Papers

Conditional diffusion models have become a powerful and flexible framework for learning complex conditional distributions from labeled data. In practice, however, acquiring high-quality labels is costly and time-consuming, leaving large volumes of unlabeled data unused. To address this, we introduce label-augmented conditional diffusion (LACD), a simple and effective approach that incorporates unlabeled examples by assigning them a designated trivial label and performing joint denoising score matching over the augmented dataset. We provide sufficient conditions guaranteeing population-level identifiability of the target conditional distribution under this scheme. Moreover, we establish rigorous statistical guarantees: when sufficiently many unlabeled samples are available, the sampling distribution produced by LACD converges strictly faster than the purely supervised estimator in total variation distance, and at least as fast in Wasserstein-1 distance. Extensive experiments on synthetic, image, and tabular benchmarks corroborate our theory and show substantial gains in sample efficiency and generative performance compared with the purely supervised estimator.

How deep does a graph neural network need to be on a sparse graph? We study its purest statistical form: node classification on the sparse contextual stochastic block model (CSBM) with average degree $\Delta=O(1)$, whose local weak limit is a broadcast-labelled Poisson Galton-Watson tree. Prior work derived a message-passing classifier $h_\ell$ that aggregates from each vertex at distance $k\le\ell$ the attenuated evidence $2\operatorname{artanh}(\gamma^k t(X_v))$, with $\gamma$ the edge signal and $t$ a bounded likelihood-ratio transform of the feature. We prove that the value of depth is governed by a single number, the Kesten-Stigum ratio $\kappa=\gamma^2\Delta$. Below the threshold ($\kappa<1$), the error sequence is Cauchy at a geometric rate, $|\mathcal{E}(\ell)-\mathcal{E}(\ell')|\le C\kappa^{(\ell+1)/3}$ for all $\ell'>\ell$, so all layers beyond depth $O(\log(1/\epsilon))$ change the error by less than $\epsilon$; conversely, under mild regularity each sufficiently deep layer still flips the decision with probability at least $c\kappa^{\ell/2}$, the empirically sharp exponent. Above the threshold ($\kappa>1$), depth is geometrically productive: $\mathcal{E}(\ell)$ is driven to a branching-process floor of order at most $1/(\kappa-1)$ at any geometric rate $\kappa^{-s\ell}$, $s<1$ (this bound has content only for $\kappa>17$). No local classifier of any depth beats the universal floor $e^{-\Delta}\Phi(-\zeta)$ set by isolated roots ($\zeta$ the feature signal-to-noise ratio), while the first layer provably helps by an explicit total-variation amount. Simulations with an exact belief-propagation baseline on the same trees show that the pairwise rule's error curve is mildly non-monotone in $\ell$, so an optimal finite depth exists (an exact instance is certified in the appendix), while BP saturates strictly faster, at an effective per-layer ratio below $\kappa$ that we identify.

A point prediction that is well calibrated on average can still be systematically biased conditional on its own value, undermining its use in downstream decision-making. We consider two objectives for reliable uncertainty quantification: self-calibration, requiring a point prediction to be unbiased conditional on its own value, and prediction-conditional validity, requiring a prediction interval to attain nominal coverage conditional on the prediction. Self-Calibrating Conformal Prediction (SC-CP) attains both objectives exactly in finite samples, but requires refitting its calibrator for every candidate outcome, which is computationally prohibitive for continuous outcomes. We propose Isotonic Conformal Prediction (ICP), a framework that decouples calibration from prediction-set construction by fitting a single isotonic recalibration map and constructing prediction intervals within strata of similar recalibrated predictions. Within this framework we develop two procedures. Split Isotonic Conformal Prediction (SICP) attains prediction-conditional validity in finite samples and self-calibration asymptotically, at the computational cost of split conformal prediction. Transductive Isotonic Conformal Prediction (TICP) attains both objectives exactly in finite samples through a per-test-point inner loop that avoids refitting the isotonic calibrator. On synthetic heteroscedastic regression problems and a real-world healthcare-utilization dataset, both procedures match the coverage of SC-CP at substantially lower computational cost.

