We consider the problem of estimating a causal effect in a multi-domain setting. The causal effect of interest is confounded by an unobserved confounder and can change between the different domains. We assume that we have access to a proxy of the hidden confounder and that all variables are discrete or categorical. We propose methodology to estimate the causal effect in the target domain, where we assume to observe only the proxy variable. Under these conditions, we prove identifiability (even when treatment and response variables are continuous). We introduce two estimation techniques, prove consistency, and derive confidence intervals. The theoretical results are supported by simulation studies and a real-world example studying the causal effect of website rankings on consumer choices.

Polynomial chaos expansions (PCE) are widely used for uncertainty quantification (UQ) tasks, particularly in the applied mathematics community. However, PCE has received comparatively less attention in the statistics literature, and fully Bayesian formulations remain rare--especially with implementations in R. Motivated by the success of adaptive Bayesian machine learning models such as BART, BASS, and BPPR, we develop a new fully Bayesian adaptive PCE method with an efficient and accessible R implementation: khaos. Our approach includes a novel proposal distribution that enables data-driven interaction selection, and supports a modified g-prior tailored to PCE structure. Through simulation studies and real-world UQ applications, we demonstrate that Bayesian adaptive PCE provides competitive performance for surrogate modeling, global sensitivity analysis, and ordinal regression tasks.

Physics-Informed Machine Learning (PIML) has successfully integrated mechanistic understanding into machine learning, particularly in domains governed by well-known physical laws. This success has motivated efforts to apply PIML to biology, a field rich in dynamical systems but shaped by different constraints. Biological modeling, however, presents unique challenges: multi-faceted and uncertain prior knowledge, heterogeneous and noisy data, partial observability, and complex, high-dimensional networks. In this position paper, we argue that these challenges should not be seen as obstacles to PIML, but as catalysts for its evolution. We propose Biology-Informed Machine Learning (BIML): a principled extension of PIML that retains its structural grounding while adapting to the practical realities of biology. Rather than replacing PIML, BIML retools its methods to operate under softer, probabilistic forms of prior knowledge. We outline four foundational pillars as a roadmap for this transition: uncertainty quantification, contextualization, constrained latent structure inference, and scalability. Foundation Models and Large Language Models will be key enablers, bridging human expertise with computational modeling. We conclude with concrete recommendations to build the BIML ecosystem and channel PIML-inspired innovation toward challenges of high scientific and societal relevance.

We study a sequential decision-making problem on a $n$-node graph $\mathcal{G}$ where each node has an unknown label from a finite set $\mathbfΩ$, drawn from a joint distribution $\mathcal{P}$ that is Markov with respect to $\mathcal{G}$. At each step, selecting a node reveals its label and yields a label-dependent reward. The goal is to adaptively choose nodes to maximize expected accumulated discounted rewards. We impose a frontier exploration constraint, where actions are limited to neighbors of previously selected nodes, reflecting practical constraints in settings such as contact tracing and robotic exploration. We design a Gittins index-based policy that applies to general graphs and is provably optimal when $\mathcal{G}$ is a forest. Our implementation runs in $\mathcal{O}(n^2 \cdot |\mathbfΩ|^2)$ time while using $\mathcal{O}(n \cdot |\mathbfΩ|^2)$ oracle calls to $\mathcal{P}$ and $\mathcal{O}(n^2 \cdot |\mathbfΩ|)$ space. Experiments on synthetic and real-world graphs show that our method consistently outperforms natural baselines, including in non-tree, budget-limited, and undiscounted settings. For example, in HIV testing simulations on real-world sexual interaction networks, our policy detects nearly all positive cases with only half the population tested, substantially outperforming other baselines.

Regression tasks, notably in safety-critical domains, require proper uncertainty quantification, yet the literature remains largely classification-focused. In this light, we introduce a family of measures for total, aleatoric, and epistemic uncertainty based on proper scoring rules, with a particular emphasis on kernel scores. The framework unifies several well-known measures and provides a principled recipe for designing new ones whose behavior, such as tail sensitivity, robustness, and out-of-distribution responsiveness, is governed by the choice of kernel. We prove explicit correspondences between kernel-score characteristics and downstream behavior, yielding concrete design guidelines for task-specific measures. Extensive experiments demonstrate that these measures are effective in downstream tasks and reveal clear trade-offs among instantiations, including robustness and out-of-distribution detection performance.

