Papers

Kernel methods have been extensively utilized in machine learning for classification and prediction tasks due to their ability to capture complex non-linear data patterns. However, single kernel approaches are inherently limited, as they rely on a single type of kernel function (e.g., Gaussian kernel), which may be insufficient to fully represent the heterogeneity or multifaceted nature of real-world data. Multiple kernel learning (MKL) addresses these limitations by constructing composite kernels from simpler ones and integrating information from heterogeneous sources. Despite these advances, traditional MKL methods are primarily designed for continuous outcomes. We extend MKL to accommodate the outcome variable belonging to the exponential family, representing a broader variety of data types, and refer to our proposed method as generalized linear models with integrated multiple additive regression with kernels (GLIMARK). Empirically, we demonstrate that GLIMARK can effectively recover or approximate the true data-generating mechanism. We have applied it to a COVID-19 chest X-ray dataset, predicting binary outcomes of ICU escalation and extracting clinically meaningful features, underscoring the practical utility of this approach in real-world scenarios.

We prove new, general versions of Bernstein-von Mises theorem for both well-specified and misspecified models when the log-likelihood is concave in the parameter and the prior distribution is log-concave. Unlike classical versions of Bernstein-von Mises theorem, our versions do not require technical smoothness assumptions, and they solely rely on convex analysis.

This article critically assesses the utility of the classical statistical technique of Canonical Correlation Analysis (CCA) for studying spatial associations and proposes a new approach to enhance it. Unlike bivariate correlation analysis, which focuses on the relationship between two individual variables, CCA investigates associations between two sets of variables by identifying pairs of linear combinations that are maximally correlated. CCA has strong potential for uncovering complex multivariate relationships that vary across geographic space. We propose Geographically Weighted Canonical Correlation Analysis (GWCCA) as a new technique for exploring local spatial associations between two sets of variables. GWCCA localizes standard CCA by weighting each observation according to its spatial distance from a target location, thereby estimating location-specific canonical correlations. The effectiveness of GWCCA in recovering spatial structure and capturing spatial effects is evaluated using synthetic data. A case study of US county-level health outcomes and social determinants of health further demonstrates the empirical capabilities of the proposed method. The results indicate that GWCCA has broad potential applications in spatial data-intensive fields such as urban planning, environmental science, public health, and transportation, where understanding local multivariate spatial associations is critical.

Probabilistic forecasting is increasingly critical across high-stakes domains, from finance and epidemiology to climate science. However, current evaluation frameworks lack a consensus metric and suffer from two critical flaws: they often assume independence across time steps or variables, and they demonstrably lack sensitivity to tail events, the very occurrences that are most pivotal in real-world decision-making. To address these limitations, we propose two kernel-based metrics: the signature maximum mean discrepancy (Sig-MMD) and our novel censored Sig-MMD (CSig-MMD). By leveraging the signature kernel, these metrics capture complex inter-variate and inter-temporal dependencies and remain robust to missing data. Furthermore, CSig-MMD introduces a censoring scheme that prioritizes a forecaster's capability to predict tail events while strictly maintaining properness, a vital property for a good scoring rule. These metrics enable a more reliable evaluation of direct multi-step forecasting, facilitating the development of more robust probabilistic algorithms.

The field of performative prediction had its beginnings in 2020 with the seminal paper "Performative Prediction" by Perdomo et al., which established a novel machine learning setup where the deployment of a predictive model causes a distribution shift in the environment, which in turn causes a mismatch between the distribution expected by the predictive model and the real distribution. This shift is defined by a so-called distribution map. In the half-decade since, a literature has emerged which has, among other things, introduced new solution concepts to the original setup, extended the setup, offered new theoretical analyses, and examined the intersection of performative prediction and other established fields. In this survey, we first lay out the performative prediction setting and explain the different optimization targets: performative stability and performative optimality. We introduce a new way of classifying different performative prediction settings, based on how much information is available about the distribution map. We survey existing implementations of distribution maps and existing methods to address the problem of performative prediction, while examining different ways to categorize them. Finally, we point out known and previously unknown connections that can be drawn to other fields, in the hopes of stimulating future research.

