noisy discrete observations. The formulation includes mean estimation, covariance estimation, and higher-order moment estimation in a single framework. Our main result gives a precise $1+o_{\mathbb{P}}(1)$ expansion for the $L^2$ error at each admissible regularization parameter. The expansion consists of bias and three variance terms corresponding respectively to variation across the independent sample paths, latent signal variation at each sample point, and variation from measurement errors, identifying the refined error structure underlying functional data. As applications, we show that KRR attains the minimax rate for source smoothness $s \leq 2$ but becomes suboptimal in the sparse regime for $s>2$ due to saturation. A technical ingredient is a set of concentration inequalities for $U$-statistics suited to the dependent product structure of functional observations.

I critique a set of entrenched methodological conventions that collectively create systemic dysfunction in statistical research for clinical decisions. These include: (1) the prevalent use of hypothesis tests to compare treatments, (2) remoteness from patient care of the methods used to evaluate the accuracy of predictions of patient outcomes, (3) poor practice of meta-analysis to combine findings across studies, and (4) widespread research with incredible certitude. It appears that the dysfunction is held in place by three factors: (i) rudimentary instruction in statistical methodology received by medical students and residents, (ii) reliance of clinical researchers on consulting biostatisticians, wo act as statistical gatekeepers in evaluation of grant proposals and paper submissions, and (iii) institutional practices of research funding agencies, medical journals, and governmental bodies that regulate medical treatment. I conjecture that systemic changes are necessary to break the existing impasse, moving statistical research to a better equilibrium.

This paper studies convolution rank regression (CRR) over decentralized distributed learning networks. We propose a novel decentralized CRR framework, in which estimators are obtained by solving consensus-constrained optimization with kernel-smoothed rank loss. The developed estimation scheme relies solely on local node data and information shared by neighboring nodes, thereby achieving privacy preservation and high communication efficiency. For heterogeneous network settings, we establish finite-sample error bounds for the decentralized CRR estimator and derive exact support recovery guarantees for the sparse decentralized CRR Lasso estimator. To facilitate numerical implementation, we adopt a generalized consensus ADMM to efficiently solve local subproblems across all network nodes. We verify the favorable performance of our developed approach via extensive numerical simulations and real-data experiments.

Statistical analysis of collections of networks, where each network is treated as the primary unit of observation, is of growing importance across a wide range of application domains, including gene regulatory, social, and financial networks. As networks consist of vertices and edges that do not naturally reside in Euclidean space, the direct application of conventional statistical methodologies, such as the computation of means and covariances, principal component analysis, and hypothesis testing, to samples of networks is not straightforward. A central challenge lies in defining meaningful measures of similarity or distance between networks of potentially varying sizes and structural types (e.g., directed, undirected, weighted or unweighted), particularly when no predefined node correspondence exists. To address these challenges, we introduce a framework termed functional topological data analysis (funTDA), which integrates tools from functional data analysis and topological data analysis to facilitate exploratory data analysis and inference on samples of networks. The proposed framework enables the computation of summary statistics, including means and variances, and supports the application of principal component analysis and hypothesis testing to topological features extracted from network data. Through simulation studies involving networks with varying connectivity structures, we demonstrate the ability of funTDA to distinguish between distinct network configurations. The methodology is illustrated through two real-data applications: networks constructed from pairwise word co-occurrences in novels by Jane Austen and Charles Dickens, and gene regulatory networks derived from gene expression measurements for seventeen individuals exposed to H3N2 influenza. In both applications, differences in network topology are assessed using principal component analysis and hypothesis testing.

