Papers

The efficient simulation of Gaussian random fields with flexible correlation structures is fundamental in spatial statistics, machine learning, and uncertainty quantification. In this work, we revisit the spectral turning-bands (STB) method as a versatile and scalable framework for simulating isotropic Gaussian random fields with a broad range of covariance models. Beyond the classical Matérn family, we show that the STB approach can be extended to two recent and flexible correlation classes that generalize the Matérn model: the Bummer-Tricomi model, which allows for polynomially decaying correlations and long-range dependence, and the Gauss-Hypergeometric model, which admits compactly supported correlations, including the Generalized Wendland family as a special case. We derive exact stochastic representations for both families: a Beta-prime mixture formulation for the Kummer-Tricomi model and complementary Beta- and Gasper-mixture representations for the Gauss-Hypergeometric model. These formulations enable exact, numerically stable, and computationally efficient simulation with linear complexity in the number of spectral components. Numerical experiments confirm the accuracy and computational stability of the proposed algorithms across a wide range of parameter configurations, demonstrating their practical viability for large-scale spatial modeling. As an application, we use the proposed STB simulators to perform parametric bootstrap for standard error estimation and model selection under weighted pairwise composite likelihood in the analysis of a large climate dataset.

The progression-free survival ratio (PFSr) is a widely used measure in personalized oncology trials. It evaluates the effectiveness of treatment by comparing two consecutive event times - one under standard therapy and one under an experimental treatment. However, most proposed tests based on the PFSr cannot control the nominal type I error rate, even under mild assumptions such as random right-censoring. Consequently the results of these tests are often unreliable. As a remedy, we propose to estimate the relevant probabilities related to the PFSr by adapting recently developed methodology for the relative treatment effect between paired event times. As an additional alternative, we develop inference procedures based on differences and ratios of restricted mean survival times. An extensive simulation study confirms that the proposed novel methodology provides reliable inference, whereas previously proposed techniques break down in many realistic settings. The utility of our methods is further illustrated through an analysis of real data from a molecularly aided tumor trial.

Rating systems play a crucial role in evaluating player skill across competitive environments. The Elo rating system, originally designed for deterministic and information-complete games such as chess, has been widely adopted and modified in various domains. However, the traditional Elo rating system only considers game outcomes for rating calculation and assumes uniform initial states across players. This raises important methodological challenges in skill modelling for popular partially randomized incomplete-information games such as Rummy. In this paper, we examine the limitations of conventional Elo ratings when applied to luck-driven environments and propose a modified Elo framework specifically tailored for Rummy. Our approach incorporates score-based performance metrics and explicitly models the influence of initial hand quality to disentangle skill from luck. Through extensive simulations involving 270,000 games across six strategies of varying sophistication, we demonstrate that our proposed system achieves stable convergence, superior discriminative power, and enhanced predictive accuracy compared to traditional Elo formulations. The framework maintains computational simplicity while effectively capturing the interplay of skill, strategy, and randomness, with broad applicability to other stochastic competitive environments.

The adapted Wasserstein distance $\mathcal{AW}$ is a modification of the classical Wasserstein metric, that provides robust and dynamically consistent comparisons of laws of stochastic processes, and has proved particularly useful in the analysis of stochastic control problems, model uncertainty, and mathematical finance. In applications, the law of a stochastic process $\mu$ is not directly observed, and has to be inferred from a finite number of samples. As the empirical measure is not $\mathcal{AW}$-consistent, Backhoff, Bartl, Beiglböck and Wiesel introduced the adapted empirical measure $\widehat{\mu}^N$, a suitable modification, and proved its $\mathcal{AW}$-consistency when observations are i.i.d. In this paper we study $\mathcal{AW}$-convergence of the adapted empirical measure $\widehat{\mu}^N$ to the population distribution $\mu$, for observations satisfying a generalization of the $\eta$-mixing condition introduced by Kontorovich and Ramanan. We establish moment bounds and sub-exponential concentration inequalities for $\mathcal{AW}(\mu,\widehat{\mu}^N)$, and prove consistency of $\widehat{\mu}^N$. In addition, we extend the Bounded Differences inequality of Kontorovich and Ramanan for $\eta$-mixing observations to uncountable spaces, a result that may be of independent interest. Numerical simulations illustrating our theory are also provided.