This paper considers the asymptotic behavior in $\beta$-Hölder spaces, and under $L^p$ loss, of the non-modified gamma kernel density estimator introduced by Chen (Annals of the Institute of Statistical Mathematics, 52, 471-480, 2000) for the analysis of nonnegative data, in the situation where the target may have a finite effective or true upper endpoint but the estimator itself is left untruncated and treats the support as $[0,\infty)$. The finite endpoint is an analytical device in the definition of the function class and the risk, not information supplied to the estimator. The functional classes are chosen so that the target density matches smoothly to zero at the upper endpoint, which isolates the behavior at the origin and avoids additional upper-endpoint leakage bias. The estimator attains the minimax rate for $(p,\beta)\in [1,3)\times(0,2]$ and part of $[3,4)\times(0,2]$, but fails for $p\geq 4$ or $\beta>2$, leaving a small region unresolved.

We introduce WildCat, a high-accuracy, low-cost approach to compressing the attention mechanism in neural networks. While attention is a staple of modern network architectures, it is also notoriously expensive to deploy due to resource requirements that scale quadratically with the input sequence length $n$. WildCat avoids these quadratic costs by only attending over a small weighted coreset. Crucially, we select the coreset using a fast but spectrally-accurate subsampling algorithm – randomly pivoted Cholesky – and weight the elements optimally to minimise reconstruction error. Remarkably, given bounded inputs, WildCat approximates exact attention with super-polynomial $O(n^{-\sqrt{\log(\log(n))}})$ error decay while running in near-linear $O(n^{1+o(1)})$ time. In contrast, prior practical approximations either lack error guarantees or require quadratic runtime to guarantee such high fidelity. We couple this advance with a GPU-optimized PyTorch implementation and a suite of benchmark experiments demonstrating the benefits of WildCat for image generation, image classification, and language model KV cache compression.

The friendship paradox index is a network summary statistic used to quantify the friendship paradox, which describes the tendency for an individual's friends to have more friends than the individual. In this paper, we utilize Markov's inequality to derive the weak law of large numbers for the friendship paradox index in a random geometric graph, a widely-used model for networks with spatial dependence and geometry. For uniform random geometric graph, where the nodes are uniformly distributed in a space, the friendship paradox index is asymptotically equal to $1/4$. On the contrary, in nonuniform random geometric graphs, the nonuniform node distribution leads to distinct limiting properties for the index. In the relatively sparse regime, the friendship paradox index is still asymptotically equal to $1/4$, the same as in the uniform case. In the intermediate sparse regime, however, the index converges in probability to $1/4$ plus a constant that is explicitly dependent on the node distribution. Finally, in the relatively dense case, the index diverges to infinity as the graph size increases. Our results highlight the sharp contrast between the uniform case and its nonuniform counterpart.

Current instance segmentation models achieve high performance on average predictions, but lack principled uncertainty quantification: their outputs are not calibrated, and there is no guarantee that a predicted mask is close to the ground truth. To address this limitation, we introduce a conformal prediction algorithm to generate adaptive confidence sets for instance segmentation. Given an image and a pixel coordinate query, our algorithm generates a confidence set of instance predictions for that pixel, with a provable guarantee for the probability that at least one of the predictions has high Intersection-Over-Union (IoU) with the true object instance mask. We apply our algorithm to instance segmentation examples in agricultural field delineation, cell segmentation, and vehicle detection. Empirically, we find that our prediction sets vary in size based on query difficulty and attain the target coverage, outperforming baselines (naive best parameter and morphological dilation-based methods). We provide versions of the algorithm with asymptotic and finite sample guarantees. Our work is the first to capture structural uncertainty in instance segmentation by constructing confidence sets of diverse segmentation predictions.

