A popular approach to neurosymbolic AI is to take the output of the last layer of a neural network, e.g. a softmax activation, and pass it through a sparse computation graph encoding certain logical constraints one wishes to enforce. This induces a probability distribution over a set of random variables, which happen to be conditionally independent of each other in many commonly used neurosymbolic AI models. Such conditionally independent random variables have been deemed harmful as their presence has been observed to co-occur with a phenomenon dubbed deterministic bias, where systems learn to determinis-tically prefer one of the valid solutions from the solution space over the others. We provide evidence contesting this conclusion and show that the phenomenon of deterministic bias is an artifact of improperly applying neurosymbolic AI. Keywords: neurosymbolic AI partial label learning 1 Introduction Neurosymbolic (NeSy) AI is an approach to AI which seeks to combine logic and neural networks [13].

We propose a robust and scalable framework for variational Bayes (VB) that effectively handles outliers and contamination of arbitrary nature in large datasets. Our approach divides the dataset into disjoint subsets, computes the posterior for each subset, and applies VB approximation independently to these posteriors. The resulting variational posteriors with respect to the subsets are then aggregated using the geometric median of probability measures, computed with respect to the Wasserstein distance. This novel aggregation method yields the Variational Median Posterior (VM-Posterior) distribution. We rigorously demonstrate that the VM-Posterior preserves contraction properties akin to those of the true posterior, while accounting for approximation errors or the variational gap inherent in VB methods. We also provide provable robustness guarantee of the VM-Posterior. Furthermore, we establish a variational Bernstein-von Mises theorem for both multivariate Gaussian distributions with general covariance structures and the mean-field variational family. To facilitate practical implementation, we adapt existing algorithms for computing the VM-Posterior and evaluate its performance through extensive numerical experiments. The results highlight its robustness and scalability, making it a reliable tool for Bayesian inference in the presence of complex, contaminated datasets.

Constraint-based causal discovery algorithms utilize many statistical tests for conditional independence to uncover networks of causal dependencies. These approaches to causal discovery rely on an assumed correspondence between the graphical properties of a causal structure and the conditional independence properties of observed variables, known as the causal Markov condition and faithfulness. Finite data yields an empirical distribution that is "close" to the actual distribution. Across these many possible empirical distributions, the correspondence to the graphical properties can break down for different conditional independencies, and multiple violations can occur at the same time. We study this "meta-dependence" between conditional independence properties using the following geometric intuition: each conditional independence property constrains the space of possible joint distributions to a manifold. The "meta-dependence" between conditional independences is informed by the position of these manifolds relative to the true probability distribution. We provide a simple-to-compute measure of this meta-dependence using information projections and consolidate our findings empirically using both synthetic and real-world data.

Recommending items to users has long been a fundamental task, and studies have tried to improve it ever since. Most well-known models commonly employ representation learning to map users and items into a unified embedding space for matching assessment. These approaches have primary limitations, especially when dealing with explicit feedback and sparse data contexts. Two primary limitations are their proneness to overfitting and failure to incorporate epistemic uncertainty in predictions. To address these problems, we propose a novel Bayesian Deep Ensemble Collaborative Filtering method named BDECF. To improve model generalization and quality, we utilize Bayesian Neural Networks, which incorporate uncertainty within their weight parameters. In addition, we introduce a new interpretable non-linear matching approach for the user and item embeddings, leveraging the advantages of the attention mechanism. Furthermore, we endorse the implementation of an ensemble-based supermodel to generate more robust and reliable predictions, resulting in a more complete model. Empirical evaluation through extensive experiments and ablation studies across a range of publicly accessible real-world datasets with differing sparsity characteristics confirms our proposed method's effectiveness and the importance of its components.

The notion of relative universality with respect to a {\sigma}-field was introduced to establish the unbiasedness and Fisher consistency of an estimator in nonlinear sufficient dimension reduction. However, there is a gap in the proof of this result in the existing literature. The existing definition of relative universality seems to be too strong for the proof to be valid. In this note we modify the definition of relative universality using the concept of \k{o}-measurability, and rigorously establish the mentioned unbiasedness and Fisher consistency. The significance of this result is beyond its original context of sufficient dimension reduction, because relative universality allows us to use the regression operator to fully characterize conditional independence, a crucially important statistical relation that sits at the core of many areas and methodologies in statistics and machine learning, such as dimension reduction, graphical models, probability embedding, causal inference, and Bayesian estimation.

