We propose a Likelihood Matching approach for training diffusion models by first establishing an equivalence between the likelihood of the target data distribution and a likelihood along the sample path of the reverse diffusion. To efficiently compute the reverse sample likelihood, a quasi-likelihood is considered to approximate each reverse transition density by a Gaussian distribution with matched conditional mean and covariance, respectively. The score and Hessian functions for the diffusion generation are estimated by maximizing the quasi-likelihood, ensuring a consistent matching of both the first two transitional moments between every two time points. A stochastic sampler is introduced to facilitate computation that leverages on both the estimated score and Hessian information. We establish consistency of the quasi-maximum likelihood estimation, and provide non-asymptotic convergence guarantees for the proposed sampler, quantifying the rates of the approximation errors due to the score and Hessian estimation, dimensionality, and the number of diffusion steps. Empirical and simulation evaluations demonstrate the effectiveness of the proposed Likelihood Matching and validate the theoretical results.

Single-photon Lidar imaging offers a significant advantage in 3D imaging due to its high resolution and long-range capabilities, however it is challenging to apply in noisy environments with multiple targets per pixel. To tackle these challenges, several methods have been proposed. Statistical methods demonstrate interpretability on the inferred parameters, but they are often limited in their ability to handle complex scenes. Deep learning-based methods have shown superior performance in terms of accuracy and robustness, but they lack interpretability or they are limited to a single-peak per pixel. In this paper, we propose a deep unrolling algorithm for dual-peak single-photon Lidar imaging. We introduce a hierarchical Bayesian model for multiple targets and propose a neural network that unrolls the underlying statistical method. To support multiple targets, we adopt a dual depth maps representation and exploit geometric deep learning to extract features from the point cloud. The proposed method takes advantages of statistical methods and learning-based methods in terms of accuracy and quantifying uncertainty. The experimental results on synthetic and real data demonstrate the competitive performance when compared to existing methods, while also providing uncertainty information.

Belief systems are often treated as globally consistent sets of propositions or as scalar-valued probability distributions. Such representations tend to obscure the internal structure of belief, conflate external credibility with internal coherence, and preclude the modeling of fragmented or contradictory epistemic states. This paper introduces a minimal formalism for belief systems as directed, weighted graphs. In this framework, nodes represent individual beliefs, edges encode epistemic relationships (e.g., support or contradiction), and two distinct functions assign each belief a credibility (reflecting source trust) and a confidence (derived from internal structural support). Unlike classical probabilistic models, our approach does not assume prior coherence or require belief updating. Unlike logical and argumentation-based frameworks, it supports fine-grained structural representation without committing to binary justification status or deductive closure. The model is purely static and deliberately excludes inference or revision procedures. Its aim is to provide a foundational substrate for analyzing the internal organization of belief systems, including coherence conditions, epistemic tensions, and representational limits. By distinguishing belief structure from belief strength, this formalism enables a richer classification of epistemic states than existing probabilistic, logical, or argumentation-based approaches.

Out-of-distribution (OOD) detection lies at the heart of robust artificial intelligence (AI), aiming to identify samples from novel distributions beyond the training set. Recent approaches have exploited feature representations as distinguishing signatures for OOD detection. However, most existing methods rely on restrictive assumptions on the feature space that limit the separability between in-distribution (ID) and OOD samples. In this work, we propose a novel OOD detection framework based on a pseudo-label-induced subspace representation, that works under more relaxed and natural assumptions compared to existing feature-based techniques. In addition, we introduce a simple yet effective learning criterion that integrates a cross-entropy-based ID classification loss with a subspace distance-based regularization loss to enhance ID-OOD separability. Extensive experiments validate the effectiveness of our framework.

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.

Deep learning has revolutionized modern data science. However, how to accurately quantify the uncertainty of predictions from large-scale deep neural networks (DNNs) remains an unresolved issue. To address this issue, we introduce a novel post-processing approach. This approach feeds the output from the last hidden layer of a pre-trained large-scale DNN model into a stochastic neural network (StoNet), then trains the StoNet with a sparse penalty on a validation dataset and constructs prediction intervals for future observations. We establish a theoretical guarantee for the validity of this approach; in particular, the parameter estimation consistency for the sparse StoNet is essential for the success of this approach. Comprehensive experiments demonstrate that the proposed approach can construct honest confidence intervals with shorter interval lengths compared to conformal methods and achieves better calibration compared to other post-hoc calibration techniques. Additionally, we show that the StoNet formulation provides us with a platform to adapt sparse learning theory and methods from linear models to DNNs.