Papers

We propose an adaptable testing procedure for independence between two high-dimensional random vectors. The method incorporates ridge regularization and principal component-based dimension reduction into the canonical correlation analysis (CCA) framework, thereby stabilizing classical test statistics in high-dimensional settings. Depending on the reduced dimension, we develop both a regularized likelihood ratio test and a regularized largest-root test to accommodate different testing scenarios. We establish the asymptotic behavior of the proposed procedures under both the null hypothesis and representative alternatives, and further develop a data-driven method for selecting the regularization parameter. Extensive simulation studies demonstrate favorable finite-sample performance across a broad range of settings.

Survival models can model time-to-event outcomes using partially observed data. They are widely used in clinical prediction, including cancer risk, disease progression, treatment response, and mortality. Recent models often rely on rich inputs collected at a specific clinical encounter, such as medical images, laboratory tests, electronic health record snapshots, or sensor measurements. In large retrospective datasets, these inputs are usually collected over many calendar years. As a result, they may contain clues about when they were acquired through changes in devices, protocols, documentation, patient mix, or clinical practice. This creates a potential failure mode when outcomes are observed only up to a fixed study end date. More recent records necessarily have less potential follow-up than older records. A model that can infer the record date from the input may therefore learn to predict how much follow-up was available rather than the patient's true risk of experiencing the event. We call this failure mode administrative-cutoff leakage. In this paper, we characterize when this leakage can occur, distinguish it from classical informative censoring and genuine temporal changes in risk, and propose practical ways to detect it. In simulations, we show that administrative-cutoff leakage can inflate fixed-horizon AUC and can also affect Harrell's C-index under realistic follow-up patterns. We then demonstrate the same behavior in a real mammography cohort. These results motivate a simple design principle for survival prediction: for an n-year prediction task, the dataset should provide at least n years of potential follow-up after the latest input date. Otherwise, the models may be subject to bias induced by administrative-cutoff leakage.

Expert background knowledge is often available in practical applications of causal discovery. Such constraints on the true causal graph can help causal discovery in terms of identifiability of causal effects and accuracy of the learned structure, but also in reducing the space of candidate causal graphs. As causal discovery can become computationally expensive for large number of variables, it is crucial to utilize background knowledge effectively during the causal discovery process. However, most current methods only use background knowledge in a postprocessing step after causal discovery to refine the learned graph. In this work, we develop a framework for utilizing background knowledge during the causal discovery process, focusing especially on scalable causal discovery methods that recover only a subset of the whole graph. We implement our framework for multiple algorithms and empirically show that utilizing background knowledge can both reduce computational requirements and increase the quality of the learned structures.

We propose EM-INLA, a scalable algorithm for empirical-Bayes hierarchical quantile regression that combines the Expectation-Maximization (EM) algorithm with Integrated Nested Laplace Approximations (INLA). The method exploits the normal-exponential mixture representation of the Asymmetric Laplace Distribution (ALD) to reformulate each M-step as a weighted Gaussian regression handled by INLA, with the ALD scale and random-effect variances updated in closed form at each iteration. The scale and variance hyperparameters are estimated by marginal maximum likelihood via the EM algorithm, and posterior marginals for the regression and random-effect parameters are obtained from a final INLA call conditional on the converged hyperparameter estimates. The result is an algorithm that avoids all MCMC sampling while scaling to large datasets and complex hierarchical structures that are intractable for standard fully Bayesian approaches. In a simulation study across four error scenarios, EM-INLA recovers the true parameters with accuracy comparable to Hamiltonian Monte Carlo (HMC) while achieving speedups ranging from 15x to over 50x depending on the scenario. The method is applied to the 2023 Prueba Saber 11 standardized test in Colombia to model the conditional quantiles of student scores as a function of socioeconomic covariates under a hierarchical structure of students nested within schools and municipalities, with over 400,000 observations.

