This monograph studies the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis. Connections and equivalences between them are reviewed for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as for sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. The monograph serves as a basis to bridge many other methods based on Gaussian processes and reproducing kernels, which are developed in parallel by the two research communities.

Group structures are common in regression analysis. They can appear in the form of categorical predictors represented by groups of dummy variables or in the context of additive modeling, where each predictor can be expressed as a set of basis functions forming a group; in applications such as gene expression analysis and financial market modeling, groupings exist naturally in the data. For instance, genes that influence similar traits form groups in gene expression data, while stocks from the same sector form groups in financial data. In these scenarios, group shrinkage plays an important role: when there is insufficient evidence to suggest the significance of predictors within a group, the entire group of predictors is shrunk towards zero. This reduces the noise from individual "spurious predictors", which tend to appear more frequently in high-dimensional settings, and decreases model complexity, thereby reducing the risk of overfitting. 1 Within the Bayesian framework, there has been extensive research focusing on the application of continuous shrinkage priors for linear regression problems involving group predictor variables. Traditional approaches, such as the group lasso[31, 24], the group bridge [16], and the group horseshoe [29] primarily apply shrinkage at the group level and do not consider within-group shrinkage.

Parameter estimation is a foundational step in statistical modeling, enabling us to extract knowledge from data and apply it effectively. Bayesian estimation of parameters incorporates prior beliefs with observed data to infer distribution parameters probabilistically and robustly. Moreover, it provides full posterior distributions, allowing uncertainty quantification and regularization, especially useful in small or truncated samples. Utilizing the left-truncated log-logistic (LTLL) distribution is particularly well-suited for modeling time-to-event data where observations are subject to a known lower bound such as precipitation data and cancer survival times. In this paper, we propose a Bayesian approach for estimating the parameters of the LTLL distribution with a fixed truncation point \( x_L > 0 \). Given a random variable \( X \sim LL(α, β; x_L) \), where \( α> 0 \) is the scale parameter and \( β> 0 \) is the shape parameter, the likelihood function is derived based on a truncated sample \( X_1, X_2, \dots, X_N \) with \( X_i > x_L \). We assume independent prior distributions for the parameters, and the posterior inference is conducted via Markov Chain Monte Carlo sampling, specifically using the Metropolis-Hastings algorithm to obtain posterior estimates \( \hatα \) and \( \hatβ \). Through simulation studies and real-world applications, we demonstrate that Bayesian estimation provides more stable and reliable parameter estimates, particularly when the likelihood surface is irregular due to left truncation. The results highlight the advantages of Bayesian inference outperform the estimation of parameter uncertainty in truncated distributions for time to event data analysis.

Progress of AI has led to a creation of very successful, but by no means humble models and tools, especially regarding (i) the huge and further exploding costs and resources they demand, and (ii) the over-confidence of these tools with the answers they provide. Here we introduce a novel mathematical framework for a non-equilibrium entropy-optimizing reformulation of Boltzmann machines based on the exact law of total probability. It results in the highly-performant, but much cheaper, gradient-descent-free learning framework with mathematically-justified existence and uniqueness criteria, and answer confidence/reliability measures. Comparisons to state-of-the-art AI tools in terms of performance, cost and the model descriptor lengths on a set of synthetic problems with varying complexity reveal that the proposed method results in more performant and slim models, with the descriptor lengths being very close to the intrinsic complexity scaling bounds for the underlying problems. Applying this framework to historical climate data results in models with systematically higher prediction skills for the onsets of La Niña and El Niño climate phenomena, requiring just few years of climate data for training - a small fraction of what is necessary for contemporary climate prediction tools.

Large language models excel at generating fluent text but frequently struggle with structured reasoning involving temporal constraints, causal relationships, and probabilistic reasoning. To address these limitations, we propose Temporal Causal Probabilistic Description Logic (T-CPDL), an integrated framework that extends traditional Description Logic with temporal interval operators, explicit causal relationships, and probabilistic annotations. We present two distinct variants of T-CPDL: one capturing qualitative temporal relationships through Allen's interval algebra, and another variant enriched with explicit timestamped causal assertions. Both variants share a unified logical structure, enabling complex reasoning tasks ranging from simple temporal ordering to nuanced probabilistic causation. Empirical evaluations on temporal reasoning and causal inference benchmarks confirm that T-CPDL substantially improves inference accuracy, interpretability, and confidence calibration of language model outputs. By delivering transparent reasoning paths and fine-grained temporal and causal semantics, T-CPDL significantly enhances the capability of language models to support robust, explainable, and trustworthy decision-making. This work also lays the groundwork for developing advanced Logic-Retrieval-Augmented Generation (Logic-RAG) frameworks, potentially boosting the reasoning capabilities and efficiency of knowledge graph-enhanced RAG systems.

