Data assimilation will be essential for the management and expansion of geological carbon storage operations. In traditional data assimilation approaches a fixed set of geological hyperparameters, such as mean and standard deviation of log-permeability, is often assumed. Such hyperparameters, however, may be highly uncertain in practical CO2 storage applications. In this study, we develop a hierarchical data assimilation framework for carbon storage that treats hyperparameters as uncertain variables characterized by hyperprior distributions. To deal with the computationally intractable likelihood function in hyperparameter estimation, we apply a likelihood-free (or simulation-based) inference algorithm, specifically sequential Monte Carlo-based approximate Bayesian computation (SMC-ABC), to draw independent posterior samples of hyperparameters given dynamic monitoring-well data. In the second step we use an ensemble smoother with multiple data assimilation (ESMDA) procedure to provide posterior realizations of grid-block permeability. To reduce computational costs, a 3D recurrent R-U-Net deep-learning surrogate model is applied for forward function evaluations. The accuracy of the surrogate model is established through comparisons to high-fidelity simulation results. A rejection sampling (RS) procedure for data assimilation is applied to provide reference posterior results. Detailed data assimilation results from SMC-ABC-ESMDA are compared to those from the reference RS method. These include marginal posterior distributions of hyperparameters, pairwise posterior samples, and history matching results for pressure and saturation at the monitoring location. Close agreement is achieved with 'converged' RS results, for two synthetic true models, in all quantities considered. Importantly, the SMC-ABC-ESMDA procedure provides speedup of 1-2 orders of magnitude relative to RS for the two cases.

Concept Bottleneck Models (CBMs) have been proposed as a compromise between white-box and black-box models, aiming to achieve interpretability without sacrificing accuracy. The standard training procedure for CBMs is to predefine a candidate set of human-interpretable concepts, extract their values from the training data, and identify a sparse subset as inputs to a transparent prediction model. However, such approaches are often hampered by the tradeoff between enumerating a sufficiently large set of concepts to include those that are truly relevant versus controlling the cost of obtaining concept extractions. This work investigates a novel approach that sidesteps these challenges: BC-LLM iteratively searches over a potentially infinite set of concepts within a Bayesian framework, in which Large Language Models (LLMs) serve as both a concept extraction mechanism and prior. BC-LLM is broadly applicable and multi-modal. Despite imperfections in LLMs, we prove that BC-LLM can provide rigorous statistical inference and uncertainty quantification. In experiments, it outperforms comparator methods including black-box models, converges more rapidly towards relevant concepts and away from spuriously correlated ones, and is more robust to out-of-distribution samples.

Amortized meta-learning methods based on pre-training have propelled fields like natural language processing and vision. Transformer-based neural processes and their variants are leading models for probabilistic meta-learning with a tractable objective. Often trained on synthetic data, these models implicitly capture essential latent information in the data-generation process. However, existing methods do not allow users to flexibly inject (condition on) and extract (predict) this probabilistic latent information at runtime, which is key to many tasks. We introduce the Amortized Conditioning Engine (ACE), a new transformer-based meta-learning model that explicitly represents latent variables of interest. ACE affords conditioning on both observed data and interpretable latent variables, the inclusion of priors at runtime, and outputs predictive distributions for discrete and continuous data and latents. We show ACE's modeling flexibility and performance in diverse tasks such as image completion and classification, Bayesian optimization, and simulation-based inference.

Count data is prevalent in various fields like ecology, medical research, and genomics. In high-dimensional settings, where the number of features exceeds the sample size, feature selection becomes essential. While frequentist methods like Lasso have advanced in handling high-dimensional count data, Bayesian approaches remain under-explored with no theoretical results on prediction performance. This paper introduces a novel probabilistic machine learning framework for high-dimensional count data prediction. We propose a pseudo-Bayesian method that integrates a scaled Student prior to promote sparsity and uses an exponential weight aggregation procedure. A key contribution is a novel risk measure tailored to count data prediction, with theoretical guarantees for prediction risk using PAC-Bayesian bounds. Our results include non-asymptotic oracle inequalities, demonstrating rate-optimal prediction error without prior knowledge of sparsity. We implement this approach efficiently using Langevin Monte Carlo method. Simulations and a real data application highlight the strong performance of our method compared to the Lasso in various settings.

Bayesian inference provides a principled probabilistic framework for quantifying uncertainty by updating beliefs based on prior knowledge and observed data through Bayes' theorem. In Bayesian deep learning, neural network weights are treated as random variables with prior distributions, allowing for a probabilistic interpretation and quantification of predictive uncertainty. However, Bayesian methods lack theoretical generalization guarantees for unseen data. PAC-Bayesian analysis addresses this limitation by offering a frequentist framework to derive generalization bounds for randomized predictors, thereby certifying the reliability of Bayesian methods in machine learning. Temperature $T$, or inverse-temperature $\lambda = \frac{1}{T}$, originally from statistical mechanics in physics, naturally arises in various areas of statistical inference, including Bayesian inference and PAC-Bayesian analysis. In Bayesian inference, when $T < 1$ (``cold'' posteriors), the likelihood is up-weighted, resulting in a sharper posterior distribution. Conversely, when $T > 1$ (``warm'' posteriors), the likelihood is down-weighted, leading to a more diffuse posterior distribution. By balancing the influence of observed data and prior regularization, temperature adjustments can address issues of underfitting or overfitting in Bayesian models, bringing improved predictive performance.