Linear mixed models are widely used for pharmaceutical stability trending when sufficient lots are available. Expiry support is often based on whether lot-specific conditional-mean confidence limits remain within specification through a proposed expiry, and these limits depend on the denominator degrees-of-freedom (DDF) method used for $t$-based inference. We document an operationally important boundary-proximal phenomenon: when a fitted random-effect variance component is close to zero, Satterthwaite DDF—and, in sensitivity analyses, Kenward–Roger DDF—for conditional-mean predictions can collapse, inflating $t$ critical values and producing very wide and sometimes nonmonotone pointwise confidence limits on scheduled time grids. In the settings studied here, containment DDF yields stable degrees of freedom and avoids sharp discontinuities as variance components approach the boundary. In a balanced random-intercept design, a closed-form calculation gives the conditional prediction variance and delta-method Satterthwaite DDF, showing why DDF can be extremely small near the boundary when the prediction point is near the design centroid. A worked example and simulation studies show that DDF choice can materially change pass/fail conclusions even when the observed measurements themselves are not close to specification limits. Containment-based inference with the full random-effects model provides a single framework without model switching. When containment is unavailable, a 10% variance-contribution reduction procedure mitigates extreme Satterthwaite behavior by simplifying only when fitted contributions at the proposed expiry are negligible. AICc step-down is best treated as a sensitivity analysis because it can be too aggressive when the proposed-expiry margin is small.

Selective conformal prediction aims to construct prediction sets with valid coverage for a test unit conditional on it being selected by a data-driven mechanism. While existing methods in the offline setting handle any selection mechanism that is permutation invariant to the labeled data, their extension to the online setting – where data arrives sequentially and later decisions depend on earlier ones – is challenged by the fact that the selection mechanism is naturally asymmetric. As such, existing methods only address a limited collection of selection mechanisms. In this paper, we propose PErmutation-based Mondrian Conformal Inference (PEMI), a general permutation-based framework for selective conformal prediction with arbitrary asymmetric selection rules. Motivated by full and Mondrian conformal prediction, PEMI identifies all permutations of the observed data (or a Monte-Carlo subset thereof) that lead to the same selection event, and calibrates a prediction set using conformity scores over this selection-preserving reference set. Under standard exchangeability conditions, our prediction sets achieve finite-sample exact selection-conditional coverage for any asymmetric selection mechanism and any prediction model. PEMI naturally incorporates additional offline labeled data, extends to selection mechanisms with multiple test samples, and achieves FCR control with fine-grained selection taxonomies. We further work out several efficient instantiations for commonly-used online selection rules, including covariate-based rules, conformal p/e-values-based procedures, and selection based on earlier outcomes. Finally, we demonstrate the efficacy of our methods across various selection rules on a real drug discovery dataset and investigate their performance via simulations.

Iterative self-improvement fine-tunes an autoregressive large language model (LLM) on reward-verified outputs generated by the LLM itself. In contrast to the empirical success of self-improvement, the theoretical foundation of this generative, iterative procedure in a practical, finite-sample setting remains limited. We make progress toward this goal by modeling each round of self-improvement as maximum-likelihood fine-tuning on a reward-filtered distribution and deriving finite-sample guarantees for the expected reward. Our analysis reveals an explicit feedback loop where better models accept more data per iteration, supporting sustained self-improvement while explaining eventual saturation of such improvement. Adopting a task-centric view by considering reasoning tasks with multiple difficulty levels, we further prove quantifiable conditions on model initialization, task difficulty, and sample budget where easy-to-hard curricula provably achieve better guarantees than training on fixed mixtures of tasks. Our analyses are validated through Monte-Carlo simulations and experiments spanning a synthetic graph-based reasoning task and multiple standard mathematical reasoning benchmarks.

