Matrix completion focuses on recovering missing or incomplete information in matrices. This problem arises in various applications, including image processing and network analysis. Previous research proposed Poisson matrix completion for count data with noise that follows a Poisson distribution, which assumes that the mean and variance are equal. Since overdispersed count data, whose variance is greater than the mean, is more likely to occur in realistic settings, we assume that the noise follows the negative binomial (NB) distribution, which can be more general than the Poisson distribution. In this paper, we introduce NB matrix completion by proposing a nuclear-norm regularized model that can be solved by proximal gradient descent. In our experiments, we demonstrate that the NB model outperforms Poisson matrix completion in various noise and missing data settings on real data.

We consider a Bayesian budgeted multi-armed bandit problem, in which each arm consumes a different amount of resources when selected and there is a budget constraint on the total amount of resources that can be used. Budgeted Thompson Sampling (BTS) offers a very effective heuristic to this problem, but its arm-selection rule does not take into account the remaining budget information. We adopt \textit{Information Relaxation Sampling} framework that generalizes Thompson Sampling for classical $K$-armed bandit problems, and propose a series of algorithms that are randomized like BTS but more carefully optimize their decisions with respect to the budget constraint. In a one-to-one correspondence with these algorithms, a series of performance benchmarks that improve the conventional benchmark are also suggested. Our theoretical analysis and simulation results show that our algorithms (and our benchmarks) make incremental improvements over BTS (respectively, the conventional benchmark) across various settings including a real-world example.

In this paper we introduce a kernel-based measure for detecting differences between two conditional distributions. Using the `kernel trick' and nearest-neighbor graphs, we propose a consistent estimate of this measure which can be computed in nearly linear time (for a fixed number of nearest neighbors). Moreover, when the two conditional distributions are the same, the estimate has a Gaussian limit and its asymptotic variance has a simple form that can be easily estimated from the data. The resulting test attains precise asymptotic level and is universally consistent for detecting differences between two conditional distributions. We also provide a resampling based test using our estimate that applies to the conditional goodness-of-fit problem, which controls Type I error in finite samples and is asymptotically consistent with only a finite number of resamples. A method to de-randomize the resampling test is also presented. The proposed methods can be readily applied to a broad range of problems, ranging from classical nonparametric statistics to modern machine learning. Specifically, we explore three applications: testing model calibration, regression curve evaluation, and validation of emulator models in simulation-based inference. We illustrate the superior performance of our method for these tasks, both in simulations as well as on real data. In particular, we apply our method to (1) assess the calibration of neural network models trained on the CIFAR-10 dataset, (2) compare regression functions for wind power generation across two different turbines, and (3) validate emulator models on benchmark examples with intractable posteriors and for generating synthetic `redshift' associated with galaxy images.

We introduce an information theoretic criterion for Bayesian network structure learning which we call quotient normalized maximum likelihood (qNML). In contrast to the closely related factorized normalized maximum likelihood criterion, qNML satisfies the property of score equivalence. It is also decomposable and completely free of adjustable hyperparameters. For practical computations, we identify a remarkably accurate approximation proposed earlier by Szpankowski and Weinberger. Experiments on both simulated and real data demonstrate that the new criterion leads to parsimonious models with good predictive accuracy.

This study proposes a novel approach to ensemble prediction, called ``covariate-dependent stacking'' (CDST). Unlike traditional stacking methods, CDST allows model weights to vary flexibly as a function of covariates, thereby enhancing predictive performance in complex scenarios. We formulate the covariate-dependent weights through combinations of basis functions, estimate them by optimizing cross-validation, and develop an expectation-maximization algorithm, ensuring computational efficiency. To analyze the theoretical properties, we establish an oracle inequality regarding the expected loss to be minimized for estimating model weights. Through comprehensive simulation studies and an application to large-scale land price prediction, we demonstrate that the CDST consistently outperforms conventional model averaging methods, particularly on datasets where some models fail to capture the underlying complexity. Our findings suggest that the CDST is especially valuable for, but not limited to, spatio-temporal prediction problems, offering a powerful tool for researchers and practitioners in various data analysis fields.