Reliable forecasting of several interrelated environmental variables - such as regional precipitation and temperature, or other correlated geophysical fields - across many locations calls for accurate predictions accompanied by trustworthy statements of their uncertainty. Modern deep-learning models forecast such variables accurately but usually report no uncertainty, and forcing them to output uncertainty through maximum likelihood tends to degrade their accuracy, especially when the variables are strongly correlated. Motivated by this tension, we develop TSCoNet, a two-stage convolutional-recurrent model coupled with a Gaussian copula that jointly forecasts multiple variables over space and time while quantifying predictive uncertainty. The method first learns accurate mean forecasts and then, holding the mean fixed, refines a shared representation to estimate the predictive variance, yielding calibrated prediction intervals after a standard recalibration, so that uncertainty is added without sacrificing point accuracy. We study the approach on simulated non-stationary spatial fields on the sphere and on a real dataset of monthly precipitation and temperature for fifty cities over 2000-2020. The model matches the accuracy of a strong deterministic forecaster while supplying calibrated prediction intervals that the deterministic model cannot, giving a single tool that provides both accurate point forecasts and reliable uncertainty for multivariate spatio-temporal data.

In this work we lay the groundwork for automating the process of finding and proving the validity of lower confidence bounds of the mean. Our key finding is based on the observation that finding an optimal confidence bound under certain conditions can be formulated as an optimization problem. We use this observation to show that any valid confidence bound (such as Hoeffding's) must be a relaxation of a certain optimization problem. To automatically find and prove confidence bounds, we need to automate the process of defining and finding such a relaxation. We define a family of relaxations parameterized by a function called the order function. This family of relaxations can approximate any other target relaxation such as Hoeffding's by using the target as the order function. When the order function is linear (such as the sample mean in the case of Hoeffding's), our relaxation is a linear-size mixed-integer linear program.

AI visibility measurement is comparative: practitioners want to know which domains generative search engines cite most often and whether observed differences are large enough to support decisions. Yet the industry lacks a principled way to determine whether enough data has been collected. Collection budgets vary widely across studies and platforms, and conclusions are often drawn from rankings whose stability and precision are unknown. We introduce a sequential convergence framework based on two complementary criteria: rank stability evaluates whether the rank-correlation trajectory has reached a structural plateau, while structural sufficiency evaluates whether the spread of citation shares among established domains – those whose confidence intervals exclude zero – exceeds the uncertainty of those estimates. Together, these criteria distinguish rankings that have merely stabilized from those sufficiently resolved to support inference. Both are derived from regularities in the observed citation distribution, including its rank structure, uncertainty profile, and the boundary between observed and established domains. The framework retains a small number of structural constants but requires no externally specified query count, correlation target, or confidence-interval width target; stopping is driven by observed measurement uncertainty and remains robust across a range of sufficiency thresholds. Applied across 30 platform-topic combinations spanning Gemini, SearchGPT, and Perplexity, the framework adapts to platform- and topic-specific citation distributions. Results show that no fixed collection budget can be justified across contexts and that convergence can instead be evaluated from the structure of the observed distribution. The framework provides a practical basis for determining when AI visibility measurements are ready to support comparative analysis.

The explosive growth of complex data has catalyzed the emergence of graph statistics as a fundamentally new discipline in data science. Unlike traditional statistics, which operates primarily within the comfortable confines of Euclidean spaces, graph statistics confronts the reality that modern data naturally organize themselves as dynamic networks composed of complex interconnections. In this article, we present an overview of graph statistics as an emerging discipline, tracing its theoretical foundations, methodological innovations, and transformative applications. We examine how the integration of evolutionary game theory, ecological niche theory, topological data analysis, and graph theory through quasi-dynamic nonlinear modeling has created a new norm of statistical thinking capable of analyzing non-Euclidean data. We show how graph statistics is poised to revolutionize fields ranging from quantitative genetics and systems biology to materials science and artificial intelligence, offering a principled framework for transforming big data into practical knowledge.

We develop a sensitivity-analysis workflow for causal panel estimators, covering synthetic difference-in-differences, matrix completion, fixed-effect imputation, and group-time average treatment effects. The workflow combines Riesz-representation omitted-variable-bias bounds with partial-$R^2$ robustness values and separates two reporting routes. Route A gives a direct sensitivity profile for additive or projected confounding summarized by outcome-side and Riesz-side partial $R^2$ values. Route B treats observed-covariate benchmarks as auxiliary data only when benchmark-count, alpha-side alignment, model-check, dependence, and dominance diagnostics are credible; otherwise its main role is demotion. We derive estimator-specific Riesz diagnostics and clarify which are fixed-weight, target-level, or first-stage-conditional rather than full derivatives of regularized training maps. Monte Carlo stress tests distinguish calibrated benchmark settings from dominance failure, coarse alpha-side benchmarks, benchmark dependence, noisy covariates, and concentrated SDID weights. In the California tobacco-control panel, the SDID estimate is $-15.60$ packs per capita; corrected finite-donor placebo inference gives standard error 9.49 and add-one $p=0.051$. A refit-weight finite-difference audit changes the Route A nullification robustness value from 0.054 to 0.045, leaving the low-single-digit conclusion unchanged. A county-level minimum-wage application applies the same profile to a multi-cohort staggered panel.

