Linear regression is one of the most widely used statistical methods, especially in the social sciences [Agresti and Finlay, 2009] and other domains where data comes from humans.

This paper considers a new family of variational distributions motivated by Sklar's theorem. This family is based on new copula-like densities on the hypercube with non-uniform marginals which can be sampled efficiently, i.e. with a complexity linear in the dimension d of the state space. Then, the proposed variational densities that we suggest can be seen as arising from these copula-like densities used as base distributions on the hypercube with Gaussian quantile functions and sparse rotation matrices as normalizing flows. The latter correspond to a rotation of the marginals with complexity O (d log d) . We provide some empirical evidence that such a variational family can also approximate non-Gaussian posteriors and can be beneficial compared to Gaussian approximations. Our method performs largely comparably to state-of-the-art variational approximations on standard regression and classification benchmarks for Bayesian Neural Networks.

However, after artefact-suppression, M/EEG data is often rank deficient which limits the application of Riemannian concepts.

Sparse Group Lasso is a method of linear regression analysis that finds sparse parameters in terms of both feature groups and individual features.

Our objective is again to recover all (or some, or one) of them from the samples. In this paper, we consider MLR with the additional presence of corruptions - i.e. adversarial additive errors in the

We study theoretical predictive performance of ridge and ridge-less least-squares regression when covariate vectors arise from evaluating $p$ random, means-square continuous functions over a latent metric space at $n$ random and unobserved locations, subject to additive noise. This leads us away from the standard assumption of i.i.d. data to a setting in which the $n$ covariate vectors are exchangeable but not independent in general. Under an assumption of independence across dimensions, $4$-th order moment, and other regularity conditions, we obtain probabilistic bounds on a notion of predictive excess risk adapted to our random functional covariate setting, making use of recent results of Barzilai and Shamir. We derive convergence rates in regimes where $p$ grows suitably fast relative to $n$, illustrating interplay between ingredients of the model in determining convergence behaviour and the role of additive covariate noise in benign-overfitting.

Using more test-time computation during language model inference, such as generating more intermediate thoughts or sampling multiple candidate answers, has proven effective in significantly improving model performance. This paper takes an initial step toward bridging the gap between practical language model inference and theoretical transformer analysis by incorporating randomness and sampling. We focus on in-context linear regression with continuous/binary coefficients, where our framework simulates language model decoding through noise injection and binary coefficient sampling. Through this framework, we provide detailed analyses of widely adopted inference techniques. Supported by empirical results, our theoretical framework and analysis demonstrate the potential for offering new insights into understanding inference behaviors in real-world language models.

Causal inference often hinges on strong assumptions - such as no unmeasured confounding or perfect compliance - that are rarely satisfied in practice. Partial identification offers a principled alternative: instead of relying on unverifiable assumptions to estimate causal effects precisely, it derives bounds that reflect the uncertainty inherent in the data. Despite its theoretical appeal, partial identification remains underutilized in applied work, in part due to the fragmented nature of existing methods and the lack of practical guidance. This thesis addresses these challenges by systematically comparing a diverse set of bounding algorithms across multiple causal scenarios. We implement, extend, and unify state-of-the-art methods - including symbolic, optimization-based, and information-theoretic approaches - within a common evaluation framework. In particular, we propose an extension of a recently introduced entropy-bounded method, making it applicable to counterfactual queries such as the Probability of Necessity and Sufficiency (PNS). Our empirical study spans thousands of randomized simulations involving both discrete and continuous data-generating processes. We assess each method in terms of bound tightness, computational efficiency, and robustness to assumption violations. To support practitioners, we distill our findings into a practical decision tree for algorithm selection and train a machine learning model to predict the best-performing method based on observable data characteristics. All implementations are released as part of an open-source Python package, CausalBoundingEngine, which enables users to apply and compare bounding methods through a unified interface.

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