We present the $Y_0$ Python package, which implements causal identification algorithms that apply interventional, counterfactual, and transportability queries to data from (randomized) controlled trials, observational studies, or mixtures thereof. $Y_0$ focuses on the qualitative investigation of causation, helping researchers determine whether a causal relationship can be estimated from available data before attempting to estimate how strong that relationship is. Furthermore, $Y_0$ provides guidance on how to transform the causal query into a symbolic estimand that can be non-parametrically estimated from the available data. $Y_0$ provides a domain-specific language for representing causal queries and estimands as symbolic probabilistic expressions, tools for representing causal graphical models with unobserved confounders, such as acyclic directed mixed graphs (ADMGs), and implementations of numerous identification algorithms from the recent causal inference literature. The $Y_0$ source code can be found under the MIT License at https://github.com/y0-causal-inference/y0 and it can be installed with pip install y0.

Distributionally robust evaluation estimates the worst-case expected return over an uncertainty set of possible covariate and reward distributions, and distributionally robust learning finds a policy that maximizes that worst-case return across that uncertainty set. Unfortunately, current methods for distributionally robust evaluation and learning create overly conservative evaluations and policies. In this work, we propose a practical bandit evaluation and learning algorithm that tailors the uncertainty set to specific problems using mathematical programs constrained by structural equation models. Further, we show how conditional independence testing can be used to detect shifted variables for modeling. We find that the structural equation model (SEM) approach gives more accurate evaluations and learns lower-variance policies than traditional approaches, particularly for large shifts. Further, the SEM approach learns an optimal policy, assuming the model is sufficiently well-specified.

Large Language Models (LLMs) have demonstrated transformative potential in reshaping the world. As these models are pretrained on general corpora, they often require domain-specific fine-tuning to optimize performance in specialized business applications. Due to their massive scale, parameter-efficient fine-tuning (PEFT) methods are widely used to reduce training costs. Among them, hybrid PEFT methods that combine multiple PEFT techniques have achieved the best performance. However, existing hybrid PEFT methods face two main challenges when fine-tuning LLMs for specialized applications: (1) relying on point estimates, lacking the ability to quantify uncertainty for reliable decision-making, and (2) struggling to dynamically adapt to emerging data, lacking the ability to suit real-world situations. We propose Bayesian Hybrid Parameter-Efficient Fine-Tuning (BH-PEFT), a novel method that integrates Bayesian learning into hybrid PEFT. BH-PEFT combines Adapter, LoRA, and prefix-tuning to fine-tune feedforward and attention layers of the Transformer. By modeling learnable parameters as distributions, BH-PEFT enables uncertainty quantification. We further propose a Bayesian dynamic fine-tuning approach where the last posterior serves as the prior for the next round, enabling effective adaptation to new data. We evaluated BH-PEFT on business tasks such as sentiment analysis, news categorization, and commonsense reasoning. Results show that our method outperforms existing PEFT baselines, enables uncertainty quantification for more reliable decisions, and improves adaptability in dynamic scenarios. This work contributes to business analytics and data science by proposing a novel BH-PEFT method and dynamic fine-tuning approach that support uncertainty-aware and adaptive decision-making in real-world situations.

We propose a novel classification framework grounded in symbolic dynamics and data compression using chaotic maps. The core idea is to model each class by generating symbolic sequences from thresholded real-valued training data, which are then evolved through a one-dimensional chaotic map. For each class, we compute the transition probabilities of symbolic patterns (e.g., `00', `01', `10', and `11' for the second return map) and aggregate these statistics to form a class-specific probabilistic model. During testing phase, the test data are thresholded and symbolized, and then encoded using the class-wise symbolic statistics via back iteration, a dynamical reconstruction technique. The predicted label corresponds to the class yielding the shortest compressed representation, signifying the most efficient symbolic encoding under its respective chaotic model. This approach fuses concepts from dynamical systems, symbolic representations, and compression-based learning. We evaluate the proposed method: \emph{ChaosComp} on both synthetic and real-world datasets, demonstrating competitive performance compared to traditional machine learning algorithms (e.g., macro F1-scores for the proposed method on Breast Cancer Wisconsin = 0.9531, Seeds = 0.9475, Iris = 0.8469 etc.). Rather than aiming for state-of-the-art performance, the goal of this research is to reinterpret the classification problem through the lens of dynamical systems and compression, which are foundational perspectives in learning theory and information processing.

Using diffusion priors to solve inverse problems in imaging have significantly matured over the years. In this chapter, we review the various different approaches that were proposed over the years. We categorize the approaches into the more classic explicit approximation approaches and others, which include variational inference, sequential monte carlo, and decoupled data consistency. We cover the extension to more challenging situations, including blind cases, high-dimensional data, and problems under data scarcity and distribution mismatch. More recent approaches that aim to leverage multimodal information through texts are covered. Through this chapter, we aim to (i) distill the common mathematical threads that connect these algorithms, (ii) systematically contrast their assumptions and performance trade-offs across representative inverse problems, and (iii) spotlight the open theoretical and practical challenges by clarifying the landscape of diffusion model based inverse problem solvers.