Bayesian inference of Bayesian network structures is often performed by sampling directed acyclic graphs along an appropriately constructed Markov chain. We present two techniques to improve sampling. First, we give an efficient implementation of basic moves, which add, delete, or reverse a single arc. Second, we expedite summing over parent sets, an expensive task required for more sophisticated moves: we devise a preprocessing method to prune possible parent sets so as to approximately preserve the sums. Our empirical study shows that our techniques can yield substantial efficiency gains compared to previous methods.

Causal discovery aims to recover graphs that represent causal relations among given variables from observations, and new methods are constantly being proposed. Increasingly, the community raises questions about how much progress is made, because properly evaluating discovered graphs remains notoriously difficult, particularly under latent confounding. We propose a graph distance measure for acyclic directed mixed graphs (AD-MGs) based on the downstream task of cause-effect estimation under unobserved confounding. Our approach uses identification via fixing and a symbolic verifier to quantify how graph differences distort cause-effect esti-mands for different treatment-outcome pairs. We analyze the behavior of the measure under different graph perturbations and compare it against existing distance metrics.

Modeling complex, non-linear, and uncertain relationships between input and output variables remains a central challenge in modern statistical learning and artificial intelligence. Traditional neural networks, trained via point estimation, have demonstrated remarkable success in a variety of domains but inherently provide deterministic predictions - that is, single-valued outputs without accompanying measures of uncertainty. This limitation becomes critical in domains characterized by limited, noisy, or ambiguous data, such as medicine, climate science, or finance, where quantifying uncertainty is as important as producing accurate predictions (Gal & Ghahramani, 2016; Kendall & Gal, 2017; Abdar et al., 2021). Bayesian Neural Networks (BNNs) provide a probabilistic extension of standard neural networks by treating weights and biases as random variables endowed with prior distributions (MacKay, 1992; Neal, 2012). Through Bayes' theorem, BNNs infer a posterior distribution over weights, allowing predictions to reflect epistemic uncertainty - the uncertainty arising from limited data and model knowledge. However, the exact posterior is analytically intractable for deep models, motivating approximate inference methods such as variational inference (Graves, 2011; Blundell et al., 2015) and Monte Carlo dropout (Gal & Ghahramani, 2016). Despite their appeal, these approaches may yield biased or overconfident posteriors due to restrictive variational families (Hern andez-Lobato & Adams, 2015a; Osband et al., 2023), often resulting in over-smoothed predictive distributions. An alternative paradigm for probabilistic modeling is the Mixture Density Network (MDN), introduced by Bridle (1990) and developed further by Jacobs et al. (1991).

Uncertain knowledge graph embedding (UnKGE) methods learn vector representations that capture both structural and uncertainty information to predict scores of unseen triples. However, existing methods produce only point estimates, without quantifying predictive uncertainty-limiting their reliability in high-stakes applications where understanding confidence in predictions is crucial. To address this limitation, we propose \textsc{UnKGCP}, a framework that generates prediction intervals guaranteed to contain the true score with a user-specified level of confidence. The length of the intervals reflects the model's predictive uncertainty. \textsc{UnKGCP} builds on the conformal prediction framework but introduces a novel nonconformity measure tailored to UnKGE methods and an efficient procedure for interval construction. We provide theoretical guarantees for the intervals and empirically verify these guarantees. Extensive experiments on standard benchmarks across diverse UnKGE methods further demonstrate that the intervals are sharp and effectively capture predictive uncertainty.

The heterogeneity of multimodal data leads to inconsistencies and imbalance, allowing a dominant modality to steer gradient updates. Existing solutions mainly focus on optimization- or data-based strategies but rarely exploit the information inherent in multimodal imbalance or conduct its quantitative analysis. To address this gap, we propose a novel quantitative analysis framework for Multimodal Imbalance and design a sample-level adaptive loss function. We define the Modality Gap as the Softmax score difference between modalities for the correct class and model its distribution using a bimodal Gaussian Mixture Model(GMM), representing balanced and imbalanced samples. Using Bayes' theorem, we estimate each sample's posterior probability of belonging to these two groups. Based on this, our adaptive loss (1) minimizes the overall Modality Gap, (2) aligns imbalanced samples with balanced ones, and (3) adaptively penalizes each according to its imbalance degree. A two-stage training strategy-warm-up and adaptive phases,yields state-of-the-art performance on CREMA-D (80.65%), AVE (70.40%), and KineticSound (72.42%). Fine-tuning with high-quality samples identified by the GMM further improves results, highlighting their value for effective multimodal fusion.