In this paper, we study the nonparametric estimation of the squared diffusion coefficient of an ergodic diffusion process on non-compact supports, when the two coefficients of the stochastic differential equation belong to a space larger than the space of Hölder continuous functions. The estimators are constructed by projection onto finite-dimensional spaces, thereby avoiding truncation of the dimension. We establish risk bounds of non-adaptive estimators and explicit rates of convergence for bounded and unbounded diffusion coefficients. A model selection procedure is performed, followed by the derivation of risk bounds using Talagrand's inequality. The theoretical results are completed with a numerical study over simulated data.

Structure learning of directed acyclic graphs (DAGs) from observational data is a foundational task in causal discovery and is widely used to infer regulatory networks from medical and genomic measurements. The Bayesian formulation quantifies model uncertainty and admits prior biological knowledge, but its practical use has been hampered by the super-exponential growth of the DAG space and by the intractability of the node-marginal likelihood under flexible, non-conjugate priors. Existing closed-form solutions are largely confined to the conjugate Normal–Inverse-Gamma prior. We develop a Laplace-approximated Bayesian scoring function for the non-conjugate Normal–Gamma prior on the modified Cholesky parameterisation of the precision matrix, embed it in a Metropolis–Hastings sampler over DAGs, and couple the latent Gaussian network to a binary clinical outcome through a probit link. We show that the node-marginal integral is of generalised inverse-Gaussian form, so that its exact value is a modified Bessel function of the second kind and the proposed scoring function is its leading large-argument asymptotic; the posterior of each conditional variance is likewise generalised inverse-Gaussian and is sampled exactly. In simulation, the proposed prior improves on the conjugate baseline and on the PC, greedy-equivalence-search, NOTEARS, and DAGMA benchmarks at sample sizes typical of clinical cohorts. On two real datasets, the Sachs protein-signalling network, scored against its validated consensus graph, and the Wisconsin Diagnostic Breast Cancer data, the method recovers known structure and, through the DAG-probit extension, predicts malignancy from nuclear morphometry with a cross-validated ROC-AUC of $0.94$ using a sparse, interpretable set of direct predictors.

The Neural Tangent Kernel (NTK) is one powerful tool for analyzing the training dynamics of neural networks in the over-parameterized regime. Recently, the theoretical framework has been extended to physics-informed neural networks (PINNs) for solving linear PDEs, one highly popular class of neural PDE solvers. In the analysis, the positivity of the associated NTK plays a fundamental role. However, establishing the positivity of the NTK for PINNs is highly challenging, due to the presence of multiple differential operators. In this work, we propose a new theoretical framework, called Differential Neural Tangent Kernel (DNTK), for analyzing PINNs through the lens of the NTK, and establish the positivity of the infinite width DNTK for both shallow and deep neural networks for a wide class of activation functions, including RePU and smooth but non-polynomial activations, for all linear differential operators. These theoretical results lay the foundation for the analysis of gradient type algorithms for training PINNs.

Recursive models show promise on reasoning and language tasks, yet their test-time scaling lacks a principled criterion for selecting trajectories or determining recurrent depth. We introduce Energy-guided Recursive Model (ERM), which uses Hopfield-type memories of valid local and global structures to assign intrinsic energies to candidate trajectories. These energies guide candidate selection and suggest an effective range of recurrent depths, implying that deeper recurrence does not necessarily improve reasoning accuracy. They also enable sampling methods such as parallel tempering to improve exploration. For reasoning tasks, ERM achieves optimal solutions on Sudoku ($98.97\%$), Pencil Puzzle Bench (PPBench, $88.04\%$) and Maze ($99.30\%$), reaching the best accuracy in recursive modeling. On language modeling, ERM reduces RedPajama-V2 perplexity by $1.74\%$ with marginal inference overhead. The results support energy guidance as a practical framework for improving test-time scaling in recursive models.