Efron's bootstrap is the default tool for estimating the sampling distribution of a statistic, yet it is provably inconsistent for maxima of bounded-support distributions, means under infinite variance, extreme quantiles, and tail-index estimators. The classical remedies, the m-out-of-n bootstrap and subsampling, require rate corrections that depend on unknown parameters and behave erratically at realistic sample sizes. We propose an amortized alternative: a neural network is trained on simulated datasets drawn from a prior over a distribution family, using single independent draws of the root T_n - T(F) scored by the pinball loss, a proper scoring rule whose population minimizer is the posterior-predictive law of the root. At test time, a single forward pass maps one dataset of n = 200 observations to its full sampling-distribution estimate, from which confidence intervals follow directly. On four canonical bootstrap-failure problems (bounded-support maximum, alpha-stable mean, Pareto tail index, and 99% value-at-risk under tempered stable returns), the method attains nominal 95% coverage, beats every feasible classical method in Wasserstein distance to the true sampling distribution, and captures over 97% of the achievable improvement where the exact Bayes-optimal answer is computable. For the value-at-risk problem no distribution-free method can reach nominal coverage at all; the learned method attains 94.7%. A single universal network with a statistic token matches all four specialists, and on real daily market returns the unchanged model averages 0.87 coverage against 0.73 for the bootstrap, as predicted by our out-of-family analysis.

Principal component regression (PCR) regularizes high-dimensional prediction by choosing a spectral cutoff, but rank selection cannot correct systematic inflation of the retained empirical eigenvalues. We study clean Gaussian random designs in which the aggregate covariance tail creates a nearly scalar sample-space floor comparable to the predictive head scale. De-floored principal component regression (dPCR) retains the cutoff and subtracts an estimated floor from the retained denominators. We prove an ordinary-PCR prediction-risk lower bound uniform over all ranks and a high-probability dPCR upper bound. When the floor is sharp and inexpensive to remove in population prediction risk, the conditional risk of dPCR is asymptotically negligible relative to that of the best ordinary PCR rank. An exact risk decomposition explains the separation: denominator inflation is governed by first spectral mass, whereas the clean prediction cost of correction is governed by squared spectral mass. A same-sample trimmed-mean floor estimate attains the oracle dPCR upper-bound rate at a prespecified rank, and the separation persists under approximate predictive alignment when the tail prediction-energy fraction vanishes. Separate pointwise fixed-aspect formulas show that the risk-optimal positive scalar correction improves rank-$1$ PCR, whereas mean-floor subtraction is generally not optimal for a broad Marchenko–Pastur bulk.

Backpropagation makes training deep networks memory intensive because it must store intermediate activations. Forward-mode methods avoid this cost, but their gradient estimates become increasingly noisy as the number of trained parameters grows. We introduce Split Forward Gradient (Split-FG), which splits a network at an intermediate representation: it computes the output head gradient exactly and estimates only the trunk gradient with a Jacobian–vector product. This reduces estimator variance and requires no backward pass through the trunk, while retaining an Adam-style convergence guarantee. Our experiments reveal an important practical failure mode. On WikiText-103, naive forward-gradient training of the trunk performs worse than leaving a randomly initialized trunk frozen, likely because Adam updates every noisy, under-determined trunk coordinate too aggressively. Simply using a much smaller learning rate for the trunk reverses this result: a $16$M-parameter GPT-2-style model reaches validation perplexity $387$, compared with $668$ for the frozen-trunk control and $2{,}885$ for a matched pure forward-gradient baseline (backpropagation reaches $150$). Split-FG also produces the strongest backprop-free results on our tabular benchmarks and reaches $60.5\%$ on CIFAR-10 and $35.2\%$ on CIFAR-100 with a heavy-head design. It reduces peak memory by up to $35\%$ relative to matched backpropagation, although the performance gap widens as the forward-mode trunk grows.

Partial-mastery (PM) cognitive diagnostic models (CDMs) extend traditional CDMs by replacing binary latent attribute mastery indicators with continuous mastery scores for multiple latent attributes. In PM-CDMs, each subject is characterized by a fixed continuous latent mastery vector, from which item-specific binary attribute profiles are independently generated. This formulation provides a bridge between classical CDMs and continuous latent variable models. Despite growing interest in PM-CDMs, their identifiability properties remain unexplored. In this work, we establish the first identifiability results for PM-CDMs. We derive sufficient conditions for identifiability that are direct analogues of established conditions for traditional CDMs. To develop the main argument, we use symbolic computation on a minimal example with five items and two latent attributes to show that the Jacobian of the model parameterization is generically nonzero. Combining tools from real analysis and algebraic statistics, we prove that this local property implies generic finite-to-one identifiability of the item parameters and the marginal distributions of the relevant latent attributes. We further show that if the $Q$-matrix contains such identifiable local structures for all attribute pairs, identifiability extends to the full PM-CDM. These findings provide a rigorous theoretical foundation for estimation and inference in partial-mastery cognitive diagnostic models.