Clinical trials commonly report effects on the full study population as well as for patient subgroups. Random-effects meta-analyses of subgroup-specific effects and treatment-by-subgroup interactions might be misaligned, especially when subgroups contribute differently to study populations. We show that, after contribution adjustment, subgroup meta-analysis can recover the same interaction inference regardless of whether interaction or subgroup data are used. Our framework for subgroup-data interaction meta-analysis has four advantages. First, it adjusts for variation in relative subgroup contribution, quantified by the within-trial information fraction. Second, it is robust to prevalence imbalance and variation. Third, it provides a self-contained model-based approach. Fourth, it can incorporate prior information into Bayesian interaction meta-analysis. The method demonstration uses an example application including 7 trials of disease-modifying therapies in relapsing-remitting multiple sclerosis; the contribution-adjusted model manages to reproduce the results from the interaction meta-analysis while yielding matching estimates for the subgroup effects. By controlling subgroup contribution while retaining subgroup interpretability, this approach allows for interaction decision-making when published subgroup data are available. Although the proposed model is presented in the Bayesian context, it can also be implemented in frequentist or likelihood frameworks.

We construct a Gaussian random field (GRF) that combines fractional smoothness with spatially varying anisotropy. The GRF is defined through a stochastic partial differential equation (SPDE), where the range, marginal variance, and anisotropy vary spatially according to a spectral parametrization of the SPDE coefficients. Priors are constructed to reduce overfitting in this flexible covariance model, and we estimate parameters using a gradient-based optimization approach based on automatic differentiation. In a simulation study, we investigate how many observations are required to reliably estimate fractional smoothness and non-stationarity, and find that one realization containing 500 observations or more is needed in the scenario considered. We also find that the proposed penalization prevents overfitting across varying numbers of observation locations. Two case studies demonstrate that the relative importance of fractional smoothness and non-stationarity is application dependent. Non-stationarity improves predictions in an application to ocean salinity, whereas fractional smoothness improves predictions in an application to precipitation. Predictive ability is assessed using mean squared error and the continuous ranked probability score. In addition to prediction, the proposed approach can be used as a tool to explore the presence of fractional smoothness and non-stationarity.

Deficits in working memory, which includes both the ability to learn and to retain information short-term, are a hallmark of many cognitive disorders. Our study analyzes data from a neuroscience experiment on animal subjects, where performance on a working memory task was recorded as repeated binary success or failure data. We estimate continuous probability of success curves from this binary data in the context of functional data analysis, which is largely used in biological processes that are intrinsically continuous. We then register these curves to decompose each function into its amplitude, representing overall performance, and its phase, representing the speed of learning or response. Because we are able to separate speed from performance, we can address the crucial question of whether a cognitive disorder impacts not only how well subjects can learn and remember, but also how fast. This allows us to analyze the components jointly to uncover how speed and performance co-vary, and to compare them separately to pinpoint whether group differences stem from a deficit in peak performance or a change in speed.

Concept learning becomes possible only when existing representations fail to account for experience. Most models of learning and inference, however, presuppose a fixed representational basis within which belief updating occurs. In this paper, I address a prior question: under what structural conditions can the representational basis itself expand in a principled and selective way? I propose a geometric framework in which conceptual growth is modeled as admissible basis extension evaluated under a Minimum Description Length (MDL) criterion. Experience, whether externally observed or internally simulated, is represented as vectors relative to a current conceptual subspace. Residual components capture systematic representational failure, and candidate conceptual extensions are restricted to low-rank, admissible transformations. I show that any MDL-accepted extension can be chosen so that its novel directions lie entirely within the residual span induced by experience, while extensions orthogonal to this span strictly increase description length and are therefore rejected. This yields a conservative account of imagination and conceptual innovation. Internally generated counterfactual representations contribute to learning only insofar as they expose or amplify structured residual error, and cannot introduce arbitrary novelty. I further distinguish representational counterfactuals–counterfactuals over an agent's conceptual basis–from causal or value-level counterfactuals, and show how MDL provides a normative selection principle governing representational change. Overall, the framework characterizes conceptual development as an error-driven, geometry-constrained process of basis extension, clarifying both the role and the limits of imagination in learning and theory change.