This paper outlines a comprehensive theoretical and architectural framework for constructing epistemically grounded artificial intelligence systems capable of propositional commitment, metacognitive reasoning, contradiction detection, and normative truth maintenance. Moving beyond the constraints of stochastic language generation, we propose a model in which artificial agents engage in structured, rule-governed reasoning that adheres to explicit epistemic norms. The approach integrates insights from epistemology, formal logic, inferential semantics, knowledge graph structuring, probabilistic justification, and immutable blockchain evidence to create systems that do not merely simulate knowledge, but operate under explicit, verifiable constraints on belief, justification, and truth. We begin with an analysis of epistemic norms in artificial reasoning, contrasting evi-dentialist, Bayesian, and logical foundations, and establishing a requirement for internal consistency and constraint against falsehood. Central to the proposed system is a prohibition against internal deception: no model component may assert what it internally contradicts.

The wisdom of crowds is an umbrella term for phenomena suggesting that the collective judgment or decision of a large group can be more accurate than the individual judgments or decisions of the group members. A well-known example illustrating this concept is the competition at a country fair described by Galton, where the median value of the individual guesses about the weight of an ox resulted in an astonishingly accurate estimate of the actual weight. This phenomenon resembles classical results in probability theory and relies on independent decision-making. The accuracy of the group's final decision can be significantly reduced if the final agents' opinions are driven by a few influential agents. In this paper, we consider a group of agents who initially possess uncorrelated and unbiased noisy measurements of a common state of the world. Assume these agents iteratively update their estimates according to a simple non-Bayesian learning rule, commonly known in mathematical sociology as the French-DeGroot dynamics or iterative opinion pooling. As a result of this iterative distributed averaging process, each agent arrives at an asymptotic estimate of the state of the world, with the variance of this estimate determined by the matrix of weights the agents assign to each other. Every agent aims at minimizing the variance of her asymptotic estimate of the state of the world; however, such variance is also influenced by the weights allocated by other agents. To achieve the best possible estimate, the agents must then solve a game-theoretic, multi-objective optimization problem defined by the available sets of influence weights. We characterize both the Pareto frontier and the set of Nash equilibria in the resulting game. Additionally, we examine asynchronous best-response dynamics for the group of agents and prove their convergence to the set of strict Nash equilibria.

High-quality training data is critical to the performance of machine learning models, particularly Large Language Models (LLMs). However, obtaining real, high-quality data can be challenging, especially for smaller organizations and early-stage startups. Synthetic data generators provide a promising solution by replicating the statistical and structural properties of real data while preserving privacy and scalability. This study evaluates the performance of six tabular synthetic data generators from two widely used open-source libraries: SDV (Gaussian Copula, CTGAN, TVAE) and Synthicity (Bayesian Network, CTGAN, TVAE). Using a real-world dataset from the UCI Machine Learning Repository, comprising energy consumption and environmental variables from Belgium, we simulate a low-data regime by training models on only 1,000 rows. Each generator is then tasked with producing synthetic datasets under two conditions: a 1:1 (1,000 rows) and a 1:10 (10,000 rows) input-output ratio. Evaluation is conducted using two criteria: statistical similarity, measured via classical statistics and distributional metrics; and predictive utility, assessed using a "Train on Synthetic, Test on Real" approach with four regression models. While statistical similarity remained consistent across models in both scenarios, predictive utility declined notably in the 1:10 case. The Bayesian Network from Synthicity achieved the highest fidelity in both scenarios, while TVAE from SDV performed best in predictive tasks under the 1:10 setting. Although no significant performance gap was found between the two libraries, SDV stands out for its superior documentation and ease of use, making it more accessible for practitioners.

Social reasoning - inferring unobservable beliefs and intentions from partial observations of other agents - remains a challenging task for large language models (LLMs). We evaluate the limits of current reasoning language models in the social deduction game Avalon and find that while the largest models demonstrate strong performance, they require extensive test-time inference and degrade sharply when distilled to smaller, real-time-capable variants. To address this, we introduce a hybrid reasoning framework that externalizes belief inference to a structured probabilistic model, while using an LLM for language understanding and interaction. Our approach achieves competitive performance with much larger models in Agent-Agent play and, notably, is the first language agent to defeat human players in a controlled study - achieving a 67% win rate and receiving higher qualitative ratings than both reasoning baselines and human teammates. We release code, models, and a dataset to support future work on social reasoning in LLM agents, which can be found at https://camp-lab-purdue.github.io/bayesian-social-deduction/

Large language models (LLMs) have shown significant potential to change how we write, communicate, and create, leading to rapid adoption across society. This dissertation examines how individuals and institutions are adapting to and engaging with this emerging technology through three research directions. First, I demonstrate how the institutional adoption of AI detectors introduces systematic biases, particularly disadvantaging writers of non-dominant language varieties, highlighting critical equity concerns in AI governance. Second, I present novel population-level algorithmic approaches that measure the increasing adoption of LLMs across writing domains, revealing consistent patterns of AI-assisted content in academic peer reviews, scientific publications, consumer complaints, corporate communications, job postings, and international organization press releases. Finally, I investigate LLMs' capability to provide feedback on research manuscripts through a large-scale empirical analysis, offering insights into their potential to support researchers who face barriers in accessing timely manuscript feedback, particularly early-career researchers and those from under-resourced settings.