Structural reliability analysis supports the design and safety assessment of buildings and civil infrastructure but requires specialized expertise throughout the workflow. This study presents a multi-agent large language model framework that automates component-level reliability analysis from a natural-language problem statement to interpreted estimates of the reliability index and failure probability. Specialized agents handle problem formulation, method planning, code generation, execution, and result interpretation, with human confirmation at key decision points. The Method Planner is fine-tuned using QLoRA for a priori reliability-method category selection. Analysis results are not generated directly by an LLM; instead, validated deterministic solvers compute the reliability estimates, improving reproducibility and reducing hallucination risk. The framework uses open-weight models and supports local execution without closed APIs. Results show that it lowers the expertise barrier to structural reliability assessment while preserving computational trustworthiness.

We introduce a novel approach that integrates compositional data analysis (CoDA) in the forecasting of hierarchical and grouped time series and apply it to improve mortality forecasts. By leveraging the inherent decomposition of mortality data by cause of death and sex, our framework addresses key challenges in mortality forecasting, ensuring internal coherence across different levels of aggregation while maintaining high predictive accuracy. Our empirical results on Italian cause-specific mortality data demonstrate that the proposed approach significantly improves the reliability of forecasts compared to the traditional CoDA, especially for total mortality forecasts. Consistent improvements in forecasting accuracy are also found for total deaths by cause and by sex.

Maximum-entropy reference distributions are usually constructed on the normalized probability simplex. This formulation is less natural for unnormalized statistical models, in which positive multiples represent the same shape, and it does not directly explain how a prescribed admissible region should determine the deformation parameter of a bounded-support reference distribution. We formulate maximum entropy on the projective space of nonnegative measures and establish three results of statistical relevance. First, a universality theorem shows that every admissible monotone transform of the same normalized power functional has exactly the same optimizer under linear moment constraints. The result unifies the maximum-entropy implications of Tsallis and Rényi entropies, Hölder composite scores, pseudo-spherical scores, Bregman–Hölder constructions, and related homogeneous divergences without asserting a new distribution family. Second, the common optimizer is characterized as a $q$-exponential density; under mean and covariance constraints it is a compactly supported $q$-Gaussian for positive deformation and a Student-type density for negative deformation. Third, a prescribed Mahalanobis acceptance region with squared radius $R^2>d+2$ uniquely determines the deformation parameter $\gamma_R=2/(R^2-d-2)$. The resulting affine-equivariant reference density is the unique projective maximum-entropy solution, and its support coincides with the specified ellipsoid without an additional support constraint. This provides a principled method for constructing bounded-support statistical reference distributions from robust location and scatter estimates or from externally specified admissible regions.

We consider data-driven bandwidth selection for a kernel spectral estimator at zero frequency based on a local-to-zero version of the cross validated log-likelihood (CVLL) criterion. The modified version is $\mbox{CVLL}_c$, based on a sum over Fourier frequencies from $1$ to $n^c$ with $0<c<1$, where $n$ is the sample size. We focus on the expectation of a key term in a Taylor series expansion for $\mbox{CVLL}_c$ and show that in the case $4/5 < c < 1$ it converges to the corresponding asymptotic mean squared error of the spectral estimator at zero frequency. This provides some justification for the use of the local CVLL criterion for Heteroskedasticity and Autocorrelation Consistent (HAC) standard error estimation. Our theoretical results do not follow from existing literature on CVLL because those results exploit the fact that CVLL is global, summing over all frequencies in $(0,\pi)$ rather than local-to-zero frequency, as is the case for $\mbox{CVLL}_c$.