Effective feature selection is essential for high-dimensional data analysis and machine learning. Unsupervised feature selection (UFS) aims to simultaneously cluster data and identify the most discriminative features. Most existing UFS methods linearly project features into a pseudo-label space for clustering, but they suffer from two critical limitations: (1) an oversimplified linear mapping that fails to capture complex feature relationships, and (2) an assumption of uniform cluster distributions, ignoring outliers prevalent in real-world data. To address these issues, we propose the Robust Autoencoder-based Unsupervised Feature Selection (RAEUFS) model, which leverages a deep autoencoder to learn nonlinear feature representations while inherently improving robustness to outliers. We further develop an efficient optimization algorithm for RAEUFS. Extensive experiments demonstrate that our method outperforms state-of-the-art UFS approaches in both clean and outlier-contaminated data settings.

We study the high-dimensional linear model with noise distribution known up to a scale parameter. With an $\ell_1$-penalty on the regression coefficients, we show that a transformation of the log-likelihood allows for a choice of the tuning parameter not depending on the scale parameter. This transformation is a generalization of the square root Lasso for quadratic loss. The tuning parameter can asymptotically be taken at the detection edge. We establish an oracle inequality, variable selection and asymptotic efficiency of the estimator of the scale parameter and the intercept. The examples include Subbotin distributions and the Gumbel distribution.

Transformers can implement both generalizable algorithms (e.g., induction heads) and simple positional shortcuts (e.g., memorizing fixed output positions). In this work, we study how the choice of pretraining data distribution steers a shallow transformer toward one behavior or the other. Focusing on a minimal trigger-output prediction task – copying the token immediately following a special trigger upon its second occurrence – we present a rigorous analysis of gradient-based training of a single-layer transformer. In both the infinite and finite sample regimes, we prove a transition in the learned mechanism: if input sequences exhibit sufficient diversity, measured by a low “max-sum” ratio of trigger-to-trigger distances, the trained model implements an induction head and generalizes to unseen contexts; by contrast, when this ratio is large, the model resorts to a positional shortcut and fails to generalize out-of-distribution (OOD). We also reveal a trade-off between the pretraining context length and OOD generalization, and derive the optimal pretraining distribution that minimizes computational cost per sample. Finally, we validate our theoretical predictions with controlled synthetic experiments, demonstrating that broadening context distributions robustly induces induction heads and enables OOD generalization. Our results shed light on the algorithmic biases of pretrained transformers and offer conceptual guidelines for data-driven control of their learned behaviors.

Uniform confidence bands are widely used to quantify uncertainty about unknown functions in nonparametric inference. Although various methods, including the multiplier bootstrap, have been developed and theoretically justified, practical implementations typically replace the supremum over a continuum by the maximum over a finite evaluation grid. This paper investigates how this discretization affects coverage accuracy and how fine the evaluation grid must be to ensure valid uniform inference. We first illustrate through simulations of a kernel-based nonparametric estimator that coarse grids can substantially reduce uniform coverage. Surprisingly, increasing the sample size can worsen coverage on a fixed coarse grid, even as the remaining approximation error on the grid decreases. We then derive general upper bounds on the coverage probability and coverage error of uniform confidence bands based on multiplier-bootstrap critical values on the finite grid points, separating discretization error from the remaining approximation error on the grid. For local empirical processes arising from kernel smoothing, we establish primitive conditions under which coverage converges to zero even when the number of grid points diverges, and conditions ensuring asymptotically nominal coverage. The latter results yield rate-sharp grid resolution requirements in terms of the bandwidth. Furthermore, we derive an explicit leading constant in the bound on coverage error and develop a quantitative grid selection rule for a user-specified discretization tolerance level, which can be implemented through a simple one-dimensional search.

