This paper studies the probability of error associated with the social machine learning framework, which involves an independent training phase followed by a cooperative decision-making phase over a graph. This framework addresses the problem of classifying a stream of unlabeled data in a distributed manner. We consider two kinds of classification tasks with limited observations in the prediction phase, namely, the statistical classification task and the single-sample classification task. For each task, we describe the distributed learning rule and analyze the probability of error accordingly. To do so, we first introduce a stronger consistent training condition that involves the margin distributions generated by the trained classifiers. Based on this condition, we derive an upper bound on the probability of error for both tasks, which depends on the statistical properties of the data and the combination policy used to combine the distributed classifiers. For the statistical classification problem, we employ the geometric social learning rule and conduct a non-asymptotic performance analysis. An exponential decay of the probability of error with respect to the number of unlabeled samples is observed in the upper bound. For the single-sample classification task, a distributed learning rule that functions as an ensemble classifier is constructed. An upper bound on the probability of error of this ensemble classifier is established.

Despite the successes of probabilistic models based on passing noise through neural networks, recent work has identified that such methods often fail to capture tail behavior accurately--unless the tails of the base distribution are appropriately calibrated. To overcome this deficiency, we propose a systematic approach for analyzing the tails of random variables, and we illustrate how this approach can be used during the static analysis (before drawing samples) pass of a probabilistic programming language compiler. To characterize how the tails change under various operations, we develop an algebra which acts on a three-parameter family of tail asymptotics and which is based on the generalized Gamma distribution. Our algebraic operations are closed under addition and multiplication; they are capable of distinguishing sub-Gaussians with differing scales; and they handle ratios sufficiently well to reproduce the tails of most important statistical distributions directly from their definitions. Our empirical results confirm that inference algorithms that leverage our heavy-tailed algebra attain superior performance across a number of density modeling and variational inference tasks.

Multiple measures, such as WEAT or MAC, attempt to quantify the magnitude of bias present in word embeddings in terms of a single-number metric. However, such metrics and the related statistical significance calculations rely on treating pre-averaged data as individual data points and employing bootstrapping techniques with low sample sizes. We show that similar results can be easily obtained using such methods even if the data are generated by a null model lacking the intended bias. Consequently, we argue that this approach generates false confidence. To address this issue, we propose a Bayesian alternative: hierarchical Bayesian modeling, which enables a more uncertainty-sensitive inspection of bias in word embeddings at different levels of granularity. To showcase our method, we apply it to Religion, Gender, and Race word lists from the original research, together with our control neutral word lists. We deploy the method using Google, Glove, and Reddit embeddings. Further, we utilize our approach to evaluate a debiasing technique applied to Reddit word embedding. Our findings reveal a more complex landscape than suggested by the proponents of single-number metrics. The datasets and source code for the paper are publicly available.

Probabilistic programming combines general computer programming, statistical inference, and formal semantics to help systems make decisions when facing uncertainty. Probabilistic programs are ubiquitous, including having a significant impact on machine intelligence. While many probabilistic algorithms have been used in practice in different domains, their automated verification based on formal semantics is still a relatively new research area. In the last two decades, it has attracted much interest. Many challenges, however, remain. The work presented in this paper, probabilistic relations, takes a step towards our vision to tackle these challenges. Our work is based on Hehner's predicative probabilistic programming, but there are several obstacles to the broader adoption of his work. Our contributions here include (1) the formalisation of its syntax and semantics by introducing an Iverson bracket notation to separate relations from arithmetic; (2) the formalisation of relations using Unifying Theories of Programming (UTP) and probabilities outside the brackets using summation over the topological space of the real numbers; (3) the constructive semantics for probabilistic loops using Kleene's fixed-point theorem; (4) the enrichment of its semantics from distributions to subdistributions and superdistributions to deal with the constructive semantics; (5) the unique fixed-point theorem to simplify the reasoning about probabilistic loops; and (6) the mechanisation of our theory in Isabelle/UTP, an implementation of UTP in Isabelle/HOL, for automated reasoning using theorem proving. We demonstrate our work with six examples, including problems in robot localisation, classification in machine learning, and the termination of probabilistic loops.