Estimating multiple precision matrices in high-dimension presents significant challenges, particularly when distinct datasets share a common conditional dependency structure but exhibit population-specific interaction strengths. We address this problem by introducing the Multiplicative Graphical Lasso (Mglasso), a method for jointly estimating precision matrices across multiple Gaussian graphical models under a shared sparsity constraint. Each precision matrix is decomposed as a Schur-Hadamard product of a shared structural matrix $\boldsymbol{\Theta}$, which encodes the common conditional independence graph, and a population-specific matrix $\boldsymbol{\Gamma}_{l}$, which captures variation in edge strengths across populations. We optimize a penalized log-likelihood that utilizes an $\ell_1$-penalty to enforce common sparsity and a Frobenius norm penalty to regulate population-specific variations. The optimization is efficiently performed using the Alternating Direction Method of Multipliers (ADMM) algorithm integrated with gradient descent. Theoretically, we establish the local strict convexity of the objective function and provide rigorous high-dimensional consistency guarantees, including supremum norm error bounds and exact support recovery under sub-Gaussian tail conditions. Extensive simulations show superior model selection consistency at smaller sample sizes compared to the benchmark Group Graphical Lasso (GGL). Finally, the method's practical utility is further validated through real-world applications.

We present two approaches for predicting tennis match outcomes using topological data analysis and graph theory on ATP singles matches from 2000-2025. The first method applies lower-star filtration to player competitive networks, extracting topological features through persistent homology using four summary methods (VAB, HNAV, HWNAV, OW-HNPV) combined with Modified Band Depth analysis. Algorithmic optimizations including ego graph approximations and triangle elimination enable analysis of about 66k matches. Our Random Forest model achieves 66.2% accuracy (AUC = 0.719) using topological, graph-theoretic, and ranking features. Feature importance analysis reveals that rankings contribute 36.3%, centralities 25.5%, and TDA features 24.0%, with topological features providing complementary signal. When rankings are unavailable, the topology-only model maintains 63.56% accuracy, demonstrating that network-derived features alone capture meaningful competitive structure. The second method uses a modified Katz similarity index with temporal edge weighting, achieving 62.48% accuracy on held-out test data. This work represents the first application of lower-star filtration to tennis prediction, provides systematic comparison of four topological summary methods in sports analytics, and demonstrates that TDA can achieve above-chance prediction using network topology alone while providing additional value when combined with traditional features.

We study empirical risk minimization for learning non-linear dynamical systems whose transition dynamics may switch over time. Under stability assumptions, and i.i.d switching over a set of $K$ modes, we derive non-asymptotic bounds on the prediction risk expressed in terms of the metric entropy of the underlying function class. We instantiate our general result for Hölder and linear function classes, obtaining explicit convergence rates that depend on the effective sample size $Tp_i$, where $T$ is the trajectory length and $p_i$ is the probability of observing mode $i$. Numerical simulations support our theoretical findings. To the best of our knowledge, these results are the first non-asymptotic guarantees for learning switched nonlinear dynamical systems from a single trajectory.

Variational autoencoders (VAEs) transform high-dimensional, often noisy data into a compact latent representation, making downstream optimization more tractable. Three challenges persist in VAE-based constrained optimization: (i) sampling effectively within the latent space, (ii) identifying the active decision variables that actually influence the objective and constraints, and (iii) enforcing constraints without destabilizing training. We propose a Multi-stage Constrained Optimization Framework (MCOF). First, an entropy-constrained VAE (EC-VAE) coupled with a feature selector embeds objective and constraint information into a designated subset of latent variables, so that optimization proceeds over a low-dimensional subspace while the remaining coordinates supply solution diversity. Second, a Uniform Transformation (UT) module applies a per-dimension probability integral transform, replacing the irregular aggregate posterior with a uniform distribution over a bounded box and mitigating posterior collapse and Gaussian mixture bias. Third, a constraint-priority filter method (CPFM) solves the resulting surrogate problem by alternating violation-reduction and objective-reduction steps under a filter acceptance test, returning solutions that are feasible for the learned surrogate to a specified tolerance without requiring multiplier estimation. Finally, unselected latent coordinates are resampled to generate diverse decodings of a single optimized solution. We validate MCOF on a synthetic problem, where we ablate each stage and recover the analytic optimum, and on a ZINC250k drug design task, where the generated molecules satisfy the imposed constraints and are entirely novel relative to the training set.