We present Points2Plans, a framework for composable planning with a relational dynamics model that enables robots to solve long-horizon manipulation tasks from partial-view point clouds. Given a language instruction and a point cloud of the scene, our framework initiates a hierarchical planning procedure, whereby a language model generates a high-level plan and a sampling-based planner produces constraint-satisfying continuous parameters for manipulation primitives sequenced according to the high-level plan. Key to our approach is the use of a relational dynamics model as a unifying interface between the continuous and symbolic representations of states and actions, thus facilitating language-driven planning from high-dimensional perceptual input such as point clouds. Whereas previous relational dynamics models require training on datasets of multi-step manipulation scenarios that align with the intended test scenarios, Points2Plans uses only single-step simulated training data while generalizing zero-shot to a variable number of steps during real-world evaluations. We evaluate our approach on tasks involving geometric reasoning, multi-object interactions, and occluded object reasoning in both simulated and real-world settings. Results demonstrate that Points2Plans offers strong generalization to unseen long-horizon tasks in the real world, where it solves over 85% of evaluated tasks while the next best baseline solves only 50%. Qualitative demonstrations of our approach operating on a mobile manipulator platform are made available at sites.google.com/stanford.edu/points2plans.

The standard approach to answering an identifiable causal-effect query (e.g., $P(Y|do(X)$) when given a causal diagram and observational data is to first generate an estimand, or probabilistic expression over the observable variables, which is then evaluated using the observational data. In this paper, we propose an alternative paradigm for answering causal-effect queries over discrete observable variables. We propose to instead learn the causal Bayesian network and its confounding latent variables directly from the observational data. Then, efficient probabilistic graphical model (PGM) algorithms can be applied to the learned model to answer queries. Perhaps surprisingly, we show that this \emph{model completion} learning approach can be more effective than estimand approaches, particularly for larger models in which the estimand expressions become computationally difficult. We illustrate our method's potential using a benchmark collection of Bayesian networks and synthetically generated causal models.

Bayesian learning provides a unified skeleton to solve the electrophysiological source imaging task. From this perspective, existing source imaging algorithms utilize the Gaussian assumption for the observation noise to build the likelihood function for Bayesian inference. However, the electromagnetic measurements of brain activity are usually affected by miscellaneous artifacts, leading to a potentially non-Gaussian distribution for the observation noise. Hence the conventional Gaussian likelihood model is a suboptimal choice for the real-world source imaging task. In this study, we aim to solve this problem by proposing a new likelihood model which is robust with respect to non-Gaussian noises. Motivated by the robust maximum correntropy criterion, we propose a new improper distribution model concerning the noise assumption. This new noise distribution is leveraged to structure a robust likelihood function and integrated with hierarchical prior distributions to estimate source activities by variational inference. In particular, the score matching is adopted to determine the hyperparameters for the improper likelihood model. A comprehensive performance evaluation is performed to compare the proposed noise assumption to the conventional Gaussian model. Simulation results show that, the proposed method can realize more precise source reconstruction by designing known ground-truth. The real-world dataset also demonstrates the superiority of our new method with the visual perception task. This study provides a new backbone for Bayesian source imaging, which would facilitate its application using real-world noisy brain signal.

This paper introduces EVINCE (Entropy and Variation IN Conditional Exchanges), a dialogue framework advancing Artificial General Intelligence (AGI) by enhancing versatility, adaptivity, and reasoning in large language models (LLMs). Leveraging adversarial debate and a novel dual entropy theory, EVINCE improves prediction accuracy, robustness, and stability in LLMs by integrating statistical modeling, information theory, and machine learning to balance diverse perspective exploration with strong prior exploitation. The framework's effectiveness is demonstrated through consistent convergence of information-theoretic metrics, particularly improved mutual information, fostering productive LLM collaboration. We apply EVINCE to healthcare, showing improved disease diagnosis, and discuss its broader implications for decision-making across domains. This work provides theoretical foundations and empirical validation for EVINCE, paving the way for advancements in LLM collaboration and AGI development.

Real-time dynamic path planning in complex traffic environments presents challenges, such as varying traffic volumes and signal wait times. Traditional static routing algorithms like Dijkstra and A* compute shortest paths but often fail under dynamic conditions. Recent Reinforcement Learning (RL) approaches offer improvements but tend to focus on local optima, risking dead-ends or boundary issues. This paper proposes a novel approach based on causal inference for real-time dynamic path planning, balancing global and local optimality. We first use the static Dijkstra algorithm to compute a globally optimal baseline path. A distributed control strategy then guides vehicles along this path. At intersections, DynamicRouteGPT performs real-time decision-making for local path selection, considering real-time traffic, driving preferences, and unexpected events. DynamicRouteGPT integrates Markov chains, Bayesian inference, and large-scale pretrained language models like Llama3 8B to provide an efficient path planning solution. It dynamically adjusts to traffic scenarios and driver preferences and requires no pre-training, offering broad applicability across road networks. A key innovation is the construction of causal graphs for counterfactual reasoning, optimizing path decisions. Experimental results show that our method achieves state-of-the-art performance in real-time dynamic path planning for multiple vehicles while providing explainable path selections, offering a novel and efficient solution for complex traffic environments.