In many networks each vertex has a position measured from outside the network: a neuron's location along the body axis, a residue's index along a protein sequence, a genomic bin's position in base pairs. The chance that two vertices connect is then a surface over pairs of positions, and questions about the network become questions about small regions of that surface. A region on the diagonal covers pairs inside one stretch of the axis, and a departure there is a community with a boundary. A region off the diagonal covers pairs spanning two separated stretches, and a departure there is a bridge. Existing methods address one part of this at a time: community detection returns groups without placing them on the axis, block models fix the width in advance, and scan statistics test one window at one scale. We expand the surface in a wavelet dictionary whose elements are exactly such regions, at every position and width. We derive in closed form the modularity, edge length, transitivity and degree spread each element produces, and the generating element is recovered from those summaries. A scan estimates every coefficient against a background fitted on separate edges and controls the error rate across all positions and widths at once. Applied to a connectome, a protein contact map and a chromatin contact map, it recovers known anatomy in the first and is calibrated in all three. When positions are estimated with error near the width sought, the location of a departure is not identified at any signal strength, and a check decides this before any analysis.

Analysts often fit several models to the same network. Each estimates a graphon, the function giving the probability of a link between two nodes. Reporting one predictive graphon means weighting these estimates, and the weights can be constrained in several ways. We determine which constraint is correct and when the combination improves on the best single model. We introduce Bayesian predictive synthesis at the graphon level. The models enter as agents, a prior is placed on the weights combining their link probabilities, and the posterior returns one predictive graphon with credible intervals. We study this rule when the network is a union of overlapping mechanisms, so a pair is linked if at least one mechanism links it. Weights forced to sum to one cannot reproduce a union. Free and nonnegative weights do far better and do equally well. A noisy-OR rule matches the union exactly. From one observed graph we give an exact variance formula for the fitted weights, and the usual credible intervals are too narrow. On polblogs, email-Eu-core, Cora, ca-GrQc, wiki-Vote and soc-Epinions1, combining adds under one percent once a flaw in the usual benchmark is corrected, and a larger single model often wins. On the Amazon and YouTube multiplex networks, where the layers are recorded separately, it beats every competitor we tuned, including spectral estimation, blockmodels, random forests and node2vec.

Crime counts in city neighbourhoods, disease counts in counties, and sales at firms joined by trade are naturally represented as counts on the nodes of a network. In each case a high count at one node can raise the counts at the nodes linked to it next period. The strength of that spillover changes over time, and standard network autoregressions hold it fixed. We therefore use a network state-space model, in which the spillover is a coefficient that drifts and a filter estimates its value in each period. What the data reveal about that coefficient depends on the network. It is learned by contrasting nodes whose neighbours have high counts with nodes whose neighbours have low ones. If every node is linked to every other, all nodes share the same neighbours, the contrasts vanish, and the spillover is not identified. Robustness is often checked by refitting with the links spread evenly, and a stable coefficient is read as reassurance. That refit is the same model with the spillover rescaled, so it cannot disagree. Forecasts carry a second warning: beyond two steps ahead, simulation averages a quantity with no finite mean, and the output gives no sign of it. We give a measure of what a network and a data set reveal about the spillover, the accuracy the filter can reach, and an exact test of whether the network matters. Burglaries in Chicago, COVID-19 cases in Texas counties, and measles cases in the Weser–Ems districts illustrate all three. On both disease datasets the model outperforms every competing forecast in the comparison.