Graphical models are ubiquitous in social and empirical science as they are intuitive and easy to use. These models belong to the broader class of Markov models, defined using solely conditional independence (CI) restrictions. In order to estimate finite-dimensional target parameters in such models efficiently, semi-parametric theory provides a principled framework for constructing regular and asymptotically linear estimators via influence functions (IFs). These estimators are asymptotically normal and root-$n$ consistent. Characterizing the class of all influence functions for a target parameter is crucial for statistically efficient inference in these models. For models that are Markov relative to directed acyclic graphs (DAGs), the orthogonal complement of the tangent space is known, implying that for any target the class of all influence functions can be derived once an influence function is obtained. On the other hand, for Markov models not equivalent to a DAG model – such as ordinary Markov models associated with undirected graphs, chain graphs, or acyclic directed mixed graphs – the orthogonal complement has not been characterized, impeding semi-parametric inference in these models. We derive closed form expressions for the orthogonal complement of the tangent space for general Markov models and illustrate our results by characterizing the class of influence functions for the conditional mean parameter in several graphical models.

Unexpected shocks recur in global operations, requiring decision rules that adapt as market and operating conditions change. Many operational systems also have hierarchical structures in which long-term and short-term decisions pursue a shared objective. We study how hierarchical reinforcement learning can strengthen resilience by adapting these interdependent rules jointly. We develop a two-timescale hierarchical reinforcement learning framework that adapts long-term and short-term policies at their respective time scales. Because the policies are interdependent, we synchronize their updates and prove, to our knowledge, the first convergence guarantees for coupled two-timescale learning. Over $T$ periods, our policies' average gap from an optimal policy pair is $O(T^{-1/2})$, improving to $O(\log T/T)$ when poor decisions produce clearer profit losses. In a used-car case study, inventory replenishment is the long-term decision and customer-arrival pricing the short-term decision. Relative to the strongest partially adaptive benchmark, the framework increases mean profit by $9.2\%$ under joint demand-supply shocks and by $11.8\%$ under a prolonged shock scenario, while maintaining a more stable profit trajectory over time. Short-term adaptation addresses routine seasonality and one-sided disruptions by responding immediately to changing conditions. Under joint demand-supply shocks, however, it is insufficient alone; long-term adaptation is also needed to create favorable conditions for short-term decisions. Joint adaptation thus yields higher and more stable profits through disruption and recovery. Because many organizations already use hierarchical planning, the framework strengthens operational resilience without altering existing decision structures.

Continuous cuffless blood pressure (BP) monitoring is essential for connected health systems and wearable devices, enabling early detection, longitudinal tracking, and personalized management of cardiovascular disease. Many prior approaches attempt to estimate BP indirectly by reconstructing electrocardiography (ECG) from photoplethysmography (PPG), assuming ECG provides a stronger physiological link to BP. However, ECG sensing is less accessible in wearable settings and may introduce unnecessary complexity. In this work, we first perform a large-scale physiological correlation analysis on the MIMIC-III waveform database, revealing that PPG exhibits substantially stronger coupling with arterial blood pressure (ABP) ($|r|=0.247$, $p<0.001$) than ECG does ($r=0.018$, $p=0.187$), challenging the assumption that ECG provides a superior intermediate representation. Motivated by this insight, we conduct a systematic comparison between direct PPG-to-BP prediction and ECG-mediated pipelines using multiple state-of-the-art deep learning models. Across 1.74M segments from 3,127 patients, direct PPG-to-BP prediction achieves British Hypertension Society Grade A performance ($\mathrm{MAE}_{\mathrm{SBP}} = 4.82 mmHg$, $\mathrm{MAE}_{\mathrm{DBP}} = 4.31 mmHg$), outperforming all ECG-mediated approaches, which achieve only Grade B accuracy. Our findings suggest that accurate continuous BP monitoring can be achieved directly from wearable PPG signals, enabling simpler, more efficient pipelines for real-world connected health systems.

We derive a general distribution formula applicable to a wide variety of Gibbs-type partitions and use it to obtain large sample results for linear combinations of the component frequency spectrum $(M_{jn})_{1\le j\le n}$ (in genetics, the allele frequency spectrum) associated with a random partitioning of $\{1,2,\ldots, n\}$. The two-parameter Pitman-Yor sampling model is analysed in detail and asymptotic distributions of sums of the form $\sum _{j=\lf \lambda n\rf}^{\lf \mu n\rf} M_{jn}$, $0<\lambda\le \mu\le 1$, are obtained. Our results suggest a possible functional limit theorem for $\sum _{j=\lf \lambda n\rf}^{n} M_{jn}$. Useful connections with limit shapes for random structures on the set of partitions and other applications are suggested.