We study a class of diffusion processes on $\mathbb{R}^n$ interacting through a symmetric matrix $X\in\mathbb{R}^{n\times n}$. When eigenvectors of $X$ are Haar-uniform on the orthogonal group, we derive a dynamical mean-field limit for the empirical law of sample paths, extending the classical Sompolinsky–Zippelius characterization for $X\sim\mathrm{GOE}$. The limit takes the form of a generalized Langevin equation with correlated Gaussian noise and memory, whose correlation and response kernels relate to those of the original dynamics through convolution equations involving the free cumulants of the eigenvalue distribution of $X$. For the overdamped Langevin diffusion associated with $\mu(\boldsymbol{\theta})\propto \exp\!\big(\frac12\boldsymbol{\theta}^{\top}X\boldsymbol{\theta}\big)\prod_{i=1}^n\nu(\mathrm{d}\theta_i)$, we analyze the mean-field limit under a rapid-mixing assumption. The correlation and response kernels admit time-translation-invariant approximants satisfying a fluctuation-dissipation relation. The generalized Langevin equation admits a Markovian approximation coupled to an auxiliary multivariate OU process and converges to a replica-symmetric prediction for the empirical coordinate law under $\mu$. This auxiliary correlation structure is characterized through the infinitesimal generator of a Markov semigroup for the lifted path-history process. Consequently, the free energy converges to a replica-symmetric limit under an explicit high-temperature condition, which for an Ising model is $\|X\|_{\mathrm{op}}<1/2$. By recent dynamical universality results, the same free-energy characterization holds for deterministic models without random disorder when $X$ satisfies a set of deterministic delocalization conditions.

While autoregressive models optimize the exact data likelihood via the chain rule, diffusion models are typically trained with denoising objectives. We develop conservation laws based on generalized extrinsic information transfer (GEXIT) functions for a broad class of memoryless noise processes, showing that the data–model cross-entropy (CE) can be characterized exactly as an integral of local information-theoretic derivatives along the noise path. This yields a unified characterization of the likelihood for discrete and continuous diffusion, with the Gaussian case reducing to the well-known mutual information–minimum mean-square error (I-MMSE) relationship. An immediate implication is a locality property: one can compute the information-theoretic derivatives using only the marginal posteriors along the noise path. As a result, training reduces to learning the marginal posteriors by minimizing the negative log-likelihood. While the conservation law implies that the entropy does not depend on the noise path, finite-capacity denoisers approximate the posteriors with varying accuracy across noise types, leading to differences in performance. We validate these predictions on synthetic Markov sources and standard benchmarks, including text8 and CIFAR-10.

We prove a full Adam theorem for spectral heavy-tail onset in a closed Gaussian Stein-Hermite teacher-student state-evolution model. The theorem begins with the actual full-batch Adam recurrences, derives the population gradient by Stein-Hermite calculus, proves finite-width covariance concentration, converts multi-step Adam momentum into an exact non-centered Gaussian sign kernel, controls the diagonal Adam denominator by a basis-homogenization theorem, derives a regularly varying projected update response from a Hermite edge-transfer theorem, pushes the response through the exact Gram update, and proves approximate-target KL contraction with matching upper and lower hitting bounds. The final law is (\tau_\varepsilon=\Theta(\Delta_1^{-\gamma}d^\rho\log(\Psi_0/\varepsilon))), where (\Delta_1) is the first spike-bulk spectral gap. The result is full in the following precise sense: every step from Adam's momentum and denominator to the spectral hitting law is formalized inside the closed state-evolution model. We also prove that a stronger arbitrary-gradient Adam theorem is impossible, and that exact two-step linear-network loss dynamics do not identify factor spectra or heavy-tail hitting times.

Heavy-tailed empirical spectral densities of neural-network weight matrices are widely used as diagnostics of implicit self-regularization, but the step complexity of heavy-tail emergence remains poorly understood. We formulate spectral heavy-tail formation as a right-censored hitting-time problem: a run that does not reach a heavy-tail diagnostic within the observation horizon is treated as censored rather than discarded. In controlled full-batch teacher–student dynamics, we find that the first-step spike–bulk gap alone does not explain onset time. Instead, finite-onset regression supports a dimension-corrected spectral-gap law, (\tau_{\mathrm{HT}}\approx C\Delta_1^{-\gamma}d^\rho), with (R^2=0.683), (\gamma=0.626), and (\rho=0.772) across 330 completed runs. Right-censored lognormal accelerated-failure-time models further favor the dimension-corrected model over a gap-only model, improving AIC from 706.62 to 628.70. Theoretically, we prove that exact early loss dynamics in linear networks do not determine factor spectral tails, that Adam recurrences alone do not imply spectral redistribution, and that projected singular-basis spreading implies contraction of a spectral-tail potential and hence a dimension-corrected hitting-time bound. Empirically, projected-kernel profiles support the sufficient spreading mechanism, Adam and AdamW agree under tested grids, GD and signGD do not reach onset in the same regimes, and real pretrained Qwen2.5-0.5B and Pythia-70M transformer weights show non-Gaussian spectral-tail structure relative to matched Gaussian nulls. The result is a reproducible spectral hitting-time law with rigorous conditional theory, not a claim that Adam necessarily generates heavy tails from first principles.