In observational studies, adjusting for confounders is required if a treatment comparison is planned. A crude comparison of the primary endpoint without covariate adjustment will suffer from biases, and the addition of regression models could improve precision by incorporating imbalanced covariates and thus help make correct inference. Desirability of outcome ranking (DOOR) is a patient-centric benefit-risk evaluation methodology designed for randomized clinical trials. Still, robust covariate adjustment methods could further expand the compatibility of this method in observational studies. In DOOR analysis, each participant's outcome is ranked based on pre-specified clinical criteria, where the most desirable rank represents a good outcome with no side effects and the least desirable rank is the worst possible clinical outcome. We develop a causal framework for estimating the population-level DOOR probability, via the inverse probability of treatment weighting method, G-Computation method, and a Doubly Robust method that combines both. The performance of the proposed methodologies is examined through simulations. We also perform a causal analysis of the Multi-Drug Resistant Organism (MDRO) network within the Antibacterial Resistant Leadership Group (ARLG), comparing the benefit:risk between Mono-drug therapy and Combination-drug therapy.

This paper proposes a new way of evaluating the accuracy and validity of probabilistic forecasts that change over time (such as an in-game win probability model, or an election forecast). Under this approach, each model to be evaluated is treated as a canonical Kelly bettor, and the models are pitted against each other in an iterative betting contest. The growth or decline of each model's bankroll serves as the evaluation metric. Under this approach, market consensus probabilities and implied model credibilities can be updated real time as each model updates, and do not require one to wait for the final outcome. Using a simulation model, it will be shown that this method is in general more accurate than traditional average log-loss and Brier score methods at distinguishing a correct model from an incorrect model. This Kelly approach is shown to have a direct mathematical and conceptual analogue to Bayesian inference, with bankroll serving as a proxy for Bayesian credibility.

We study a canonical multi-task demand-learning problem motivated by retail pricing, where a firm seeks to estimate heterogeneous linear price-response functions across multiple decision contexts. Each context is described by rich covariates but exhibits limited price variation, motivating transfer learning across tasks. A central challenge in leveraging cross-task transfer is endogeneity: prices may be arbitrarily correlated with unobserved task-level demand determinants across tasks. We propose a new meta-learning framework that identifies the conditional mean of task-specific causal demand parameters given a subset of task-specific observables despite such confounding, assuming that each task contains at least two distinct locally exogenous price points. This subset is carefully designed to include all of the prices to address cross-task confounding, while masking two demand outcomes that provide randomized supervision to address identifiability issues arising from the inclusion of all prices. We show that this information design is maximally uniformly valid, in that any refinement of the conditioning set that reveals withheld-outcome information is not guaranteed to identify the conditional mean causal target. We validate our method on real and synthetic data, demonstrating improved recovery of demand responses relative to standard transfer-learning baselines.

We study the problem of learning multi-index models (MIMs), where the label depends on the input $\boldsymbol{x} \in \mathbb{R}^d$ only through an unknown $\mathsf{s}$-dimensional projection $\boldsymbol{W}_*^\mathsf{T} \boldsymbol{x} \in \mathbb{R}^\mathsf{s}$. Exploiting the equivariance of this problem under the orthogonal group $\mathcal{O}_d$, we obtain a sharp harmonic-analytic characterization of the learning complexity for MIMs with spherically symmetric inputs – which refines and generalizes previous Gaussian-specific analyses. Specifically, we derive statistical and computational complexity lower bounds within the Statistical Query (SQ) and Low-Degree Polynomial (LDP) frameworks. These bounds decompose naturally across spherical harmonic subspaces. Guided by this decomposition, we construct a family of spectral algorithms based on harmonic tensor unfolding that sequentially recover the latent directions and (nearly) achieve these SQ and LDP lower bounds. Depending on the choice of harmonic degree sequence, these estimators can realize a broad range of trade-offs between sample and runtime complexity. From a technical standpoint, our results build on the semisimple decomposition of the $\mathcal{O}_d$-action on $L^2 (\mathbb{S}^{d-1})$ and the intertwining isomorphism between spherical harmonics and traceless symmetric tensors.

Lloyd's k-means algorithm is one of the most widely used clustering methods. We prove that in high-dimensional, high-noise settings, the algorithm exhibits catastrophic failure: with high probability, essentially every partition of the data is a fixed point. Consequently, Lloyd's algorithm simply returns its initial partition - even when the underlying clusters are trivially recoverable by other methods. In contrast, we prove that Hartigan's k-means algorithm does not exhibit this pathology. Our results show the stark difference between these algorithms and offer a theoretical explanation for the empirical difficulties often observed with k-means in high dimensions.