Hierarchical neural networks are widely used in artificial intelligence, yet their mathematical properties remain incompletely understood. In the infinite-width limit, two different theoretical frameworks have been proposed. One reduces deep learning to kernel regression with a fixed kernel by assuming that the parameters remain close to their initialization, whereas the other allows the parameters to move away from their initialization, requiring the kernel itself to be optimized. In this paper, we study a three-layer neural network with a finite but large number of hidden units. We show that training the input-to-hidden weights yields a smaller generalization error than keeping them fixed. Furthermore, the latter setting exhibits singularities in the parameter space, whereas the former does not. These findings indicate that singularities play an essential role even in wide neural networks.

We prove monotonicity properties for medians of gamma sums and differences of the form $Z_{\alpha} = \alpha X_1 + (2 - \alpha)X_2$, where $X_1$ and $X_2$ are independent gamma random variables with common shape parameter. By combining these monotonicity properties with known bounds for the median of the gamma distribution, we establish sharp bounds for the median of the variance-gamma and McKay Type I distributions. Also, by exploiting the normal variance-mean mixture representation of the generalized hyperbolic distribution together with bounds for a ratio of modified Bessel functions of the second kind, we obtain sharp bounds for the median of the generalized hyperbolic distribution. We thus resolve all five conjectures of Gaunt and Merkle (2021). As a by-product of our analysis, we show that the variance-gamma distributions with positive asymmetry parameter and the McKay Type I distributions satisfy the “mode-median-mean” inequality for all admissible parameter values, and that the same is true of the generalized hyperbolic distribution with positive asymmetry parameter.

Large language models produce prompt-dependent probabilities over words, whereas scientific systems require uncertainty over meaningful states that can be updated as evidence arrives. We develop an observable framework for determining when language-derived probabilities support such a sequential state representation. Theoretically, we define typed measurable transformations of contextual language, construct a minimal closed representation, and give necessary and sufficient conditions for semantic updates to exist uniquely. We bound irreducible nonclosure and accumulated error, and under average contraction prove existence, uniqueness and stability of an external random recursion on a probability simplex. These results define a stochastic lexical calculus without attributing an internal calculus to the language model. Empirically, frozen experiments test the observable implications. Raw prompt-conditioned probabilities fail the prespecified invariance gate; after prompt-specific calibration, a common three-state representation passes the stability gates and covers 28 of 30 untouched eight-step paths, or 0.933 at nominal level 0.90. Accordingly, language probabilities support a stochastic state only conditionally on verified closure, stability and coverage within a declared operating domain.

Accurate fracture risk prediction is important for osteoporosis management, but commonly used clinical tools may not fully use information available in electronic health records (EHRs) and dual-energy X-ray absorptiometry (DXA) reports. We developed and externally validated time-to-event fracture prediction models among adults aged 50 years or older with clinically obtained DXA reports in 2 US health care systems. The development cohort was derived from NewYork-Presbyterian/Weill Cornell Medical Center and the external validation cohort from the Indiana Network for Patient Care. Predictors included demographics, lifestyle factors, prior fracture, comorbidities, medication exposures, osteoporosis treatment history, and DXA-derived T-scores extracted from radiology reports. The outcome was time from index DXA to first incident fragility fracture identified from structured diagnosis codes. We evaluated penalized Cox regression, random survival forest, gradient-boosting survival, and XGBoost survival models using 2 prespecified predictor settings and compared discrimination with clinically reported FRAX major osteoporotic fracture probabilities. The development cohort included 11,510 adults, of whom 858 sustained incident fragility fractures; the external validation cohort included 1,932 adults, of whom 180 sustained fractures. In internal validation, the expanded Cox model achieved a mean Harrell C-index of 0.779, compared with 0.653 for FRAX. In external validation, the corresponding Cox model achieved a Harrell C-index of 0.714, compared with 0.590 for FRAX; gradient-boosting survival had the highest external discrimination (0.725). EHR- and DXA-enhanced models showed better discrimination than clinically reported FRAX scores in this DXA-tested population, but calibration assessment, prospective evaluation, and implementation workflow assessment are needed before clinical use.

