Bayesian Optimization is critically vulnerable to extreme outliers. Existing provably robust methods typically assume a bounded cumulative corruption budget, which makes them defenseless against even a single corruption of sufficient magnitude. To address this, we introduce a new adversary whose budget is only bounded in the frequency of corruptions, not in their magnitude. We then derive RCGP-UCB, an algorithm coupling the famous upper confidence bound (UCB) approach with a Robust Conjugate Gaussian Process (RCGP). We present stable and adaptive versions of RCGP-UCB, and prove that they achieve sublinear regret in the presence of up to $O(T^{1/2})$ and $O(T^{1/3})$ corruptions with possibly infinite magnitude. This robustness comes at near zero cost: without outliers, RCGP-UCB's regret bounds match those of the standard GP-UCB algorithm.

Approaches based on point processes provide a principled statistical framework for modeling neural spiking activity.

However, training EBMs on data in discrete or mixed state spaces poses significant challenges due to the lack of robust and fast sampling methods. In this work, we propose to train discrete EBMs with Energy Discrepancy, a loss function which only requires the evaluation of the energy function at data points and their perturbed counterparts, thus eliminating the need for Markov chain Monte Carlo.

Specifically, we lay the foundation within the Bayesian framework.

Graphical models are widely used to compactly model the conditional interdependence among multiple random variables Lauritzen [1996] and Pearl [2009]. The vertices of the graph represent the random variables (RVs), while the edges encode the inter-dependence among the RVs. The complete structure of the graph is analytically captured by the joint probability distribution of the random variables. Graphical models offer effective and tractable solutions to various inferential and decision-making solutions in different domains, e.g., computer vision Won and Derin [1992], genetics Chen et al. [2013], Fang et al. [2016], Dobra et al. [2004], social networks Jacob et al. [2014], and power systems Dvijotham et al. [2017].

Recently, considerable research effort has been devoted to developing deep architectures for topic models to learn topic structures.

Causal discovery combines data with knowledge provided by experts to learn the DAG representing the causal relationships between a given set of variables. When data are scarce, bagging is used to measure our confidence in an average DAG obtained by aggregating bootstrapped DAGs. However, the aggregation step has received little attention from the specialized literature: the average DAG is constructed using only the confidence in the individual edges of the bootstrapped DAGs, thus disregarding complex higher-order edge structures. In this paper, we introduce a novel theoretical framework based on higher-order structures and describe a new DAG aggregation algorithm. We perform a simulation study, discussing the advantages and limitations of the proposed approach. Our proposal is both computationally efficient and effective, outperforming state-of-the-art solutions, especially in low sample size regimes and under high dimensionality settings.

Small and medium-sized enterprises (SMEs) represent 99.9% of U.S. businesses yet remain systematically excluded from AI due to a mismatch between their operational scale and modern machine learning's data requirements. This paper introduces SmallML, a Bayesian transfer learning framework achieving enterprise-level prediction accuracy with datasets as small as 50-200 observations. We develop a three-layer architecture integrating transfer learning, hierarchical Bayesian modeling, and conformal prediction. Layer 1 extracts informative priors from 22,673 public records using a SHAP-based procedure transferring knowledge from gradient boosting to logistic regression. Layer 2 implements hierarchical pooling across J=5-50 SMEs with adaptive shrinkage, balancing population patterns with entity-specific characteristics. Layer 3 provides conformal sets with finite-sample coverage guarantees P(y in C(x)) >= 1-alpha for distribution-free uncertainty quantification. Validation on customer churn data demonstrates 96.7% +/- 4.2% AUC with 100 observations per business -- a +24.2 point improvement over independent logistic regression (72.5% +/- 8.1%), with p < 0.000001. Conformal prediction achieves 92% empirical coverage at 90% target. Training completes in 33 minutes on standard CPU hardware. By enabling enterprise-grade predictions for 33 million U.S. SMEs previously excluded from machine learning, SmallML addresses a critical gap in AI democratization. Keywords: Bayesian transfer learning, hierarchical models, conformal prediction, small-data analytics, SME machine learning

Online ratings influence customer decision-making, yet standard aggregation methods, such as the sample mean, fail to adapt to quality changes over time and ignore review heterogeneity (e.g., review sentiment, a review's helpfulness). To address these challenges, we demonstrate the value of using the Gaussian process (GP) framework for rating aggregation. Specifically, we present a tailored GP model that captures the dynamics of ratings over time while additionally accounting for review heterogeneity. Based on 121,123 ratings from Yelp, we compare the predictive power of different rating aggregation methods in predicting future ratings, thereby finding that the GP model is considerably more accurate and reduces the mean absolute error by 10.2% compared to the sample mean. Our findings have important implications for marketing practitioners and customers. By moving beyond means, designers of online reputation systems can display more informative and adaptive aggregated rating scores that are accurate signals of expected customer satisfaction.