We develop a new and powerful method to analyze time series to rigorously detect flares in the presence of an irregularly oscillatory baseline, and apply it to stellar light curves observed with TESS. First, we remove the underlying non-stochastic trend using a time-varying amplitude harmonic model. We then model the stochastic component of the light curves in a manner analogous to financial time series, as an ARMA+GARCH process, allowing us to detect and characterize impulsive flares as large deviations inconsistent with the correlation structure in the light curve. We apply the method to exemplar light curves from TIC13955147 (a G5V eruptive variable), TIC269797536 (an M4 high-proper motion star), and TIC441420236 (AU Mic, an active dMe flare star), detecting up to $145$, $460$, and $403$ flares respectively, at rates ranging from ${\approx}0.4$–$8.5$~day$^{-1}$ over different sectors and under different detection thresholds. We detect flares down to amplitudes of $0.03$%, $0.29$%, and $0.007$% of the bolometric luminosity for each star respectively. We model the distributions of flare energies and peak fluxes as power-laws, and find that the solar-like star exhibits values similar to that on the Sun ($\alpha_{E,P}\approx1.85,2.36$), while for the less- and highly-active low-mass stars $\alpha_{E,P}>2$ and $<2$ respectively.

Model reduction techniques have emerged as a powerful paradigm across different fronts of scientific computing. Despite their success, the provided tools and methodologies remain limited if high-dimensional dynamical systems subject to initial uncertainty and/or stochastic noise are encountered; in particular if rare events are of interest. We address this open challenge by borrowing ideas from Mori-Zwanzig formalism and Chorin's optimal prediction method. The novelty of our work lies on employing time-dependent optimal projection of the dynamic on a desired set of resolved variables. We show several theoretical and numerical properties of our model reduction approach. In particular, we show that the devised surrogate trajectories are consistent with the probability flow of the full-order system. Furthermore, we identify the measure underlying the projection through polynomial chaos expansion technique. This allows us to efficiently compute the projection even for trajectories that are initiated on low probability events. Moreover, we investigate the introduced model-reduction error of the surrogate trajectories on a standard setup, characterizing the convergence behaviour of the scheme. Several numerical results highlight the computational advantages of the proposed scheme in comparison to Monte-Carlo and optimal prediction method. Through this framework, we demonstrate that by tracking the measure along with the consistent projection of the dynamic we are able to access accurate estimates of different statistics including observables conditional on a given initial configuration.

Trust is the invisible glue that holds together the fabric of societies, economic systems, and political institutions. Yet, its dynamics-especially in real-world settings remain unpredictable and difficult to control. While classical trust game models largely rely on discrete frameworks with limited noise, they fall short in capturing sudden behavioral shifts, extreme volatility, or abrupt breakdowns in cooperation.Here, we propose-for the first time a comprehensive stochastic model of trust based on Lévy processes that integrates three fundamental components: Brownian motion (representing everyday fluctuations), Poissonian jump intensity (capturing the frequency of shocks), and random distributions for jump magnitudes. This framework surpasses conventional models by enabling simulations of phenomena such as "sudden trust collapse," "chaotic volatility," and "nonlinear recoveries" dynamics often neglected in both theoretical and empirical studies.By implementing four key simulation scenarios and conducting a detailed parameter sensitivity analysis via 3D and contour plots, we demonstrate that the proposed model is not only mathematically more advanced, but also offers a more realistic representation of human dynamics compared to previous approaches. Beyond its technical contributions, this study outlines a conceptual framework for understanding fragile, jump-driven behaviors in social, economic, and geopolitical systems-where trust is not merely a psychological construct, but an inherently unstable and stochastic variable best captured through Lévy based modeling.

Validation gating is a fundamental component of classical Kalman-based tracking systems. Only measurements whose normalized innovation squared (NIS) falls below a prescribed threshold are considered for state update. While this procedure is statistically motivated by the chi-square distribution, it implicitly replaces the unconditional innovation process with a conditionally observed one, restricted to the validation event. This paper shows that innovation statistics computed after gating converge to gate-conditioned rather than nominal quantities. Under classical linear–Gaussian assumptions, we derive exact expressions for the first- and second-order moments of the innovation conditioned on ellipsoidal gating, and show that gating induces a deterministic, dimension-dependent contraction of the innovation covariance. The analysis is extended to NN association, which is shown to act as an additional statistical selection operator. We prove that selecting the minimum-norm innovation among multiple in-gate measurements introduces an unavoidable energy contraction, implying that nominal innovation statistics cannot be preserved under nontrivial gating and association. Closed-form results in the two-dimensional case quantify the combined effects and illustrate their practical significance.