The rapid growth of massive and complex datasets in fields such as econometrics, environmental sciences, risk management, and public policy has reshaped statistical modeling while introducing significant computational and methodological challenges. These challenges arise not only from data scale and model complexity, but also from the sequential or streaming nature of modern applications and from data-privacy constraints that prevent sharing raw data and thus limit joint analysis. To address these challenges, we introduce a novel and comprehensive Bayesian framework for distributed and recursive inference, grounded in the Integrated Nested Laplace Approximations (INLA) methodology and implemented using the R-INLA software. Our contributions include the partitioning both data and structured model components, reducing computational complexity while preserving accuracy relative to centralized full-data inference. We demonstrate the effectiveness of the proposed framework through case studies that highlight its applicability in large-scale, streaming, and privacy-sensitive settings. By integrating distributed, federated, and recursive paradigms, this work offers scalable, adaptive, and generalizable tools for modern Bayesian inference.

Randomized experiments with ordinal outcomes are common across scientific applications, but conventional causal estimands such as the average treatment effect are difficult to interpret because ordinal categories lack meaningful numerical spacing. Two interpretable estimands, the probabilities that the treatment is beneficial and strictly beneficial, depend on the joint distribution of the potential outcomes and are therefore not identifiable from observed data. The sharp nonparametric bounds used to circumvent this are often too wide to be informative. We develop a Bayesian latent variable framework that models the join distribution by an ordered probit model and delivers coherent super population and finite population posterior inference on both estimands. The association between the potential outcomes remains non-identified and is treated as a sensitivity parameter. We characterize its impact on the estimands theoretically and empirically, showing a non-monotonic relationship governed by the location of the treatment effect relative to the roots of their partial derivatives and use this to recommend a range for applied use. Simulations show that posterior intervals are narrower than the bounds while retaining near-nominal coverage, a gain that reflects the parametric structure imposed rather than additional information in the data. Application on a rheumatoid arthritis trial shows that the method provides precise and practically relevant assessments of treatment effectiveness.

Bitcoin price prediction on sub-daily timescales is a hard open problem in computational finance. Bitcoin exhibits fat-tailed returns, non-stationary dynamics, and a price discovery process influenced by social discourse on Reddit and Twitter. Conventional approaches fuse OHLCV technical features with sentiment via static concatenation, applying identical fusion weights regardless of market state. This is inconsistent with the behavioural finance literature, which shows that retail sentiment is most predictive during volatile periods and noisy during calm ones. This paper proposes Regime-Aware Multi-Modal Learning (RAML), which conditions fusion of sentiment and price features on a dynamically detected binary market regime. Rolling 24-hour volatility partitions observations into stable and volatile regimes; a learnable sigmoid gate adjusts the weight of the sentiment embedding relative to the price embedding, trusting sentiment more during volatility and price dynamics more during stable phases. The system is evaluated on 3,491 hourly observations (July 2024-September 2025), combining Bitcoin OHLCV data with Reddit /r/Bitcoin FinBERT sentiment. Four models are compared - price-only BiLSTM, sentiment-only classifier, static-concatenation BiLSTM, and RAML - across 3-hour and 6-hour horizons, with an ablation study isolating the sentiment branch, regime detection, and adaptive fusion. RAML achieves macro-F1 of 0.5474 (3h) and 0.5513 (6h), with the highest AUC at 3 hours (0.5084), indicating better calibration. Ablation confirms every component is necessary, and replacing adaptive weighting with concatenation causes recall collapse at 6 hours (F1: 0.14). These results establish regime-conditioned adaptive fusion as a necessary design principle for multi-modal financial forecasting.