Population size estimation from capture-recapture data is central for studying hard-to-reach populations, incorporating auxiliary covariates to account for heterogeneous capture probabilities and recapture dependencies. However, missing attributes pose a critical methodological challenge due to reluctance to share sensitive information, data collection limitations, and imperfect record linkage. Existing approaches either ignore missingness or rely on a priori imputation, potentially introducing substantial bias. In this work, we develop a novel nonparametric estimation framework using a Missing at Random assumption to identify capture probabilities under missing covariates. Using semiparametric efficiency theory, we construct one-step estimators that combine efficiency, robustness, and finite-sample validity: they approximately achieve the nonparametric efficiency bound, accommodate flexible machine learning methods through a doubly robust structure, and provide approximately valid inference for any sample size. Simulations demonstrate substantial improvements over naive imputation approaches, with our doubly robust ML estimators maintaining valid inference even at high missingness rates where competing methods fail. We apply our methodology to re-estimate mortality in the Gaza Strip from October 7, 2023, to June 30, 2024, using three-list capture-recapture data with missing demographic information. Our approach yields more conservative yet precise estimates compared to previous methods, indicating the true death toll exceeds official statistics by approximately 26%. Our framework provides practitioners with principled tools for handling incomplete data in conflict settings and other applications with hard-to-reach populations.

Conditional Value-at-Risk (CVaR) is a central tail-risk measure in stochastic structural mechanics, yet its accurate evaluation under high-dimensional, spatially correlated material uncertainty remains computationally prohibitive for classical Monte Carlo methods. Leveraging bounded-expectation reformulations of CVaR compatible with quantum amplitude estimation, we develop a quantum-enhanced inference framework that casts CVaR evaluation as a statistically consistent, confidence-constrained maximum-likelihood amplitude estimation problem. The proposed method extends iterative quantum amplitude estimation (IQAE) by embedding explicit maximum-likelihood inference within a rigorously controlled interval-tracking architecture. To ensure global correctness under finite-shot noise and the non-injective oscillatory response induced by Grover amplification, we introduce a stabilized inference scheme incorporating multi-hypothesis feasibility tracking, periodic low-depth disambiguation, and a bounded restart mechanism governed by an explicit failure-probability budget. This formulation preserves the quadratic oracle-complexity advantage of amplitude estimation while providing finite-sample confidence guarantees and reduced estimator variance. The framework is demonstrated on benchmark problems with spatially correlated lognormal Young's modulus fields generated using a Nystrom low-rank Gaussian kernel model. Numerical results show that the proposed estimator achieves substantially lower oracle complexity than classical Monte Carlo CVaR estimation at comparable confidence levels, while maintaining rigorous statistical reliability. This work establishes a practically robust and theoretically grounded quantum-enhanced methodology for tail-risk quantification in stochastic continuum mechanics.

Customer lifetime value (CLV) describes a customer's long-term economic value for a business. This metric is widely used in marketing, for example, to select customers for a marketing campaign. However, modeling CLV is challenging. When relying on customers' purchase histories, the input data is sparse. Additionally, given its long-term focus, prediction horizons are often longer than estimation periods. Probabilistic models are able to overcome these challenges and, thus, are a popular option among researchers and practitioners. The latter also appreciate their applicability for both small and big data as well as their robust predictive performance without any fine-tuning requirements. Their popularity is due to three characteristics: data parsimony, scalability, and predictive accuracy. The R package CLVTools provides an efficient and user-friendly implementation framework to apply key probabilistic models such as the Pareto/NBD and Gamma-Gamma model. Further, it provides access to the latest model extensions to include time-invariant and time-varying covariates, parameter regularization, and equality constraints. This article gives an overview of the fundamental ideas of these statistical models and illustrates their application to derive CLV predictions for existing and new customers.