The analysis of data from multiple experiments, such as observations of several individuals, is commonly approached using mixed-effects models, which account for variation between individuals through hierarchical representations. This makes mixed-effects models widely applied in fields such as biology, pharmacokinetics, and sociology. In this work, we propose a novel methodology for scalable Bayesian inference in hierarchical mixed-effects models. Our framework first constructs amortized approximations of the likelihood and the posterior distribution, which are then rapidly refined for each individual dataset, to ultimately approximate the parameters posterior across many individuals. The framework is easily trainable, as it uses mixtures of experts but without neural networks, leading to parsimonious yet expressive surrogate models of the likelihood and the posterior. We demonstrate the effectiveness of our methodology using challenging stochastic models, such as mixed-effects stochastic differential equations emerging in systems biology-driven problems. However, the approach is broadly applicable and can accommodate both stochastic and deterministic models. We show that our approach can seamlessly handle inference for many parameters. Additionally, we applied our method to a real-data case study of mRNA transfection. When compared to exact pseudomarginal Bayesian inference, our approach proved to be both fast and competitive in terms of statistical accuracy.

Bayesian inference provides a principled framework for quantifying uncertainty in both parameters and models by computing full posterior distributions and model evidence (Gelman et al., 2013). However, Bayesian inference is often analytically intractable, requiring the use of approximate methods like Markov chain Monte Carlo (MCMC; Brooks, 2011) or variational inference (VI; Blei et al., 2017). These methods typically necessitate repeated evaluations of the target density, and many require differentiability of the model (Neal, 2011; Kucukelbir et al., 2017). When model evaluations are computationally expensive - for instance, involving extensive numerical methods - these requirements make standard Bayesian approaches impractical. Due to these computational demands, practitioners often resort to simpler alternatives such as maximum a posteriori (MAP) estimation or maximum likelihood estimation (MLE); 1 see for example Wilson and Collins (2019); Ma et al. (2023). While these point estimates can provide useful insights, they fail to capture parameter uncertainty, potentially leading to overconfident or biased conclusions (Gelman et al., 2013). This limitation highlights the need for efficient posterior approximation methods that avoid the computational costs of standard inference techniques.1.

Inference and estimation are fundamental aspects of statistics, system identification and machine learning. For most inference problems, prior knowledge is available on the system to be modeled, and Bayesian analysis is a natural framework to impose such prior information in the form of a prior distribution. However, in many situations, coming out with a fully specified prior distribution is not easy, as prior knowledge might be too vague, so practitioners prefer to use a prior distribution that is as `ignorant' or `uninformative' as possible, in the sense of not imposing subjective beliefs, while still supporting reliable statistical analysis. Jeffreys prior is an appealing uninformative prior because it offers two important benefits: (i) it is invariant under any re-parameterization of the model, (ii) it encodes the intrinsic geometric structure of the parameter space through the Fisher information matrix, which in turn enhances the diversity of parameter samples. Despite these benefits, drawing samples from Jeffreys prior is a challenging task. In this paper, we propose a general sampling scheme using the Metropolis-Adjusted Langevin Algorithm that enables sampling of parameter values from Jeffreys prior, and provide numerical illustrations of our approach through several examples.

Bayesian optimization (BO) is a widely used algorithm for solving expensive black-box optimization problems. However, its performance decreases significantly on high-dimensional problems due to the inherent high-dimensionality of the acquisition function. In the proposed algorithm, we adaptively dropout the variables of the acquisition function along the iterations. By gradually reducing the dimension of the acquisition function, the proposed approach has less and less difficulty to optimize the acquisition function. Numerical experiments demonstrate that AdaDropout effectively tackle high-dimensional challenges and improve solution quality where standard Bayesian optimization methods often struggle. Moreover, it achieves superior results when compared with state-of-the-art high-dimensional Bayesian optimization approaches. This work provides a simple yet efficient solution for high-dimensional expensive optimization.

This study addresses the challenge of ambiguity in knowledge graph question answering (KGQA). While recent KGQA systems have made significant progress, particularly with the integration of large language models (LLMs), they typically assume user queries are unambiguous, which is an assumption that rarely holds in real-world applications. To address these limitations, we propose a novel framework that dynamically handles both entity ambiguity (e.g., distinguishing between entities with similar names) and intent ambiguity (e.g., clarifying different interpretations of user queries) through interactive clarification. Our approach employs a Bayesian inference mechanism to quantify query ambiguity and guide LLMs in determining when and how to request clarification from users within a multi-turn dialogue framework. We further develop a two-agent interaction framework where an LLM-based user simulator enables iterative refinement of logical forms through simulated user feedback. Experimental results on the WebQSP and CWQ dataset demonstrate that our method significantly improves performance by effectively resolving semantic ambiguities. Additionally, we contribute a refined dataset of disambiguated queries, derived from interaction histories, to facilitate future research in this direction.