As generative AI is increasingly applied to automate multi-step and high-stake workflows, human judgment and involvement remain essential for ensuring the quality of AI-generated outputs. In practice, while it is desirable for human experts to provide oversight on AI regularly, often by reviewing intermediate outputs, giving feedback, making corrections, and steering subsequent steps, such oversight is constrained by the time and resources that humans can afford. This creates a tension between the need for human oversight and AI's efficiency in delivering more output with less intervention. An important but underexplored question, then, is how to optimally engage humans in human-AI coworking. This work was originally motivated by our empirical observation that in long AI workflows, human oversight often improves user satisfaction while reducing unnecessary rework and token consumption. From there, we formulate the problem of where to place oversight stages in human-AI coworking. Under reasonable assumptions, we then develop the nonuniformity principle, which states that the optimal schedule places oversight stages with non-decreasing gaps along the workflow. We empirically validate this principle in two common AI agent workflows: writing literature reviews and constructing websites.

Unit-variance standardisation is often applied routinely before principal component analysis (PCA), although it replaces covariance geometry by correlation geometry. A resulting biplot may be computed correctly yet fail to support a scientific interpretation about the original-scale phenomenon. We therefore make the scientific statement, rather than the decomposition alone, the unit of methodological assessment. A PCA-biplot claim is representationally well posed only when the declared scientific target justifies the operator analysed, the invoked axis or invariant subspace is identifiable at the stated level, and the displayed projection preserves the prespecified relationships within a substantively justified tolerance; failure of any condition makes the claim ill posed for the stated interpretation. A projector formulation yields basis-invariant diagnoses for repeated-eigenvalue blocks, and a residual-Gram bound quantifies pairwise error from omitted coordinates. Six controlled population scenarios provide exact representational truth, including zero target association with arbitrary projected angles, collapse of a full-space 60-degree relationship to 0 degrees, and non-identifiable named axes. The contribution is not another reminder that scaling matters: it establishes that computational correctness is necessary but not sufficient for scientific interpretability and provides a formal procedure for retaining, reformulating, qualifying, or rejecting a PCA-biplot statement. A real-data illustration and bootstrap extension are provided as supplementary material.

Classical measurement theory is traditionally formulated in an algebraic framework. However, it is fundamentally commutative and does not naturally represent noncommutative measurement phenomena identified in the social sciences. This study investigates whether the transformation structure of classical measurement theory preserves a canonical noncommutative geometry. A measurement state vector is introduced, and transformations relating various forms of equivalence (parallelism) between measures are expressed as a Lie matrix group. A faithful matrix representation of the Heisenberg group is introduced, and the conjugation of Heisenberg elements by a general measurement transformation is derived. Results show this conjugation defines an automorphism of the Heisenberg group, preserving its commutator structure. However, if the elements of the measurement state vector are equally scaled the Heisenberg geometry is preserved exactly. The findings establish a symmetry linking classical measurement transformations with the Heisenberg group providing a mathematical foundation for extending classical measurement theory to phenomena exhibiting noncommutative structures.

Individualized treatment rules (ITRs) formalize precision medicine by assigning treatments according to patient covariates, with the goal of maximizing expected clinical outcomes. Such rules are especially important when treatment effects vary across patients, as in chronic diseases where demographic, clinical, genetic, imaging, or biomarker information may modify the relative benefits of available therapies. Individualized treatment rules (ITRs) formalize precision medicine by assigning treatments according to patient covariates, with the goal of maximizing expected clinical outcomes. Such rules are especially important when treatment effects vary across patients, as in chronic diseases where demographic, clinical, genetic, imaging, or biomarker information may modify the relative benefits of available therapies. Outcome weighted learning (OWL) estimates ITRs by recasting treatment assignment as a weighted classification problem that directly targets clinical value. Motivated by the flexibility of modern neural networks, we extend single hidden-layer neural-network OWL (NNOWL) from ridge-type regularization to nonlinear variable selection and kernel-based approximation. We establish non-asymptotic convergence rates for these estimators, and study the global convergence and implicit bias of gradient descent for NNOWL. Finally, we extend the neural-network methods from OWL to residual weighted learning. Simulation studies illustrate the roles of over-parameterization, kernel approximation, and nonlinear variable selection, and a data application in Alzheimer's disease demonstrates the proposed methods.