The broken random sample problem was first introduced by DeGroot, Feder, and Gole (1971, Ann. Math. Statist.): in each observation (batch), a random sample of $M$ i.i.d. point pairs $ ((X_i,Y_i))_{i=1}^M$ is drawn from a joint distribution with density $p(x,y)$, but we can observe only the unordered multisets $(X_i)_{i=1}^M$ and $(Y_i)_{i=1}^M$ separately; that is, the pairing information is lost. For large $M$, inferring $p$ from a single observation has been shown to be essentially impossible. In this paper, we propose a parametric method based on a pseudo-log-likelihood to estimate $p$ from $N$ i.i.d. broken sample batches, and we prove a fast convergence rate in $N$ for our estimator that is uniform in $M$, under mild assumptions.

The Gaussian kernel is one of the most important kernels, applicable to many research fields, including scientific computing and data science. In this paper, we present asymptotic analysis of the Gaussian kernel matrix in high dimension under a statistical model of noisy data. The main result is a nice combination of Karoui's asymptotic analysis with procedures of constrained low rank matrix approximations. More specifically, Karouli clarified an important asymptotic structure of the Gaussian kernel matrix, leading to strong consistency of the eigenvectors, though the eigenvalues are inconsistent. This paper focuses on the above results and presents a consistent estimator with the use of the smallest eigenvalue, whenever the target kernel matrix tends to low rank in the asymptotic regime. Importantly, asymptotic analysis is given under a statistical model representing partial noise. Although a naive estimator is inconsistent, applying an optimization method for low rank approximations with constraints, we overcome the difficulty caused by the inconsistency, resulting in a new estimator with strong consistency in rank deficient cases.

Detecting early warning signals in climatic time series is essential for anticipating critical transitions and tipping points. Common statistical indicators include increased variance and lag-one autocorrelation prior to bifurcation points. However, these indicators are sensitive to observational noise, long-term mean trends, and long-memory dependence, all of which are prevalent in climatic time series. Such effects can easily obscure genuine signals or generate spurious detections. To address these challenges, we employ a flexible Bayesian framework for modelling time-varying autocorrelation in long-range dependent time series, also accounting for time-varying variance. The approach uses a mixture of two fractional Gaussian noise processes with a time-dependent weight function to represent fractional Gaussian noise with a time-varying Hurst exponent. Inference is performed via integrated nested Laplace approximation, enabling joint estimation of mean trends and handling of irregularly sampled observations. The strengths and limitations of detecting changes in the autocorrelation is investigated in extensive simulations. Applied to real climatic data sets, we find evidence of early warning signals in a reconstructed Atlantic multidecadal variability index, while dismissing such signals for paleoclimate records spanning the Dansgaard-Oeschger events.

Data streams are rarely static in dynamic environments like Industry 4.0. Instead, they constantly change, making traditional offline models outdated unless they can quickly adjust to the new data. This need can be adequately addressed by continual learning (CL), which allows systems to gradually acquire knowledge without incurring the prohibitive costs of retraining them from scratch. Most research on continual learning focuses on classification problems, while very few studies address regression tasks. We propose the first prototype-based generative replay framework designed for online task-free continual regression. Our approach defines an adaptive output-space discretization model, enabling prototype-based generative replay for continual regression without storing raw data. Evidence obtained from several benchmark datasets shows that our framework reduces forgetting and provides more stable performance than other state-of-the-art solutions.

Compared to theoretical frameworks that assume equal sensitivity to deviations in all features of data, the theory of anomaly detection allowing for variable sensitivity across features is less developed. To the best of our knowledge, this issue has not yet been addressed in the context of isolation-based methods, and this paper represents the first attempt to do so. This paper introduces an Extended Isolation Forest with feature sensitivities, which we refer to as the Anisotropic Isolation Forest (AIF). In contrast to the standard EIF, the AIF enables anomaly detection with controllable sensitivity to deviations in different features or directions in the feature space. The paper also introduces novel measures of directional sensitivity, which allow quantification of AIF's sensitivity in different directions in the feature space. These measures enable adjustment of the AIF's sensitivity to task-specific requirements. We demonstrate the performance of the algorithm by applying it to synthetic and real-world datasets. The results show that the AIF enables anomaly detection that focuses on directions in the feature space where deviations from typical behavior are more important.