An important goal of precision medicine is to personalize medical treatment by identifying individuals who are most likely to benefit from a specific treatment. The Likely Responder (LR) framework, which identifies a subpopulation where treatment response is expected to exceed a certain clinical threshold, plays a role in this effort. However, the LR framework, and more generally, data-driven subgroup analyses, often fail to account for uncertainty in the estimation of model-based data-driven subgrouping. We propose a simple two-stage approach that integrates subgroup identification with subsequent subgroup-specific inference on treatment effects. We incorporate model estimation uncertainty from the first stage into subgroup-specific treatment effect estimation in the second stage, by utilizing Bayesian posterior distributions from the first stage. We evaluate our method through simulations, demonstrating that the proposed Bayesian two-stage model produces better calibrated confidence intervals than naïve approaches. We apply our method to an international COVID-19 treatment trial, which shows substantial variation in treatment effects across data-driven subgroups.

Small area estimation (SAE) improves estimates for local communities or groups, such as counties, neighborhoods, or demographic subgroups, when data are insufficient for each area. This is important for targeting local resources and policies, especially when national-level or large-area data mask variation at a more granular level. Researchers often fit hierarchical Bayesian models to stabilize SAE when data are sparse. Ideally, Bayesian procedures also exhibit good frequentist properties, as demonstrated by calibrated Bayes metrics. However, hierarchical Bayesian models tend to shrink domain estimates toward the overall mean and may produce credible intervals that do not maintain nominal coverage. Hoff et al. developed the Frequentist, but Assisted by Bayes (FAB) intervals for subgroup estimates with normally distributed outcomes. However, non-normally distributed data present new challenges, and multiple types of intervals have been proposed for estimating proportions. We examine domain inference with binary outcomes and extend FAB intervals to improve nominal coverage. We describe how to numerically compute FAB intervals for a proportion and evaluate their performance through repeated simulation studies. Leveraging multilevel regression and poststratification (MRP), we further refine SAE to correct for sample selection bias, construct the FAB intervals for MRP estimates and assess their repeated sampling properties. Finally, we apply the proposed inference methods to estimate COVID-19 infection rates across geographic and demographic subgroups. We find that the FAB intervals improve nominal coverage, at the cost of wider intervals.

In fantasy sports, strategic thinking-not mere luck-often defines who wins and who falls short. As fantasy cricket grows in popularity across India, understanding whether success stems from skill or chance has become both an analytical and regulatory question. This study introduces a new limited-selection contest framework in which participants choose from four expert-designed teams and share prizes based on the highest cumulative score. By combining simulation experiments with real performance data from the 2024 Indian Premier League (IPL), we evaluate whether measurable skill emerges within this structure. Results reveal that strategic and informed team selection consistently outperforms random choice, underscoring a clear skill advantage that persists despite stochastic variability. The analysis quantifies how team composition, inter-team correlation, and participant behaviour jointly influence winning probabilities, highlighting configurations where skill becomes statistically dominant. These findings provide actionable insights for players seeking to maximise returns through strategy and for platform designers aiming to develop fair, transparent, and engaging skill-based gaming ecosystems that balance competition with regulatory compliance.

Despite being one of the earliest neural network layers, the Multilayer Perceptron (MLP) is arguably one of the least understood parts of the transformer architecture due to its dense computation and lack of easy visualization. This paper seeks to understand the MLP layers in dense LLM models by hypothesizing that these layers secretly approximately perform a sparse computation – namely, that they can be well approximated by sparsely-activating Mixture of Experts (MoE) layers. Our hypothesis is based on a novel theoretical connection between MoE models and Sparse Autoencoder (SAE) structure in activation space. We empirically validate the hypothesis on pretrained LLMs, and demonstrate that the activation distribution matters – these results do not hold for Gaussian data, but rather rely crucially on structure in the distribution of neural network activations. Our results shine light on a general principle at play in MLP layers inside LLMs, and give an explanation for the effectiveness of modern MoE-based transformers. Additionally, our experimental explorations suggest new directions for more efficient MoE architecture design based on low-rank routers.

