Coverage functions are an important class of discrete functions that capture laws of diminishing returns. In this paper, we propose a new problem of learning time-varying coverage functions which arise naturally from applications in social network analysis, machine learning, and algorithmic game theory. We develop a novel parametrization of the time-varying coverage function by illustrating the connections with counting processes. We present an efficient algorithm to learn the parameters by maximum likelihood estimation, and provide a rigorous theoretic analysis of its sample complexity. Empirical experiments from information diffusion in social network analysis demonstrate that with few assumptions about the underlying diffusion process, our method performs significantly better than existing approaches on both synthetic and real world data.

Stochastic variational inference (SVI) uses stochastic optimization to scale up Bayesian computation to massive data. We present an alternative perspective on SVI as approximate parallel coordinate ascent. SVI trades-off bias and variance to step close to the unknown true coordinate optimum given by batch variational Bayes (VB). We define a model to automate this process.

This work presents novel algorithms for learning Bayesian networks of bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in sampling k-trees (maximal graphs of treewidth k), and subsequently selecting, exactly or approximately, the best structure whose moral graph is a subgraph of that k-tree. The approaches are empirically compared to each other and to state-of-the-art methods on a collection of public data sets with up to 100 variables.

Sampling from hierarchical Bayesian models is often difficult for MCMC methods, because of the strong correlations between the model parameters and the hyperparameters. Recent Riemannian manifold Hamiltonian Monte Carlo (RMHMC) methods have significant potential advantages in this setting, but are computationally expensive. We introduce a new RMHMC method, which we call semi-separable Hamiltonian Monte Carlo, which uses a specially designed mass matrix that allows the joint Hamiltonian over model parameters and hyperparameters to decompose into two simpler Hamiltonians. This structure is exploited by a new integrator which we call the alternating blockwise leapfrog algorithm. The resulting method can mix faster than simpler Gibbs sampling while being simpler and more efficient than previous instances of RMHMC.

Bayesian observer models are very effective in describing human performance in perceptual tasks, so much so that they are trusted to faithfully recover hidden mental representations of priors, likelihoods, or loss functions from the data. However, the intrinsic degeneracy of the Bayesian framework, as multiple combinations of elements can yield empirically indistinguishable results, prompts the question of model identifiability. We propose a novel framework for a systematic testing of the identifiability of a significant class of Bayesian observer models, with practical applications for improving experimental design. We examine the theoretical identifiability of the inferred internal representations in two case studies. First, we show which experimental designs work better to remove the underlying degeneracy in a time interval estimation task. Second, we find that the reconstructed representations in a speed perception task under a slow-speed prior are fairly robust.

The prediction of time-changing variances is an important task in the modeling of financial data. Standard econometric models are often limited as they assume rigid functional relationships for the evolution of the variance. Moreover, functional parameters are usually learned by maximum likelihood, which can lead to overfitting. To address these problems we introduce GP-Vol, a novel non-parametric model for time-changing variances based on Gaussian Processes. This new model can capture highly flexible functional relationships for the variances. Furthermore, we introduce a new online algorithm for fast inference in GP-Vol. This method is much faster than current offline inference procedures and it avoids overfitting problems by following a fully Bayesian approach. Experiments with financial data show that GP-Vol performs significantly better than current standard alternatives.

The Mixture-of-Experts (MoE) architecture has enabled the creation of massive yet efficient Large Language Models (LLMs). However, the standard deterministic routing mechanism presents a significant limitation: its inherent brittleness is a key contributor to model miscalibration and overconfidence, resulting in systems that often do not know what they don't know. This thesis confronts this challenge by proposing a structured \textbf{Bayesian MoE routing framework}. Instead of forcing a single, deterministic expert selection, our approach models a probability distribution over the routing decision itself. We systematically investigate three families of methods that introduce this principled uncertainty at different stages of the routing pipeline: in the \textbf{weight-space}, the \textbf{logit-space}, and the final \textbf{selection-space}. Through a series of controlled experiments on a 3-billion parameter MoE model, we demonstrate that this framework significantly improves routing stability, in-distribution calibration, and out-of-distribution (OoD) detection. The results show that by targeting this core architectural component, we can create a more reliable internal uncertainty signal. This work provides a practical and computationally tractable pathway towards building more robust and self-aware LLMs, taking a crucial step towards making them know what they don't know.

Online controlled experiments (A/B tests) are fundamental to data - driven decision - making in the digital economy. However, their real - world application is frequently compromised by two critical shortcomings: the use of statistically flawed heuristics like " p - value peeking", which inflates false positive rates, and an over - reliance on proxy metrics like conversion rates, which can lead to decisions that inadvertently harm core business profitability. This paper addresses these challenges by introducing a comp rehensive and scalable Bayesian decision framework designed for profit optimization in multi - variant (A/B/n) experiments. We propose a hierarchical Bayesian model that simultaneously estimates the probability of conversion (using a Beta - Bernoulli model) and the monetary value of that conversion (using a robust Bayesian model for the mean transaction value). Building on this, we employ a decision - theoretic stopping rule based on Expected Loss, enabling experiments to be concluded not only when a superior variant is identified but also when it becomes clear that no variant offers a practically significant improvement (stopping f or futility). The framework successfully navigates "revenue traps" where a variant with a higher conversion rate would have resulted in a net financial loss, correctly terminates futile experiments early to conserve resources, and maintains strict statisti cal integrity throughout the monitoring process. Ultimately, this work provides a practical and principled methodology for organizations to move beyond simple A/B testing towards a mature, profit - driven experimentation culture, ensuring that statistical conclusions translate directly to strategic busines s value.

Statistical relationships in observed data can arise for several different reasons: the observed variables may be causally related, they may share a latent common cause, or there may be selection bias. Each of these scenarios can be modelled using different causal graphs. Not all such causal graphs, however, can be distinguished by experimental data. In this paper, we formulate the equivalence class of causal graphs as a novel graphical structure, the selected-marginalized directed graph (smDG). That is, we show that two directed acyclic graphs with latent and selected vertices have the same smDG if and only if they are indistinguishable, even when allowing for arbitrary interventions on the observed variables. As a substitute for the more familiar d-separation criterion for DAGs, we provide an analogous sound and complete separation criterion in smDGs for conditional independence relative to passive observations. Finally, we provide a series of sufficient conditions under which two causal structures are indistinguishable when there is only access to passive observations.

A central question in psycholinguistics is the timing of syntax in sentence processing. Much of the existing evidence comes from violation paradigms, which conflate two separable processes - syntactic category detection and phrase structure construction - and implicitly assume that phrase structure follows category detection. In this study, we use co-registered EEG and eye-tracking data from the ZuCo corpus to disentangle these processes and test their temporal order under naturalistic reading conditions. Analyses of gaze transitions showed that readers preferentially moved between syntactic heads, suggesting that phrase structures, rather than serial word order, organize scanpaths. Bayesian network modeling further revealed that structural depth was the strongest driver of deviations from linear reading, outweighing lexical familiarity and surprisal. Finally, fixation-related potentials demonstrated that syntactic surprisal influences neural activity before word onset (-184 to -10 ms) and during early integration (48 to 300 ms). These findings extend current models of syntactic timing by showing that phrase structure construction can precede category detection and dominate lexical influences, supporting a predictive "tree-scaffolding" account of comprehension.