This paper studies optimization on networks modeled as metric graphs. Motivated by applications where the objective function is expensive to evaluate or only available as a black box, we develop Bayesian optimization algorithms that sequentially update a Gaussian process surrogate model of the objective to guide the acquisition of query points. To ensure that the surrogates are tailored to the network's geometry, we adopt Whittle-Matérn Gaussian process prior models defined via stochastic partial differential equations on metric graphs. In addition to establishing regret bounds for optimizing sufficiently smooth objective functions, we analyze the practical case in which the smoothness of the objective is unknown and the Whittle-Matérn prior is represented using finite elements. Numerical results demonstrate the effectiveness of our algorithms for optimizing benchmark objective functions on a synthetic metric graph and for Bayesian inversion via maximum a posteriori estimation on a telecommunication network.

This note introduces a unified theory for causal inference that integrates Riesz regression, covariate balancing, density-ratio estimation (DRE), targeted maximum likelihood estimation (TMLE), and the matching estimator in average treatment effect (ATE) estimation. In ATE estimation, the balancing weights and the regression functions of the outcome play important roles, where the balancing weights are referred to as the Riesz representer, bias-correction term, and clever covariates, depending on the context. Riesz regression, covariate balancing, DRE, and the matching estimator are methods for estimating the balancing weights, where Riesz regression is essentially equivalent to DRE in the ATE context, the matching estimator is a special case of DRE, and DRE is in a dual relationship with covariate balancing. TMLE is a method for constructing regression function estimators such that the leading bias term becomes zero. Nearest Neighbor Matching is equivalent to Least Squares Density Ratio Estimation and Riesz Regression.

We develop a direct debiased machine learning framework comprising Neyman targeted estimation and generalized Riesz regression. Our framework unifies Riesz regression for automatic debiased machine learning, covariate balancing, targeted maximum likelihood estimation (TMLE), and density-ratio estimation. In many problems involving causal effects or structural models, the parameters of interest depend on regression functions. Plugging regression functions estimated by machine learning methods into the identifying equations can yield poor performance because of first-stage bias. To reduce such bias, debiased machine learning employs Neyman orthogonal estimating equations. Debiased machine learning typically requires estimation of the Riesz representer and the regression function. For this problem, we develop a direct debiased machine learning framework with an end-to-end algorithm. We formulate estimation of the nuisance parameters, the regression function and the Riesz representer, as minimizing the discrepancy between Neyman orthogonal scores computed with known and unknown nuisance parameters, which we refer to as Neyman targeted estimation. Neyman targeted estimation includes Riesz representer estimation, and we measure discrepancies using the Bregman divergence. The Bregman divergence encompasses various loss functions as special cases, where the squared loss yields Riesz regression and the Kullback-Leibler divergence yields entropy balancing. We refer to this Riesz representer estimation as generalized Riesz regression. Neyman targeted estimation also yields TMLE as a special case for regression function estimation. Furthermore, for specific pairs of models and Riesz representer estimation methods, we can automatically obtain the covariate balancing property without explicitly solving the covariate balancing objective.

Low-Rank Adaptation (LoRA) offers a cost-effective solution for fine-tuning large language models (LLMs), but it often produces overconfident predictions in data-scarce few-shot settings. To address this issue, several classical statistical learning approaches have been repurposed for scalable uncertainty-aware LoRA fine-tuning. However, these approaches neglect how input characteristics affect the predictive uncertainty estimates. To address this limitation, we propose Contextual Low-Rank Adaptation (C-LoRA) as a novel uncertainty-aware and parameter efficient fine-tuning approach, by developing new lightweight LoRA modules contextualized to each input data sample to dynamically adapt uncertainty estimates. Incorporating data-driven contexts into the parameter posteriors, C-LoRA mitigates overfitting, achieves well-calibrated uncertainties, and yields robust predictions. Extensive experiments on LLaMA2-7B models demonstrate that C-LoRA consistently outperforms the state-of-the-art uncertainty-aware LoRA methods in both uncertainty quantification and model generalization. Ablation studies further confirm the critical role of our contextual modules in capturing sample-specific uncertainties. C-LoRA sets a new standard for robust, uncertainty-aware LLM fine-tuning in few-shot regimes. Although our experiments are limited to 7B models, our method is architecture-agnostic and, in principle, applies beyond this scale; studying its scaling to larger models remains an open problem. Our code is available at https://github.com/ahra99/c_lora.

Large language models (LLMs) continue to advance, with an increasing number of domain-specific variants tailored for specialised tasks. However, these models often lack transparency and explainability, can be costly to fine-tune, require substantial prompt engineering, yield inconsistent results across domains, and impose significant adverse environmental impact due to their high computational demands. To address these challenges, we propose the Bayesian network LLM fusion (BNLF) framework, which integrates predictions from three LLMs, including FinBERT, RoBERTa, and BERTweet, through a probabilistic mechanism for sentiment analysis. BNLF performs late fusion by modelling the sentiment predictions from multiple LLMs as probabilistic nodes within a Bayesian network. Evaluated across three human-annotated financial corpora with distinct linguistic and contextual characteristics, BNLF demonstrates consistent gains of about six percent in accuracy over the baseline LLMs, underscoring its robustness to dataset variability and the effectiveness of probabilistic fusion for interpretable sentiment classification.