Generative Flow Networks (GFlowNets) are a family of generative models that learn to sample objects with probabilities proportional to a given reward function. The key concept behind GFlowNets is the use of two stochastic policies: a forward policy, which incrementally constructs compositional objects, and a backward policy, which sequentially deconstructs them. Recent results show a close relationship between GFlowNet training and entropy-regularized reinforcement learning (RL) problems with a particular reward design. However, this connection applies only in the setting of a fixed backward policy, which might be a significant limitation. As a remedy to this problem, we introduce a simple backward policy optimization algorithm that involves direct maximization of the value function in an entropy-regularized Markov Decision Process (MDP) over intermediate rewards. We provide an extensive experimental evaluation of the proposed approach across various benchmarks in combination with both RL and GFlowNet algorithms and demonstrate its faster convergence and mode discovery in complex environments.

Linguistic fuzzy information evolution is crucial in understanding information exchange among agents. However, different agent weights may lead to different convergence results in the classic DeGroot model. Similarly, in the Hegselmann-Krause bounded confidence model (HK model), changing the confidence threshold values of agents can lead to differences in the final results. To address these limitations, this paper proposes three new models of linguistic fuzzy information dynamics: the per-round random leader election mechanism-based DeGroot model (PRRLEM-DeGroot), the PRRLEM-based homogeneous HK model (PRRLEM-HOHK), and the PRRLEM-based heterogeneous HK model (PRRLEM-HEHK). In these models, after each round of fuzzy information updates, an agent is randomly selected to act as a temporary leader with more significant influence, with the leadership structure being reset after each update. This strategy increases the information sharing and enhances decision-making by integrating multiple agents' evaluation information, which is also in line with real life (\emph{Leader is not unchanged}). The Monte Carlo method is then employed to simulate the behavior of complex systems through repeated random tests, obtaining confidence intervals for different fuzzy information. Subsequently, an improved golden rule representative value (GRRV) in fuzzy theory is proposed to rank these confidence intervals. Simulation examples and a real-world scenario about space situational awareness validate the effectiveness of the proposed models. Comparative analysis with the other models demonstrate our ability to address the echo chamber and improve the robustness.

This study explores portfolio selection using predictive models for portfolio returns. Portfolio selection is a fundamental task in finance, and various methods have been developed to achieve this goal. For example, the mean-variance approach constructs portfolios by balancing the trade-off between the mean and variance of asset returns, while the quantile-based approach optimizes portfolios by accounting for tail risk. These traditional methods often rely on distributional information estimated from historical data. However, a key concern is the uncertainty of future portfolio returns, which may not be fully captured by simple reliance on historical data, such as using the sample average. To address this, we propose a framework for predictive portfolio selection using conformal inference, called Conformal Predictive Portfolio Selection (CPPS). Our approach predicts future portfolio returns, computes corresponding prediction intervals, and selects the desirable portfolio based on these intervals. The framework is flexible and can accommodate a variety of predictive models, including autoregressive (AR) models, random forests, and neural networks. We demonstrate the effectiveness of our CPPS framework using an AR model and validate its performance through empirical studies, showing that it provides superior returns compared to simpler strategies.

Representation learning of Text-Attributed Graphs (TAGs) has garnered significant attention due to its applications in various domains, including recommendation systems and social networks. Despite advancements in TAG learning methodologies, challenges remain in explainability due to the black-box nature of existing TAG representation learning models. This paper presents TAGExplainer, the first method designed to generate natural language explanations for TAG learning. TAGExplainer employs a generative language model that maps input-output pairs to explanations reflecting the model's decision-making process. To address the lack of annotated ground truth explanations in real-world scenarios, we propose first generating pseudo-labels that capture the model's decisions from saliency-based explanations, then the pseudo-label generator is iteratively trained based on three training objectives focusing on faithfulness and brevity via Expert Iteration, to improve the quality of generated pseudo-labels. The high-quality pseudo-labels are finally utilized to train an end-to-end explanation generator model. Extensive experiments are conducted to demonstrate the effectiveness of TAGExplainer in producing faithful and concise natural language explanations.

Determining the appropriate rank in Non-negative Matrix Factorization (NMF) is a critical challenge that often requires extensive parameter tuning and domain-specific knowledge. Traditional methods for rank determination focus on identifying a single optimal rank, which may not capture the complex structure inherent in real-world datasets. In this study, we introduce a novel approach called Residual Sensitivity to Intial Conditions (RSIC) that suggests potentially multiple ranks of interest by analyzing the sensitivity of the relative residuals (e.g. relative reconstruction error) to different initializations. By computing the Mean Coordinatewise Interquartile Range (MCI) of the residuals across multiple random initializations, our method identifies regions where the NMF solutions are less sensitive to initial conditions and potentially more meaningful. We evaluate RSIC on a diverse set of datasets, including single-cell gene expression data, image data, and text data, and compare it against current state-of-the-art existing rank determination methods. Our experiments demonstrate that RSIC effectively identifies relevant ranks consistent with the underlying structure of the data, outperforming traditional methods in scenarios where they are computationally infeasible or less accurate. This approach provides a more scalable and generalizable solution for rank determination in NMF that does not rely on domain-specific knowledge or assumptions.