The paper proposes a systematic framework for building data-driven stochastic differential equation (SDE) models from sparse, noisy observations. Unlike traditional parametric approaches, which assume a known functional form for the drift, our goal here is to learn the entire drift function directly from data without strong structural assumptions, making it especially relevant in scientific disciplines where system dynamics are partially understood or highly complex. We cast the estimation problem as minimization of the penalized negative log-likelihood functional over a reproducing kernel Hilbert space (RKHS). In the sparse observation regime, the presence of unobserved trajectory segments makes the SDE likelihood intractable. To address this, we develop an Expectation-Maximization (EM) algorithm that employs a novel Sequential Monte Carlo (SMC) method to approximate the filtering distribution and generate Monte Carlo estimates of the E-step objective. The M-step then reduces to a penalized empirical risk minimization problem in the RKHS, whose minimizer is given by a finite linear combination of kernel functions via a generalized representer theorem. To control model complexity across EM iterations, we also develop a hybrid Bayesian variant of the algorithm that uses shrinkage priors to identify significant coefficients in the kernel expansion. We establish important theoretical convergence results for both the exact and approximate EM sequences. The resulting EM-SMC-RKHS procedure enables accurate estimation of the drift function of stochastic dynamical systems in low-data regimes and is broadly applicable across domains requiring continuous-time modeling under observational constraints. We demonstrate the effectiveness of our method through a series of numerical experiments.

We characterize a notion of confidence that arises in learning or updating beliefs: the amount of trust one has in incoming information and its impact on the belief state. This learner's confidence can be used alongside (and is easily mistaken for) probability or likelihood, but it is fundamentally a different concept -- one that captures many familiar concepts in the literature, including learning rates and number of training epochs, Shafer's weight of evidence, and Kalman gain. We formally axiomatize what it means to learn with confidence, give two canonical ways of measuring confidence on a continuum, and prove that confidence can always be represented in this way. Under additional assumptions, we derive more compact representations of confidence-based learning in terms of vector fields and loss functions. These representations induce an extended language of compound "parallel" observations. We characterize Bayes Rule as the special case of an optimizing learner whose loss representation is a linear expectation.

Introduction As engineering fields mature, new technologies are emerging that are beginning to serve as the foundation of many societal improvements. For example, modern medical diagnostic equipment provides valuable information that gives medical professionals a better understanding of a patient's needs and ultimately improves quality of life [1]. Improvements to vehicle designs make transportation in cars or aircraft safer and more environmentally friendly [2]. Military equipment continues to be developed that better supports and protects personnel in the field [3]. Manufacturing practices and robotic equipment improve work safety conditions and reduce a product's price point, making amenities available to a wider range of consumers [4]. One approach to maximizing system availability is to incorporate some means of health assessment into the system itself. Doing so is often referred to as "integrated system health management" (ISHM) or "prognostics and health management" (PHM), which has been applied successfully to many complex systems [5]. By integrating health assessment into the very functioning of a system, more information can be obtained that provides a better understanding of the system as a whole, thus allowing system owners to become proactive in how they deal with system degradation. ISHM and PHM promise to focus on system conditions, thus supporting initiatives in what has become known as condition-based maintenance (CBM). This, in turn, enables maintenance events to be initiated based on specific system conditions rather than waiting until a failure occurs [6]. One of the key ingredients of ISHM/PHM is diagnostics, which corresponds to the process of determining the health state of the system based on sets of observations (or tests). Such tests are designed specifically to track system behavior and determine whether or not a failure has occurred. In many cases it is impossible to identify a single fault that explains the observations with certainty. Instead, candidate sets of faults are often indicated, and when using applicable models, probabilities or confidence values are associated with the faults to provide additional information. One historic approach to using test observations for diagnosis is to apply a decision tree - sometimes referred to as a fault tree1 [7].