Several optimism-based stochastic bandit algorithms – including UCB, UCB-V, linear UCB, and finite-arm GP-UCB – achieve logarithmic regret using proofs that, despite superficial differences, follow essentially the same structure. This note isolates the minimal ingredients behind these analyses: a single high-probability concentration condition on the estimators, after which logarithmic regret follows from two short deterministic lemmas describing radius collapse and optimism-forced deviations. The framework yields unified, near-minimal proofs for these classical algorithms and extends naturally to many contemporary bandit variants.

Brain connectivity analysis based on magnetic resonance imaging is crucial for understanding neurological mechanisms. However, edge-based connectivity inference faces significant challenges, particularly the curse of dimensionality when estimating high-dimensional covariance matrices. Existing methods often struggle to account for the unknown latent topological structure among brain edges, leading to inaccurate parameter estimation and unstable inference. To address these issues, this study proposes a Bayesian model based on a finite-dimensional Dirichlet distribution. Unlike non-parametric approaches, our method utilizes a finite-dimensional Dirichlet distribution to model the topological structure of brain networks, ensuring constant parameter dimensionality and improving algorithmic stability. We reformulate the covariance matrix structure to guarantee positive definiteness and employ a Metropolis-Hastings algorithm to simultaneously infer network topology and correlation parameters. Simulations validated the recovery of both network topology and correlation parameters. When applied to the Alzheimer's Disease Neuroimaging Initiative dataset, the model successfully identified structural subnetworks. The identified clusters were not only validated by composite anatomical metrics but also consistent with established findings in the literature, collectively demonstrating the model's reliability. The estimated covariance matrix also revealed that intragroup connection strength is stronger than intergroup connection strength. This study introduces a Bayesian framework for inferring brain network topology and high-dimensional covariance structures. The model configuration reduces parameter dimensionality while ensuring the positive definiteness of covariance matrices. As a result, it offers a reliable tool for investigating intrinsic brain connectivity in large-scale neuroimaging studies.

Organizations often have an incumbent predictive model in production when new data sources become available. Because historical training data lack the new features, a challenger model must be trained on a small but growing full-feature dataset. We study whether, and when, the organization should switch to the challenger. The decision is statistical and economic: the challenger's predictive performance improves as full-feature data accumulate, but repeated retraining is costly and delays benefits from deployment. We develop a framework linking learning-curve dynamics to model-switching economics. Under a standard power-law learning curve and finite data-collection horizon $T$, the optimal time to train and evaluate the challenger scales as $T^{1/(1+\alpha)}$: learning-curve shape (through its learning speed $\alpha$) is the primary theoretical determinant of when to stop experimenting; costs determine switching profitability. Even without knowing the learning curve, the operational problem is tractable: we show that any algorithm stopping on the $T^{2/3}$ scale and making reliable switch/discard decisions achieves $O(T^{2/3}\sqrt{\log T})$ regret relative to a full-foresight oracle. We propose a sequential evaluation algorithm that uses local learning-curve trends to anticipate improvement, and test it in a real-world credit-scoring study. Even with this local approximation, the algorithm theoretically and empirically achieves near-oracle performance. It is also more stable than greedy sequential evaluation algorithms, where noisy early estimates trigger premature discarding, or simple one-shot evaluation algorithms, which work only when their fixed evaluation time matches the (unknown in practice) theoretical timing scale. Our framework offers a step toward principled model governance when new data sources require costly collection, validation, and deployment.

Neural network optimization remains one of the most consequential yet poorly understood challenges in modern AI research, where improvements in training algorithms can lead to enhanced feature learning in foundation models, order-of-magnitude reductions in training time, and improved interpretability into how networks learn. While stochastic gradient descent (SGD) and its variants have become the de facto standard for training deep networks, their success in these over-parameterized regimes often appears more empirical than principled. This thesis investigates this apparent paradox by tracing the evolution of optimization algorithms from classical first-order methods to modern higher-order techniques, revealing how principled algorithmic design can demystify the training process. Starting from first principles with SGD and adaptive gradient methods, the analysis progressively uncovers the limitations of these conventional approaches when confronted with anisotropy that is representative of real-world data. These breakdowns motivate the exploration of sophisticated alternatives rooted in curvature information: second-order approximation techniques, layer-wise preconditioning, adaptive learning rates, and more. Next, the interplay between these optimization algorithms and the broader neural network training toolkit, which includes prior and recent developments such as maximal update parametrization, learning rate schedules, and exponential moving averages, emerges as equally essential to empirical success. To bridge the gap between theoretical understanding and practical deployment, this paper offers practical prescriptions and implementation strategies for integrating these methods into modern deep learning workflows.