For stochastic gradient descent (SGD) with a constant stepsize $\alpha$, the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons. In the strongly convex case, this invariant law has the familiar $\sqrt{\alpha}$ scaling and a Gaussian limit as $\alpha\downarrow 0$. We show that this behavior changes fundamentally for convex objectives $H$ with flat minima and (sub)quadratic tails. More specifically, we study SGD with Markovian noise generated by a contractive driving chain. For every sufficiently small constant stepsize $\alpha$, we prove existence, uniqueness, and geometric convergence to an augmented invariant law in a Wasserstein distance induced by an $\alpha$-dependent metric. When the minimizer $x_\star$ has local flatness exponent $m\ge2$, meaning that $\nabla^2 H(x)\asymp \lVert x-x_\star\rVert^{m-2} I_d$ as $x\to x_\star$, we obtain a contraction bound with factor $1-c\alpha^{m-1}$, where $c>0$ is a constant. This recovers the factor $1-c\alpha$ in the quadratic case $m=2$. We then analyze the small-stepsize scaling limit. We show that the invariant law concentrates on the scale $\alpha^{1/m}$ and that the rescaled iterates converge weakly to the stationary distribution of the stochastic differential equation $$ dY_t=-h_0(Y_t)\,dt+\Sigma^{1/2}\,dB_t , $$ where $h_0$ is the limiting drift at the minimizer and $\Sigma$ denotes the asymptotic covariance. This recovers the Gaussian limit when $m=2$ and gives generally non-Gaussian stationary limits in the flat case $m>2$. Finally, we give corresponding results for coordinate-separable objectives with unequal flatness exponents.

Empirical studies of racial discrimination vary race while holding nonracial traits fixed, a design the literature defends as what credible inference requires. This defense bundles two claims: which effect of race a study should target, and whether its design can recover that effect. I separate them. The same randomization that secures credible estimation and inference recovers a family of race estimands, from the all-else-equal effect to a within-race effect that lets associated traits vary with race. Every member of that family is causal rather than descriptive, and the choice among members is a claim about what a racial category is – a claim extending to ethnicity, religion, and other identity categories that index associated traits. I derive conditions, weaker than the literature's, for credible estimation and inference. Reanalyzing a Spanish-language campaign experiment, I find coethnic preference among Hispanic voters under the within-race effect and none under the all-else-equal effect.

A large body of Semi-supervised Learning~(SSL) algorithms encounter the threshold $\tau$ to select pseudo-labels. The value of $\tau$ across different SSL algorithms can vary depending on the learning perspective, yet they may achieve similar performance. It motivates us to establish a unified theoretical framework to explain the role of $\tau$ in SSL. We statistically explained that the unsupervised loss is affected independently by correct and incorrect pseudo-labels, while $\tau$ adjusts their numbers to balance the corresponding error term. This inherent trade-off indicates that SSL can reach the same loss with varying $\tau$, precise optimal values of $\tau$ during training may be unnecessary. With this, we treat $\tau$ as an updatable parameter and optimize it via differentiation; the new policy is named Meta-Thresholding Semi-Supervised Learning (MTSSL). Extensive experiments demonstrate the superior performance of MTSSL. We observe that the accuracy curves of SSL algorithms can overlap completely even when the values of $\tau$ differ significantly, which supports our theoretical framework and indicates that the selection of $\tau$ can be relaxed in the future design of SSL algorithms.

In many applications of matching, the point clouds to be matched are not merely unstructured sets of points but rather samples from distributions with an intrinsic cluster structure. In such cases, as individual points are often interchangeable within a coherent region, finding a robust region-to-region alignment is more desirable than establishing a precise point-to-point correspondence. To this end, we propose a novel approach for cluster-aware matching based on Laplacian Optimal Transport (LapOT). The key idea is to regularize the optimal transport problem with quadratic Laplacian terms constructed from similarity graphs of the point clouds, which encourages the optimal coupling to respect the cluster structure of both point sets. We also introduce Refined Simultaneous Clustering (RSC), a method that leverages the cluster-aware coupling obtained from LapOT to produce consistent partitions across the point sets, which can overcome the limitations of independent clustering and yield more stable and interpretable results. We demonstrate the effectiveness of our approach through theoretical analysis and empirical experiments, showing that LapOT indeed produces cluster-aware matching that leads to more consistent and meaningful alignments between point clouds.