We present Dual-Feedback Actor (DFA), a reinforcement learning algorithm that fuses both individual rewards and pairwise preferences (if available) into a single update rule. DFA uses the policy's log-probabilities directly to model the preference probability, avoiding a separate reward-modeling step. Preferences can be provided by human-annotators (at state-level or trajectory-level) or be synthesized online from Q-values stored in an off-policy replay buffer. Under a Bradley-Terry model, we prove that minimizing DFA's preference loss recovers the entropy-regularized Soft Actor-Critic (SAC) policy. Our simulation results show that DFA trained on generated preferences matches or exceeds SAC on six control environments and demonstrates a more stable training process. With only a semi-synthetic preference dataset under Bradley-Terry model, our algorithm outperforms reward-modeling reinforcement learning from human feedback (RLHF) baselines in a stochastic GridWorld and approaches the performance of an oracle with true rewards.

Instruction fine-tuned large language models (LLMs) enable a simple zero-shot or few-shot prompting paradigm, also known as in-context learning, for building prediction models. This convenience, combined with continued advances in LLM capability, has the potential to drive their adoption across a broad range of domains, including high-stakes applications where group fairness -- preventing disparate impacts across demographic groups -- is essential. The majority of existing approaches to enforcing group fairness on LLM-based classifiers rely on traditional fair algorithms applied via model fine-tuning or head-tuning on final-layer embeddings, but they are no longer applicable to closed-weight LLMs under the in-context learning setting, which include some of the most capable commercial models today, such as GPT-4, Gemini, and Claude. In this paper, we propose a framework for deriving fair classifiers from closed-weight LLMs via prompting: the LLM is treated as a feature extractor, and features are elicited from its probabilistic predictions (e.g., token log probabilities) using prompts strategically designed for the specified fairness criterion to obtain sufficient statistics for fair classification; a fair algorithm is then applied to these features to train a lightweight fair classifier in a post-hoc manner. Experiments on five datasets, including three tabular ones, demonstrate strong accuracy-fairness tradeoffs for the classifiers derived by our framework from both open-weight and closed-weight LLMs; in particular, our framework is data-efficient and outperforms fair classifiers trained on LLM embeddings (i.e., head-tuning) or from scratch on raw tabular features.

Bayesian network classifiers provide a feasible solution to tabular data classification, with a number of merits like high time and memory efficiency, and great explainability. However, due to the parameter explosion and data sparsity issues, Bayesian network classifiers are restricted to low-order feature dependency modeling, making them struggle in extrapolating the occurrence probabilities of complex real-world data. In this paper, we propose a novel paradigm to design high-order Bayesian network classifiers, by learning distributional representations for feature values, as what has been done in word embedding and graph representation learning. The learned distributional representations are encoded with the semantic relatedness between different features through their observed co-occurrence patterns in training data, which then serve as a hallmark to extrapolate the occurrence probabilities of new test samples. As a classifier design realization, we remake the K-dependence Bayesian classifier (KDB) by extending it into a neural version, i.e., NeuralKDB, where a novel neural network architecture is designed to learn distributional representations of feature values and parameterize the conditional probabilities between interdependent features. A stochastic gradient descent based algorithm is designed to train the NeuralKDB model efficiently. Extensive classification experiments on 60 UCI datasets demonstrate that the proposed NeuralKDB classifier excels in capturing high-order feature dependencies and significantly outperforms the conventional Bayesian network classifiers, as well as other competitive classifiers, including two neural network based classifiers without distributional representation learning.

Our bound does not require the prior to have any parametric form.

Our bound does not require the prior to have any parametric form.

Gaussian Cox processes are widely-used point process models that use a Gaussian process to describe the Bayesian a priori uncertainty present in latent intensity functions. In this paper, we propose a novel Bayesian inference scheme for Gaussian Cox processes by exploiting a conceptually-intuitive path integral formulation. The proposed scheme does not rely on domain discretization, scales linearly with the number of observed events, has a lower complexity than the state-of-the-art variational Bayesian schemes with respect to the number of inducing points, and is applicable to a wide range of Gaussian Cox processes with various types of link functions. Our scheme is especially beneficial under the multi-dimensional input setting, where the number of inducing points tends to be large. We evaluate our scheme on synthetic and real-world data, and show that it achieves comparable predictive accuracy while being tens of times faster than reference methods.