Generative artificial intelligence (AI) excels at producing complex data structures (text, images, videos) by learning patterns from training examples. Across scientific disciplines, researchers are now applying generative models to ``inverse problems'' to infer hidden parameters from observed data. While these methods can handle intractable models and large-scale studies, they can also produce biased or overconfident conclusions. We present a solution with Frequentist-Bayes (FreB), a mathematically rigorous protocol that reshapes AI-generated probability distributions into confidence regions that consistently include true parameters with the expected probability, while achieving minimum size when training and target data align. We demonstrate FreB's effectiveness by tackling diverse case studies in the physical sciences: identifying unknown sources under dataset shift, reconciling competing theoretical models, and mitigating selection bias and systematics in observational studies. By providing validity guarantees with interpretable diagnostics, FreB enables trustworthy scientific inference across fields where direct likelihood evaluation remains impossible or prohibitively expensive.

Continuous-time reinforcement learning (CTRL) provides a natural framework for sequential decision-making in dynamic environments where interactions evolve continuously over time. While CTRL has shown growing empirical success, its ability to adapt to varying levels of problem difficulty remains poorly understood. In this work, we investigate the instance-dependent behavior of CTRL and introduce a simple, model-based algorithm built on maximum likelihood estimation (MLE) with a general function approximator. Unlike existing approaches that estimate system dynamics directly, our method estimates the state marginal density to guide learning. We establish instance-dependent performance guarantees by deriving a regret bound that scales with the total reward variance and measurement resolution. Notably, the regret becomes independent of the specific measurement strategy when the observation frequency adapts appropriately to the problem's complexity. To further improve performance, our algorithm incorporates a randomized measurement schedule that enhances sample efficiency without increasing measurement cost. These results highlight a new direction for designing CTRL algorithms that automatically adjust their learning behavior based on the underlying difficulty of the environment.

Score matching enables the estimation of the gradient of a data distribution, a key component in denoising diffusion models used to recover clean data from corrupted inputs. In prior work, a heuristic weighting function has been used for the denoising score matching loss without formal justification. In this work, we demonstrate that heteroskedasticity is an inherent property of the denoising score matching objective. This insight leads to a principled derivation of optimal weighting functions for generalized, arbitrary-order denoising score matching losses, without requiring assumptions about the noise distribution. Among these, the first-order formulation is especially relevant to diffusion models. We show that the widely used heuristical weighting function arises as a first-order Taylor approximation to the trace of the expected optimal weighting. We further provide theoretical and empirical comparisons, revealing that the heuristical weighting, despite its simplicity, can achieve lower variance than the optimal weighting with respect to parameter gradients, which can facilitate more stable and efficient training.

Learning causal structure in complex systems is a fundamental challenge across a broad range of disciplines, from traditional scientific fields to modern engineering and technology. Unlike conventional statistical methods that focus merely on correlation, the field of causal discovery primarily considers the problem of discovering the directionality and strength of causal relationships between variables, often from observational data. Thus, it has become a critical tool for researchers aiming to predict the effects of interventions on the systems, especially where controlled experimentation may be expensive, unethical, or even infeasible. Such necessities arise not only in various areas of natural science, such as epidemiology [56], public health [65], genomics [14], neuroscience [86], and climate and environmental science [60], but also in numerous domains in social science, such as psychology [50], philosophy [26], and economics [37]. Moreover, with recent advances in science and technology and the increase in size and complexity of data generation processes, causal discovery has acquired significant relevance in the fields of machine learning [63] and artificial intelligence [81, 82] through various emerging areas such as causal representation learning [64, 85], causal transfer learning [83], causal algorithmic fairness [84], and causal reinforcement learning [5]. This work focuses on learning causal structures from purely observational data within the framework of causal Bayesian networks, which are widely used to represent causal relationships among variables through directed acyclic graphs (DAGs). This is, in general, a nontrivial and difficult task due to the vast number of potential DAG structures and multiple DAGs representing the same set of conditional independence relationships. In fact, DAGs are generally identifiable only up to their corresponding Markov equivalence class, in which all DAGs encode the same conditional independencies [31].

The use of machine learning tools has given causal inference a new lease of life, enabling complex models to be used with principled causal estimators and guarantees about statistically important quantities (Wager and Athey, 2018; Chernozhukov et al., 2018; Hahn et al., 2020). To build trustworthy causal models, however, we also need to understand when these methods may be more or less reliable, or perhaps fail completely. This implies that causal inference needs a set of good benchmarking tools. Unfortunately, real-world datasets are not ideal for this task, because they cannot give us access to the ground truth other than in a few very special circumstances. In particular, they rarely provide the counterfactual outcomes we care about, and the distribution we want to evaluate often differs from the one that produced the observations. Well-designed simulations can address this discrepancy (Neal et al., 2020; Parikh et al., 2022); they allow us to choose a ground truth, stress-test new methods, compare their generalizability and stability, and expose failure modes before deployment.