The modern manufacturing industry is investing in new technologies such as the Internet of Things (IoT), big data analytics, cloud computing and cybersecurity to cope with system complexity, increase information visibility, improve production performance, and gain competitive advantages in the global market. These advances are rapidly enabling a new generation of smart manufacturing, i.e., a cyber-physical system tightly integrating manufacturing enterprises in the physical world with virtual enterprises in cyberspace. To a great extent, realizing the full potential of cyber-physical systems depends on the development of new methodologies on the Internet of Manufacturing Things (IoMT) for data-enabled engineering innovations. This paper presents a review of the IoT technologies and systems that are the drivers and foundations of data-driven innovations in smart manufacturing. We discuss the evolution of internet from computer networks to human networks to the latest era of smart and connected networks of manufacturing things (e.g., materials, sensors, equipment, people, products, and supply chain). In addition, we present a new framework that leverages IoMT and cloud computing to develop a virtual machine network. We further extend our review to IoMT cybersecurity issues that are of paramount importance to businesses and operations, as well as IoT and smart manufacturing policies that are laid out by governments around the world for the future of smart factory. Finally, we present the challenges and opportunities arising from IoMT. We hope this work will help catalyze more in-depth investigations and multi-disciplinary research efforts to advance IoMT technologies.

Changepoint detection methods have seen considerable development in recent years, with online algorithms capable of identifying structural changes in streaming data in near real time. However, the majority of existing methods are designed under the assumption of IID observations, rendering them susceptible to either more false positives or longer detection delays when applied to data exhibiting temporal dependence, a common feature of many real-world data streams. In this article, we extend the generalised likelihood-ratio (GLR) statistic to autoregressive processes of order $p$, and adapt the focus algorithm to develop a computationally efficient online change detector. The resulting AR($p$)-focus algorithm achieves an average computational cost of $\mathcal{O}(\log n)$ per iteration, making it suitable for high-frequency data streams. Through simulation studies, the proposed approach is seen achieving greater detection power than IID-based tests when the underlying data exhibit temporal correlation. We further illustrate the practical utility of AR($p$)-focus through an application to a real-world telecommunications dataset.

We provide a proof that the empirical spatial distribution estimator in $\mathbb R^d$ as well as the corresponding plug-in estimator of the spatial depth are uniformly $L^1$-consistent. The consistency rate only depends on the sample size $n$, not on the dimension $d$ or any tuning or regularization parameters. This is a rare property. The result of this note originates from a conversation with ChatGPT 5.4 Pro as part of some of our own earlier experiments on its mathematical reasoning capabilities.

Mendelian randomization (MR) infers causal effects of exposures on outcomes using genetic variants as instrumental variables, but existing methods remain vulnerable to bias and loss of power when instruments are invalid. We hypothesize that closely related outcome traits share a large overlap in their valid instruments for a given exposure, so that borrowing instruments across such traits can yield more robust MR inference. To operationalize this, we introduce a novel coheterogeneity statistic, together with its asymptotic theory under different paradigms that allows for a growing number of instruments as the sample size increases, to identify secondary outcome traits likely to share valid instruments with the primary trait. We then propose extensions of two popular methods, MR-Mode and MR-PRESSO, that improve inference for the primary trait by exploiting a secondary trait. Extensive simulations show that the coheterogeneity statistic has the expected finite-sample properties and that instrument-borrowing methods consistently outperform their standard counterparts, with gains increasing as the overlap in valid instruments grows. Applications to established positive and negative control hypotheses indicate improved control of type I error and increased power. We further illustrate practical utility by revisiting the long-debated hypothesis of the causal effect of vitamin D on cardiometabolic traits.

Hawkes processes (HP) are a large class of stochastic point process models scientists have used to analyze contagion phenomena ranging from earthquakes, infectious diseases and biological neurons to financial trading activity, memes on social media and gun violence. We introduce applications of HP to the latter before reviewing general strategies for fitting HP to data, paying attention to the influence of model structure on computational scalability considerations. We then apply a recently developed high-performance computing powered Bayesian inference strategy for the spatiotemporal HP analysis of 412,376 acts of gun violence in the U.S. between 2014 and 2024. We finish with a discussion of model fit and directions for future research.