Uncovering hidden graph structures underlying real-world data is a critical challenge with broad applications across scientific domains. Recently, transformer-based models leveraging the attention mechanism have demonstrated strong empirical success in capturing complex dependencies within graphs. However, the theoretical understanding of their training dynamics has been limited to tree-like graphs, where each node depends on a single parent. Extending provable guarantees to more general directed acyclic graphs (DAGs) -- which involve multiple parents per node -- remains challenging, primarily due to the difficulty in designing training objectives that enable different attention heads to separately learn multiple different parent relationships. In this work, we address this problem by introducing a novel information-theoretic metric: the kernel-guided mutual information (KG-MI), based on the $f$-divergence. Our objective combines KG-MI with a multi-head attention framework, where each head is associated with a distinct marginal transition kernel to model diverse parent-child dependencies effectively. We prove that, given sequences generated by a $K$-parent DAG, training a single-layer, multi-head transformer via gradient ascent converges to the global optimum in polynomial time. Furthermore, we characterize the attention score patterns at convergence. In addition, when particularizing the $f$-divergence to the KL divergence, the learned attention scores accurately reflect the ground-truth adjacency matrix, thereby provably recovering the underlying graph structure. Experimental results validate our theoretical findings.

Uncovering cause-and-effect mechanisms from data is fundamental to scientific progress. While large language models (LLMs) show promise for enhancing causal discovery (CD) from unstructured data, their application to the increasingly prevalent multimodal setting remains a critical challenge. Even with the advent of multimodal LLMs (MLLMs), their efficacy in multimodal CD is hindered by two primary limitations: (1) difficulty in exploring intra- and inter-modal interactions for comprehensive causal variable identification; and (2) insufficiency to handle structural ambiguities with purely observational data. To address these challenges, we propose MLLM-CD, a novel framework for multimodal causal discovery from unstructured data. It consists of three key components: (1) a novel contrastive factor discovery module to identify genuine multimodal factors based on the interactions explored from contrastive sample pairs; (2) a statistical causal structure discovery module to infer causal relationships among discovered factors; and (3) an iterative multimodal counterfactual reasoning module to refine the discovery outcomes iteratively by incorporating the world knowledge and reasoning capabilities of MLLMs. Extensive experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed MLLM-CD in revealing genuine factors and causal relationships among them from multimodal unstructured data.

Multi-output Gaussian process (MOGP) regression allows modelling dependencies among multiple correlated response variables. Similarly to standard Gaussian processes, MOGPs are sensitive to model misspecification and outliers, which can distort predictions within individual outputs. This situation can be further exacerbated by multiple anomalous response variables whose errors propagate due to correlations between outputs. To handle this situation, we extend and generalise the robust and conjugate Gaussian process (RCGP) framework introduced by Altamirano et al. (2024). This results in the multi-output RCGP (MO-RCGP): a provably robust MOGP that is conjugate, and jointly captures correlations across outputs. We thoroughly evaluate our approach through applications in finance and cancer research.

Physics-informed machine learning (PIML) integrates prior physical information, often in the form of differential equation constraints, into the process of fitting machine learning models to physical data. Popular PIML approaches, including neural operators, physics-informed neural networks, neural ordinary differential equations, and neural discrete equilibria, are typically fit to objectives that simultaneously include both data and physical constraints. However, the multi-objective nature of this approach creates ambiguity in the measurement of model quality. This is related to a poor understanding of epistemic uncertainty, and it can lead to surprising failure modes, even when existing statistical metrics suggest strong fits. Working within a Gaussian process regression framework, we introduce the Physics-Informed Log Evidence (PILE) score. Bypassing the ambiguities of test losses, the PILE score is a single, uncertainty-aware metric that provides a selection principle for hyperparameters of a PIML model. We show that PILE minimization yields excellent choices for a wide variety of model parameters, including kernel bandwidth, least squares regularization weights, and even kernel function selection. We also show that, even prior to data acquisition, a special 'data-free' case of the PILE score identifies a priori kernel choices that are 'well-adapted' to a given PDE. Beyond the kernel setting, we anticipate that the PILE score can be extended to PIML at large, and we outline approaches to do so.

The InfoNCE objective, originally introduced for contrastive representation learning, has become a popular choice for mutual information (MI) estimation, despite its indirect connection to MI. In this paper, we demonstrate why InfoNCE should not be regarded as a valid MI estimator, and we introduce a simple modification, which we refer to as InfoNCE-anchor, for accurate MI estimation. Our modification introduces an auxiliary anchor class, enabling consistent density ratio estimation and yielding a plug-in MI estimator with significantly reduced bias. Beyond this, we generalize our framework using proper scoring rules, which recover InfoNCE-anchor as a special case when the log score is employed. This formulation unifies a broad spectrum of contrastive objectives, including NCE, InfoNCE, and $f$-divergence variants, under a single principled framework. Empirically, we find that InfoNCE-anchor with the log score achieves the most accurate MI estimates; however, in self-supervised representation learning experiments, we find that the anchor does not improve the downstream task performance. These findings corroborate that contrastive representation learning benefits not from accurate MI estimation per se, but from the learning of structured density ratios.