Mixed continuous–categorical data pose a representation problem for continuous generative models. Flow Matching and Gaussian diffusion operate in Euclidean spaces, whereas categorical laws lie on probability simplices and may be highly imbalanced. We study a logit-coordinate framework that encodes categorical variables as smoothed natural parameters and combines them with transformed numerical variables. This yields common formulations of Logit Flow Matching and Logit Diffusion. We introduce a mixed-distribution discrepancy separating categorical marginal error from conditional continuous Wasserstein error, and derive stability bounds and imbalance-aware nonparametric rates linking vector-field or drift error to decoded mixed-distribution error. Controlled simulations show that scaled-logit coordinates improve or match one-hot coordinates, especially under severe rare-cell imbalance. Across four real-data benchmarks and ten splits per dataset, Logit FM improves the primary distributional metrics on three datasets and is comparable on Churn2; Block-Conditional Logit FM consistently improves the flat model; and Logit Diffusion generally improves over or matches One-Hot Diffusion.

Modern predictive systems are expected to adapt their behavior to the specific situation they are facing. A clinical model should not treat every patient the same; a retrieval-augmented model should change its answer when given different evidence; a mixture-of-experts model should route different inputs to different experts. We call this capability context-adaptive inference: before predicting, the system uses information about the current context to specialize its parameters or computation for that instance. This article provides a unified view of context-adaptive inference across three traditions that are usually treated separately: (i) explicit adaptation in statistics (e.g. varying-coefficient models, local regression, hierarchical sharing), (ii) rapid task-specific adaptation in meta-learning and transfer, and (iii) implicit adaptation in large foundation models via prompting, retrieval, and expert routing. We formalize these approaches under a common objective: to map context $c$ to adapted parameters $\theta(c)$, then to predict via $f(x; \theta(c))$. Under squared loss, linear prediction heads, and fixed features, we prove that explicit parameter adaptation and implicit routing are mathematically equivalent to kernel ridge regression on joint features of inputs and context. Building on this bridge, we propose practical design principles and evaluation metrics including adaptation-efficiency, routing stability, and context-specific robustness to guide when to specialize, how to constrain that specialization, and how to audit context-adaptive models in deployment. Finally, we identify open problems in identifiability, robustness under distribution shift, and efficient large-scale adaptation, outlining design principles for methods that are scalable, reliable, and transparent in real-world settings.

Competitiveness in racing sports can be measured by comparing the start and finish rankings within races, as well as the start and finish rankings across races in a season. Since the importance of position changes is non-uniform and variance at the top of the ranking is more interesting than at the bottom of the ranking, we propose using weighted distances for this purpose. Therefore, two weighting schemes are applied and compared to the standard Kemeny distance to analyse the Formula One between 1950 and 2024. The evolution of competitive balance is unexpectedly robust to the weights, but competition is more intense if one focuses on the top positions. Competitive balance has been more unfavourable in the last two decades than ever before. Statistical tests uncover three structural breaks in competitive balance that are closely related to regulatory changes, highlighting the role of decision-makers in the evolution of competitiveness.

Interval censored continuous time multistate transition models (MSTMs) are widely used to characterize disease progression and other dynamic processes. Existing inference procedures are predominantly likelihood-based and require a separate model fit for each dataset, involving repeated evaluation of transition probability matrices. This repeated computation can become burdensome when the same model is applied across centers, time periods, or newly collected cohorts. We propose AIM, an amortized Bayesian inference framework for interval censored continuous time MSTMs. AIM learns the relationship between informative summary statistics and model parameters from simulated datasets generated during an offline training stage. Once trained for a prespecified model class, AIM provides posterior inference for new datasets without repeated likelihood optimization. Under fixed observation schedules and discretized covariate strata, we show that interval-specific transition counts are sufficient for the observed panel likelihood and establish identifiability of the corresponding population summaries. We further prove consistency of the induced summary posterior and its neural approximation, yielding a consistent posterior mean estimator. Simulation studies under progressive and competing-pathway MSTMs demonstrate accurate point estimation and reliable uncertainty quantification. AIM completed online inference in milliseconds, with median speedups ranging from approximately 154-fold to 2308-fold relative to repeated likelihood-based estimation. These results establish AIM as a scalable and theoretically grounded framework for reusable Bayesian inference in interval censored MSTMs.