Gene regulatory networks (GRNs) link transcription factor (TF) proteins to their target genes, yet reconstructing these networks from genome-wide data remains challenging under practical and methodological constraints. Many methods couple modeling assumptions to a specific inference procedure and rely on heuristic model selection, while evaluation is constrained by incomplete reference networks and point-estimate outputs that lack uncertainty. GRN reconstruction also depends on prior knowledge to constrain TF-gene interactions, yet available priors are often assay-dependent and difficult to transfer across species and less-characterized systems. In this thesis, we develop two complementary frameworks that address these limitations. In the first, PMF-GRN casts GRN inference as a probabilistic graphical model optimized by variational inference, enabling principled model selection and uncertainty-aware edge estimates. In the second, GLM-Prior addresses the prior bottleneck by fine-tuning the pretrained Nucleotide Transformer to predict TF-target gene interactions directly from nucleotide sequence, while generalizing across yeast, mouse, and human settings. Together, PMF-GRN and GLM-Prior motivate a dual-stage view of GRN reconstruction in which sequence-derived priors provide a transferable starting scaffold and probabilistic inference refines regulatory estimates with quantified uncertainty under incomplete evaluation resources.

A foundational principle in nonlinear dynamics is that the structure of a dynamical system can be recovered from a small number of generic measurements or coordinates. We develop an analogous principle for the identification of dynamic models for time series with noise, which builds on previous identification results for noiseless dynamical systems. The noise is allowed to be non-iid, non-Gaussian, and dependent on the state. Our results cover noisily observed differential equations and discrete-time dynamical systems, as well as stochastic models with process noise. We illustrate the utility of this identification principle using a Lorenz-63 model and a Hénon map model, both with observational noise.

A new approach for nonparametric estimation of the quantile density function (sparsity function) and its derivatives is suggested which is based on local polynomial estimation. The estimator has more advantageous properties at the boundaries than classical quantile density estimators. Asymptotic normality is shown and the bias, asymptotic variance as well as boundary properties are compared with other estimators.

This work develops a statistical test for the maximal rank of the deterministic instantaneous (or spot) covariance matrix of a continuous-time $\mathbb{R}^d$-valued Itô semi-martingale $X(t)$ using high-frequency observations with a particular focus on the impact of an adapted drift. We explicitly account for the presence of an adapted drift process, which, as our results demonstrate, cannot be neglected, by introducing a re-centred covariance estimator instead of relying solely on a second moment estimator. Building on this estimator, we test the null hypothesis that the rank of the spot covariance matrix is at most $r<d$ for all $t$ against local alternatives in which the $(r+1)$th eigenvalue is greater than some vanishing signal detection rate. Critical values are derived in a non-asymptotic framework and can be significantly affected by a potential drift. However, the power analysis establishes asymptotic consistency for separation rates, which depend on the Hölder regularity of both the drift and the spot covariance matrix, as well as on a potential spectral gap $\underline{\lambda}_r \geq 0$ under the null hypothesis. Simulation results indicate that the covariance-based test achieves higher power across a wider range of alternatives compared to classical second moment-based procedures.

This work presents a game-theoretic framework for interpretable hyperparameter-objective interaction analysis rather than proposing a new optimization algorithm. In the proposed framework, Shapley Effects are employed for global sensitivity analysis, while Pareto front sets are utilized to identify effective hyperparameter configurations and support early-stage model evaluation. The resulting analysis reveals which players (hyperparameters) are most influential with respect to different objectives in a given game (application). Consequently, the proposed framework provides interpretable insights into objective-aware hyperparameter interactions, enabling practitioners to guide subsequent optimization, reduce the search space, and perform early-stage model evaluation. The effectiveness of the proposed framework is demonstrated using three distinct neural network architectures across different problem domains under multi-objective settings.

We study aggregation of statistical evidence under unknown and potentially complex dependence using group-invariance. Building on permutation-based constructions that treat transformed datasets as exchangeable units, we aggregate evidence across statistics for each transformed dataset and calibrate the resulting aggregates across transformations. We develop a finite-sample power and adaptivity theory for this framework, together with extensions to sequential and data-dependent aggregation that preserve validity. For single-batch aggregation, which uses one collection of transformed datasets for both standardization and calibration, we show that the critical values uniformly improve on deterministic calibrations valid under arbitrary dependence, including Bonferroni correction, while adapting to the unknown dependence structure. We also introduce a sequential alpha-spending version that permits early rejection when evidence is strong, and a two-batch extension that separates standardization from calibration to accommodate learned aggregation rules and reduce computation. Applications to adaptive nonparametric testing and conformal prediction illustrate how these results sharpen existing aggregation methods.