Large-scale robotic swarms motivate the use of mean-field control (MFC). Classical partial differential equation (PDE)-based formulations provide a principled framework but can become computationally challenging in higher dimensions, whereas machine learning achieves scalability at the cost of approximation and guarantees. In this work, we establish an optimization-based framework that lifts the MFC problem into the space of occupation measures, resulting in a convex relaxation formulated as an optimization over measures. The resulting problem is solved using a Frank-Wolfe (FW) algorithm in the measure space, with each iteration reduced to a tractable optimal control problem. This approach retains the O(1/k) convergence rate of FW, avoids discretization of the state space, and naturally incorporates interaction and safety constraints. Numerical experiments demonstrate agreement with analytic and PDE-based baselines in two dimensions and show that the method scales to three-dimensional environments with multiple obstacles, where standard grid-based PDE solvers become impractical. A full 3D instance with ten obstacles is solved in minutes on a standard workstation, underscoring the practicality and scalability of the proposed framework.

Semi-supervised anomaly detection plays a key role in diverse fields such as process monitoring, healthcare, and finance. However, lightweight methods often struggle with high-dimensional data and typically require careful tuning of multiple hyperparameters. Among existing approaches, Christoffel Function–based methods are attractive due to their simplicity, requiring at most a single hyperparameter. They also benefit from a well-established theoretical foundation that yields several interesting results for data science. However, their main limitation is poor scalability to high-dimensional settings. In this paper, we introduce CLOE, a new method that combines an autoencoder for dimensionality reduction with a Christoffel Function–based detector applied in the latent space. To better align representation learning with anomaly detection, we design a novel loss function that leverages the Christoffel Function to guide the autoencoder toward representations that better capture the support of the normal data distribution. We further propose a principled procedure to set the detection threshold and an efficient strategy to tune the single remaining hyperparameter. Experiments on multiple high-dimensional tabular anomaly detection benchmarks demonstrate that CLOE achieves superior performance compared to existing methods, while preserving the lightweight and low-tuning advantages of Christoffel Function–based approaches.

Large language models (LLMs) are increasingly deployed in settings where fluent but incorrect answers can be costly. In these settings, accuracy alone is insufficient: models must also know when they are likely to be wrong. We present ConfidenceBench, a calibration benchmark that evaluates verbalized confidence estimates in 15 frontier LLMs using the Brier score, a proper scoring rule that incentivises truthful probability reporting. Confidence is elicited via prompting, requiring no access to model logits and making the framework applicable to both closed-source and open-source systems. The benchmark comprises 200 private multiple-choice questions across four categories: spatial reasoning, high-precision mathematics, word lookup, and unknowable questions. Across three independent runs, Claude Opus 4.6 and Gemini 3.1 Pro Preview achieve the lowest Brier scores, both reported as 0.103. Both substantially outperform the calibrated-random baseline of 0.1875, while Gemini 3.1 Flash-Lite scores 0.367, indicating severe miscalibration. Accuracy and calibration diverge substantially across model families: the most accurate model is not the best-calibrated, and several models perform worse than the calibrated-random Brier baseline despite reasonable accuracy. These results show that verbalized confidence calibration is a distinct and practically important axis of LLM reliability, complementary to standard accuracy-based evaluation.

For many years, pairwise comparison methods have been widely used for eliciting preferences and ranking alternatives in decision-making problems. These methods estimate priority weights from a pairwise comparison matrix and are now standard tools in multi-criteria decision analysis. In this paper, we present a statistical view of the pairwise comparison method using reference values and the geometric mean to calculate alternative priorities. Thanks to this approach, we can simultaneously capture the phenomenon of inconsistency in pairwise comparisons and the preference distance between different alternatives. In this paper, we define indicators that measure the quality of the obtained weight vector, which, thanks to the statistical approach, have an understandable interpretation.