Observational studies in fields such as epidemiology often rely on covariate adjustment to estimate causal effects. Classical graphical criteria, like the back-door criterion and the generalized adjustment criterion, are powerful tools for identifying valid adjustment sets in directed acyclic graphs (DAGs). However, these criteria are not directly applicable to summary causal graphs (SCGs), which are abstractions of DAGs commonly used in dynamic systems. In SCGs, each node typically represents an entire time series and may involve cycles, making classical criteria inapplicable for identifying causal effects. Recent work established complete conditions for determining whether the micro causal effect of a treatment or an exposure $X_{t-\gamma}$ on an outcome $Y_t$ is identifiable via covariate adjustment in SCGs, under the assumption of no hidden confounding. However, these identifiability conditions have two main limitations. First, they are complex, relying on cumbersome definitions and requiring the enumeration of multiple paths in the SCG, which can be computationally expensive. Second, when these conditions are satisfied, they only provide two valid adjustment sets, limiting flexibility in practical applications. In this paper, we propose an equivalent but simpler formulation of those identifiability conditions and introduce a new criterion that identifies a broader class of valid adjustment sets in SCGs. Additionally, we characterize the quasi-optimal adjustment set among these, i.e., the one that minimizes the asymptotic variance of the causal effect estimator. Our contributions offer both theoretical advancement and practical tools for more flexible and efficient causal inference in abstracted causal graphs.

Data markets are emerging as key mechanisms for trading personal and organizational data. Traditional data pricing studies – such as query-based or arbitrage-free pricing models – mainly emphasize price consistency and profit maximization but often neglect privacy constraints and strategic interactions. The widespread adoption of differential privacy (DP) introduces a fundamental privacy-utility trade-off: noise protects individuals' privacy but reduces data accuracy and market value. This paper develops a Stackelberg game framework for pricing DP data, where the market maker (leader) sets the price function and the data buyer (follower) selects the optimal query precision under DP constraints. We derive the equilibrium strategies for both parties under a balanced pricing function where the pricing decision variable enters linearly into the original pricing model. We obtain closed-form solutions for the optimal variance and pricing level, and determine the boundary conditions for market participation. Furthermore, we extend the analysis to Stackelberg games involving nonlinear power pricing functions. The model bridges DP and economic mechanism design, offering a unified foundation for incentive-compatible and privacy-conscious data pricing in data markets.

We establish verifiable necessary conditions for central limit theorems (CLTs) of ergodic averages of Markov chains on general state spaces. Our criteria are based on discrete-time $\mathbf{L}$-drift conditions that can be verified through estimates for the one-step transition kernel of the original chain. They yield an integral criterion for failure of the CLT for a class of observables that includes unbounded functions and indicators of tail events. The main technical step transfers lower bounds for excursions of the original chain to regenerative excursions of its Nummelin split chain. The same drift conditions yield lower bounds on the tails of return times and modulated excursion functionals, on invariant tails, and on rates of convergence in $f$-variation and $L^p$-Wasserstein distance. These estimates apply without an explicit formula for the invariant distribution. Applications to unadjusted Langevin algorithms show preservation of polynomial target tail indices at every fixed step size. For Euler schemes with additive infinite-variance noise, we establish restrictions on invariant tails and transience under superlinear drifts. Schemes with linear drift, multiplicative noise and logarithmic innovation tails exhibit sharp polynomial convergence exponents and exact CLT thresholds, despite the absence of positive Euclidean moments.