Diffusion models have recently driven significant breakthroughs in generative modeling. While state-of-the-art models produce high-quality samples on average, individual samples can still be low quality. Detecting such samples without human inspection remains a challenging task. To address this, we propose a Bayesian framework for estimating generative uncertainty of synthetic samples. We outline how to make Bayesian inference practical for large, modern generative models and introduce a new semantic likelihood (evaluated in the latent space of a feature extractor) to address the challenges posed by high-dimensional sample spaces. Through our experiments, we demonstrate that the proposed generative uncertainty effectively identifies poor-quality samples and significantly outperforms existing uncertainty-based methods. Notably, our Bayesian framework can be applied post-hoc to any pretrained diffusion or flow matching model (via the Laplace approximation), and we propose simple yet effective techniques to minimize its computational overhead during sampling.

The interplay of optimizers and architectures in neural networks is complicated and hard to understand why some optimizers work better on some specific architectures. In this paper, we find that the traditionally used sharpness metric does not fully explain the intricate interplay and introduces information-theoretic metrics called entropy gap to better help analyze. It is found that both sharpness and entropy gap affect the performance, including the optimization dynamic and generalization. We further use information-theoretic tools to understand a recently proposed optimizer called Lion and find ways to improve it.

We present a collaborative framework where multiple large language models, namely GPT-4-0125-preview, Meta-LLaMA-3-70B-Instruct, Claude-3-Opus, and Gemini-1.5-Flash, work together to generate and respond to complex PhD-level probability questions in the absence of definitive ground truth. This study explores how inter-model consensus enhances response reliability and serves as a proxy for assessing the quality of generated questions. To quantify agreement and consistency, we employ statistical methods including chi-square tests, Fleiss' Kappa, and confidence interval analysis, measuring both response precision and question clarity. Our findings highlight that Claude and Gemini generate well-structured and less ambiguous questions, leading to higher inter-model agreement. This is reflected in their narrower confidence intervals and stronger alignment with answering models. Conversely, LLaMA demonstrates increased variability and lower reliability in question formulation, as indicated by broader confidence intervals and reduced consensus rates. These results suggest that multi-model collaboration not only enhances the reliability of responses but also provides a valuable framework for assessing and improving question quality in the absence of explicit ground truth. This research offers meaningful insights into optimizing AI-driven reasoning through collaborative large-language model interactions.

Such designs arise when treatment assignment is based on whether a particular covariate -- referred to as the running variable -- lies above or below a known value, referred to as the cutoff value. Because treatment is deterministically assigned as a known function of the running variable, RDDs are trivially deconfounded: treatment assignment is independent of the outcome variable, given the running variable (because treatment is conditionally constant). However, estimation of treatment effects in RDDs is more complicated than simply controlling for the running variable, because doing so introduces a complete lack of overlap, which is the other key condition needed to justify regression adjustment for causal inference. Nonetheless, treatment effects at the cutoff may still be identified. Specifically, it is well-known that treatment effects at the cutoff can be estimated from RDDs as the magnitude of a discontinuity in the conditional mean response function at that point (Hahn et al., 2001). This paper investigates the use of Bayesian additive regression tree models (Chipman et al., 2010; Hahn et al., 2020) for the purpose of estimating conditional average treatments effects (CATE) at the cutoff, conditional on observed covariates other than the running variable. To the best of our knowledge, such data-driven CATE estimation has not been a focus of the existing RDD literature and we are the first to propose BART for this purpose.

Equivariant neural networks play a pivotal role in analyzing datasets with symmetry properties, particularly in complex data structures. However, integrating equivariance with Markov properties presents notable challenges due to the inherent dependencies within such data. Previous research has primarily concentrated on establishing generalization bounds under the assumption of independently and identically distributed data, frequently neglecting the influence of Markov dependencies. In this study, we investigate the impact of Markov properties on generalization performance alongside the role of equivariance within this context. We begin by applying a new McDiarmid's inequality to derive a generalization bound for neural networks trained on Markov datasets, using Rademacher complexity as a central measure of model capacity. Subsequently, we utilize group theory to compute the covering number under equivariant constraints, enabling us to obtain an upper bound on the Rademacher complexity based on this covering number. This bound provides practical insights into selecting low-dimensional irreducible representations, enhancing generalization performance for fixed-width equivariant neural networks.