Hypothesis Testing is the most contentious procedure in statistical Methodology. P values rejects Null Hypotheses far too easily, specially for large samples. On the other hand, Bayes Factors depends on assumptions, for example regarding Intrinsic Bayes Factors, which average? Arithmetic, Geometric, Median? Our bound is the infimum over all the averages. We develop a lower bound on Intrinsic Bayes Factors that adjust authomatically with the sample size. Furthermore, we introduce the new idea of {\it{Least Favorable Intrinsic Prior}}, which corresponds to the least favourable possible training samples. The bound sets a bridge between Intrinsic Bayes Factors and Adrian Smith and David Spiegelhalter methodology.

We introduce a new conformal prediction method that constructs calibrated prediction sets over collections of spatial events, such as tropical cyclone genesis and earthquake locations. Forecasting natural hazards has become increasingly important, due to their significant economic impact, and quantifying the uncertainty of predictions is critical for accurate risk assessment. Our approach works by representing spatial point clouds as empirical measures so that we can score them using (sliced) Wasserstein distance, then constraining the resulting distribution-valued prediction set to be supported only near the training data manifold. We derive a coverage lower bound for the intersected sets and show that, in practice, this gap can be made small through a simple data-adaptive selection criterion. Because the resulting set is not analytically tractable, we introduce a modified flow-based sampling procedure, which allows us to represent and apply these prediction sets in practice as ensembles. Numerical experiments on synthetic data, tropical cyclone genesis, and earthquake occurrences show that our method achieves near-nominal coverage, with significantly lower energy distance and manifold distance than highest predictive density region (HDR) baselines along with generative model baselines.

Test assembly, the process of constructing a complete test form from an item pool subject to blueprint constraints, has traditionally been treated as a static optimization problem. In AI-enabled assessment environments, however, item pools evolve continuously as newly generated items enter with uncertain psychometric parameters, and delivery is on demand. These conditions make test assembly a sequential decision-making problem under uncertainty: which form should be deployed now, given current but incomplete knowledge of item quality, to simultaneously maximize measurement precision, satisfy content-blueprint constraints, maintain pool sustainability, and accelerate calibration of uncertain new items? This paper proposes the Stochastic Constrained Hybrid (SCH) framework as a principled answer to this question. SCH recasts form-level assembly as a multi-armed bandit (MAB) problem with Fisher information as the reward, extending recent item-level approaches in computerized adaptive testing (CAT) to the form-level setting. A simulation study comparing six test assembly methods is also presented. The main contribution of this paper is a framework for incorporating items with uncertain parameters into the automatic test assembly process for linear test forms.

Sudden cardiac death (SCD) is a leading cause of death in the U.S. Patients at elevated risk of SCD are primarily treated with an implantable cardioverter-defibrillator (ICD), which may prevent death from cardiovascular causes but may cause severe side effects, such as reduced quality of life from shock-induced pain. Decisions about ICD treatment therefore involve complex personal trade-offs across multiple health events, including mortality and quality of life. While prediction tools could help weigh these trade-offs, they commonly focus on univariate outcomes; at best, they treat other clinical endpoints as inputs, so trade-offs cannot be directly informed. To address this, we propose a novel general Bayesian framework that jointly models a terminal event and a longitudinal marker as a bivariate process over discrete time, for settings where prediction is the primary goal. Discretization of study time lets the framework capture the dynamic interplay between outcomes while avoiding implicit extrapolation beyond truncation by a terminal event. The framework flexibly accommodates phenomena arising in applied contexts, including global time-invariant and local time-dependent dependence structures between the terminal event and the longitudinal marker, and latent association via a shared frailty term. Estimation proceeds via the Bayesian paradigm, yielding patient-specific joint posterior predictions for the time to terminal event and the future marker trajectory. We introduce the framework with a focus on its modeling flexibility, provide guidance on discretization and on Bayesian model construction and selection, and discuss insights from the joint posterior predictions. Finally, we demonstrate the framework's clinical relevance and practicality for SCD and ICD therapy using data from the Sudden Cardiac Death in Heart Failure Trial (SCD-HeFT), an important ICD-related benchmark trial.