Diffusion models do not recover semantic structure uniformly over time. Instead, samples transition from semantic ambiguity to class commitment within a narrow regime. Recent theoretical work attributes this transition to dynamical instabilities along class-separating directions, but practical methods to detect and exploit these windows in trained models are still limited. We show that tracking the class-conditional entropy of a latent semantic variable given the noisy state provides a reliable signature of these transition regimes. By restricting the entropy to semantic partitions, the entropy can furthermore resolve semantic decisions at different levels of abstraction. We analyze this behavior in high-dimensional Gaussian mixture models and show that the entropy rate concentrates on the same logarithmic time scale as the speciation symmetry-breaking instability previously identified in variance-preserving diffusion. We validate our method on EDM2-XS and Stable Diffusion 1.5, where class-conditional entropy consistently isolates the noise regimes critical for semantic structure formation. Finally, we use our framework to quantify how guidance redistributes semantic information over time. Together, these results connect information-theoretic and statistical physics perspectives on diffusion and provide a principled basis for time-localized control.

That Dirichlet processes are discrete with probability 1 is demonstrated once more. And yes, these two pages spent fifty years in Norwegian.

Denoising diffusion models (DDMs) are state-of-the-art methods for learning densities from data across numerous domains, yet many aspects of the training and sampling pipeline remain poorly understood. In particular, noise conditioning requires practitioners to incorporate contrived unprincipled noise embeddings into neural network architectures and to use ad hoc noise schedules for sampling. To address these drawbacks, we provide a complete theory for blind denoising diffusion models (BDDMs): a variant of DDMs where the noise amplitude is not passed into the neural network during training or sampling, obviating the need for the aforementioned design choices. We justify the correctness of BDDMs as a sampling algorithm under an assumption of low intrinsic dimensionality of the underlying data distribution relative to the ambient dimension. This assumption arises through the introduction of the Bayesian problem of estimating noise levels from a single noisy sample, which might be of independent interest. We empirically compare the performance of BDDMs to standard DDMs, showcasing the benefits of an adaptive scheme which is rigorously justified by our analysis.

This paper focuses on the affective component of a driver behavioural model (DBM). This component specifically models some drivers' mental states such as mental load and active fatigue, which may affect driving performance. We have used Bayesian networks (BNs) to explore the dependencies between various relevant random variables and assess the probability that a driver is in a particular mental state based on their physiological and demographic conditions. Through this approach, our goal is to improve our understanding of driver behaviour in dynamic environments, with potential applications in traffic safety and autonomous vehicle technologies.

We study discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics. These models are widely studied in event history analysis, and are characterized by the state space, the initial distribution and the transition probabilities. A finite path under the multistate Markov model is a particular set of states occupied at finite time instances $\{1, \dots, n\}$. The main goal of this paper is to establish a bridge between event history analysis and algebraic statistics. The joint probabilities of finite paths in these models have a natural monomial parametrization in terms of the initial distribution and the transition probabilities. We study the polynomial relations among joint path probabilities. When the statistical constraints on the parameters are disregarded, nonhomogeneous multistate Markov models of arbitrary order can be viewed as slices of decomposable hierarchical models. This yields a complete description of their vanishing ideals as toric ideals generated by explicit families of binomials. Moreover, the variety of this vanishing ideal equals the nonhomogeneous multistate Markov model on the probability simplex. In contrast, homogeneous multistate Markov models exhibit different algebraic behavior, as time homogeneity imposes additional polynomial relations, leading to vanishing ideals that are strictly larger than in the nonhomogeneous case. We also derive families of binomial relations that vanish on homogeneous multistate Markov models. We investigate maximum likelihood estimation from statistical and algebraic perspectives. For nonhomogeneous models, classical and algebraic formulas agree; in the homogeneous case, the algebraic approach is more complex. Lastly, we provide data applications where we demonstrate the statistical theory to obtain the maximum likelihood estimates of the parameters under specific multistate Markov models.