Physics-informed machine learning is one of the most commonly used methods for fusing physical knowledge in the form of partial differential equations with experimental data. The idea is to construct a loss function where the physical laws take the place of a regularizer and minimize it to reconstruct the underlying physical fields and any missing parameters. However, there is a noticeable lack of a direct connection between physics-informed loss functions and an overarching Bayesian framework. In this work, we demonstrate that Brownian bridge Gaussian processes can be viewed as a softly-enforced physics-constrained prior for the Poisson equation. We first show equivalence between the variational form of the physics-informed loss function for the Poisson equation and a kernel ridge regression objective. Then, through the connection between Gaussian process regression and kernel methods, we identify a Gaussian process for which the posterior mean function and physics-informed loss function minimizer agree. This connection allows us to probe different theoretical questions, such as convergence and behavior of inverse problems. We also connect the method to the important problem of identifying model-form error in applications.

Reinforcement learning (RL) methods frequently assume that each new observation completely reflects the environment's state, thereby guaranteeing Markovian (one-step) transitions. In practice, partial observability or sensor/actuator noise often invalidates this assumption. This paper proposes a systematic methodology for detecting such violations, combining a partial correlation-based causal discovery process (PCMCI) with a novel Markov Violation score (MVS). The MVS measures multi-step dependencies that emerge when noise or incomplete state information disrupts the Markov property. Classic control tasks (CartPole, Pendulum, Acrobot) serve as examples to illustrate how targeted noise and dimension omissions affect both RL performance and measured Markov consistency. Surprisingly, even substantial observation noise sometimes fails to induce strong multi-lag dependencies in certain domains (e.g., Acrobot). In contrast, dimension-dropping investigations show that excluding some state variables (e.g., angular velocities in CartPole and Pendulum) significantly reduces returns and increases MVS, while removing other dimensions has minimal impact. These findings emphasize the importance of locating and safeguarding the most causally essential dimensions in order to preserve effective single-step learning. By integrating partial correlation tests with RL performance outcomes, the proposed approach precisely identifies when and where the Markov assumption is violated. This framework offers a principled mechanism for developing robust policies, informing representation learning, and addressing partial observability in real-world RL scenarios. All code and experimental logs are accessible for reproducibility (https://github.com/ucsb/markovianess).

Off-policy evaluation can leverage logged data to estimate the effectiveness of new policies in e-commerce, search engines, media streaming services, or automatic diagnostic tools in healthcare. However, the performance of baseline off-policy estimators like IPS deteriorates when the logging policy significantly differs from the evaluation policy. Recent work proposes sharing information across similar actions to mitigate this problem. In this work, we propose an alternative estimator that shares information across similar contexts using clustering. We study the theoretical properties of the proposed estimator, characterizing its bias and variance under different conditions. We also compare the performance of the proposed estimator and existing approaches in various synthetic problems, as well as a real-world recommendation dataset. Our experimental results confirm that clustering contexts improves estimation accuracy, especially in deficient information settings.

Uncertainty quantification in neural networks through methods such as Dropout, Bayesian neural networks and Laplace approximations is either prone to underfitting or computationally demanding, rendering these approaches impractical for large-scale datasets. In this work, we address these shortcomings by shifting the focus from uncertainty in the weight space to uncertainty at the activation level, via Gaussian processes. More specifically, we introduce the Gaussian Process Activation function (GAPA) to capture neuron-level uncertainties. Our approach operates in a post-hoc manner, preserving the original mean predictions of the pre-trained neural network and thereby avoiding the underfitting issues commonly encountered in previous methods. We propose two methods. The first, GAPA-Free, employs empirical kernel learning from the training data for the hyperparameters and is highly efficient during training. The second, GAPA-Variational, learns the hyperparameters via gradient descent on the kernels, thus affording greater flexibility. Empirical results demonstrate that GAPA-Variational outperforms the Laplace approximation on most datasets in at least one of the uncertainty quantification metrics.

We introduce the use of Gaussian Processes (GPs) for the probabilistic forecasting of intermittent time series. The model is trained in a Bayesian framework that accounts for the uncertainty about the latent function and marginalizes it out when making predictions. We couple the latent GP variable with two types of forecast distributions: the negative binomial (NegBinGP) and the Tweedie distribution (TweedieGP). While the negative binomial has already been used in forecasting intermittent time series, this is the first time in which a fully parameterized Tweedie density is used for intermittent time series. We properly evaluate the Tweedie density, which is both zero-inflated and heavy tailed, avoiding simplifying assumptions made in existing models. We test our models on thousands of intermittent count time series. Results show that our models provide consistently better probabilistic forecasts than the competitors. In particular, TweedieGP obtains the best estimates of the highest quantiles, thus showing that it is more flexible than NegBinGP.