Runs tests have long been used as a non-parametric check if data contains a non-random signal. We derive a recursive expression for the distribution of the longest run using Markov chain theory. Next we develop a permutation test on the runs comprising a feature to get the probability of its height. This leads finally to a bootstrap test on the height using the raw, continuous data. Such a test can evaluate not only the large heights of peaks but also the small heights of flats. We can apply these tests to features in the spacing of data to detect and locate multi-modality.

In semi-competing risks data, the interest lies in the estimation of the survival function of a non-terminal event time, which is subject to dependent censoring by a terminal event. This problem has been extensively studied in the literature, but mostly focusing on unconditional settings. However, in many clinical applications incorporating covariates is necessary to control for confounding and improve survival function estimation. In this paper, we propose a conditional copula-graphic estimator that allows for covariate adjustment in the marginal survival functions of the non-terminal and terminal event times as well as in their dependence structure. The proposed estimator is semiparametric in that the conditional copula is specified parametrically using an Archimedean copula, but its dependence parameter function and margins are estimated nonparametrically. The estimator is obtained via a sequential iterative algorithm with alternating updates of the survival function of the non-terminal event and the conditional copula. The performance of the conditional copula-graphic estimator is assessed using simulated and real data, and is compared to that of the unconditional copula-graphic estimator to investigate the consequences of failing to account for covariate effects.

The prevailing approach to inferring weak gravitational lensing shear from images involves detecting galaxies, estimating their ellipticities, and calibrating these estimates to correct for image noise, selection bias, and model misspecification. Characterizing the statistical model and assumptions underlying this pipeline is challenging, which makes it difficult to propagate uncertainty through its various stages. As an alternative, we propose to infer shear using neural posterior estimation (NPE), a type of simulation-based inference. We train a deep neural network to map a simulated multiband image to a variational distribution over the underlying shear field, thereby folding galaxy detection, deblending, measurement, and calibration into a single implicit inference step. Once trained, the network accounts for all features present in the simulated images, including potential sources of bias. In experiments on simulated constant-shear images with increasingly complex observational effects, NPE produces accurate and well-calibrated posterior approximations for both shear components in the presence of blended galaxies, spatially varying point spread functions, stars, and detector artifacts. These results demonstrate that NPE can be a viable shear estimation method in settings where all anticipated features and artifacts can be simulated, a requirement that will become increasingly feasible as simulation fidelity improves in the coming decades.

We propose a framework for the Markov chain (MC) choice model with panel data, including parameter estimation, personalized choice prediction, and personalized assortment optimization. In contrast to the traditional setting, which assumes that each transaction is independently drawn from a random utility model, our framework accounts for dependencies among transactions for the same customer in historical data, captured by partial-ordering preference information. To the best of our knowledge, our framework initiates the study of choice modeling with panel data under MC. As our primary result, we propose novel expectation-maximization (EM) algorithms for MC parameter estimation by incorporating partial-ordering-based customer preference information. On synthetic datasets and the sushi dataset, our EM algorithms outperform the traditional EM algorithm of Simsek and Topaloglu (Operations Research, 66, 2018) and multinomial-logit-based partial-order benchmarks adapted from Jagabathula and Vulcano (Management Science, 64, 2018). As our secondary contribution, we present hardness and computational results for conditional choice prediction and assortment optimization problems. These results complement our estimation framework and clarify the computational landscape of conditional choice and assortment optimization, which may be of independent interest.

Many real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG). In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription factors. These functions are partially observed across the DAG, with noisy and heterogeneously sampled measurements, posing significant challenges for reconstruction, uncertainty propagation, and inference. To tackle these challenges, we place priors over functions and naturally arrive at Deep Gaussian Processes over DAGs. We theoretically study their prior-collapse behaviour, and the effect of graph topology and intermediate observations on the preservation of information. We obtain almost-sure lower bounds on the asymptotic frequency of depths at which the distinction between inputs is preserved, identify broad kernel classes for which these hold, and prove an observation by \cite{dunlop2018} on the role of input connections. We offer a structured variational approximation that retains graph dependencies, preserves compositional uncertainty, and captures the explaining-away behaviour of colliders. Finally, we empirically validate our theoretical results and our methodology, and model a latent-collider DAG, a protein signalling network, and a multi-fidelity heavy-ion collision emulation task, attaining state-of-the-art performance while recovering low-fidelity contributions and yielding interpretability of the simulator hierarchy.