Accurately detecting crack boundaries is crucial for reliability assessment and risk management of structures and materials, such as structural health monitoring, diagnostics, prognostics, and maintenance scheduling. Uncertainty quantification of crack detection is challenging due to various stochastic factors, such as measurement noises, signal processing, and model simplifications. A machine learning-based approach is proposed to quantify both epistemic and aleatoric uncertainties concurrently. We introduce a Bayesian Boundary-Aware Convolutional Network (B-BACN) that emphasizes uncertainty-aware boundary refinement to generate precise and reliable crack boundary detections. The proposed method employs a multi-task learning approach, where we use Monte Carlo Dropout to learn the epistemic uncertainty and a Gaussian sampling function to predict each sample's aleatoric uncertainty. Moreover, we include a boundary refinement loss to B-BACN to enhance the determination of defect boundaries. The proposed method is demonstrated with benchmark experimental results and compared with several existing methods. The experimental results illustrate the effectiveness of our proposed approach in uncertainty-aware crack boundary detection, minimizing misclassification rate, and improving model calibration capabilities.

Fixed-budget best-arm identification (BAI) is a bandit problem where the agent maximizes the probability of identifying the optimal arm within a fixed budget of observations. In this work, we study this problem in the Bayesian setting. We propose a Bayesian elimination algorithm and derive an upper bound on its probability of misidentifying the optimal arm. The bound reflects the quality of the prior and is the first distribution-dependent bound in this setting. We prove it using a frequentist-like argument, where we carry the prior through, and then integrate out the bandit instance at the end. We also provide a lower bound on the probability of misidentification in a $2$-armed Bayesian bandit and show that our upper bound (almost) matches it for any budget. Our experiments show that Bayesian elimination is superior to frequentist methods and competitive with the state-of-the-art Bayesian algorithms that have no guarantees in our setting.

Likelihood is a standard estimate for outlier detection. The specific role of the normalization constraint is to ensure that the out-of-distribution (OOD) regime has a small likelihood when samples are learned using maximum likelihood. Because autoencoders do not possess such a process of normalization, they often fail to recognize outliers even when they are obviously OOD. We propose the Normalized Autoencoder (NAE), a normalized probabilistic model constructed from an autoencoder. The probability density of NAE is defined using the reconstruction error of an autoencoder, which is differently defined in the conventional energy-based model. In our model, normalization is enforced by suppressing the reconstruction of negative samples, significantly improving the outlier detection performance. Our experimental results confirm the efficacy of NAE, both in detecting outliers and in generating in-distribution samples.

Symbolic Regression (SR) allows for the discovery of scientific equations from data. To limit the large search space of possible equations, prior knowledge has been expressed in terms of formal grammars that characterize subsets of arbitrary strings. However, there is a mismatch between context-free grammars required to express the set of syntactically correct equations, missing closure properties of the former, and a tree structure of the latter. Our contributions are to (i) compactly express experts' prior beliefs about which equations are more likely to be expected by probabilistic Regular Tree Expressions (pRTE), and (ii) adapt Bayesian inference to make such priors efficiently available for symbolic regression encoded as finite state machines. Our scientific case studies show its effectiveness in soil science to find sorption isotherms and for modeling hyper-elastic materials.

Thompson sampling (TS) is widely used in sequential decision making due to its ease of use and appealing empirical performance. However, many existing analytical and empirical results for TS rely on restrictive assumptions on reward distributions, such as belonging to conjugate families, which limits their applicability in realistic scenarios. Moreover, sequential decision making problems are often carried out in a batched manner, either due to the inherent nature of the problem or to serve the purpose of reducing communication and computation costs. In this work, we jointly study these problems in two popular settings, namely, stochastic multi-armed bandits (MABs) and infinite-horizon reinforcement learning (RL), where TS is used to learn the unknown reward distributions and transition dynamics, respectively. We propose batched $\textit{Langevin Thompson Sampling}$ algorithms that leverage MCMC methods to sample from approximate posteriors with only logarithmic communication costs in terms of batches. Our algorithms are computationally efficient and maintain the same order-optimal regret guarantees of $\mathcal{O}(\log T)$ for stochastic MABs, and $\mathcal{O}(\sqrt{T})$ for RL. We complement our theoretical findings with experimental results.

Shepard's universal law of generalization is a remarkable hypothesis about how intelligent organisms should perceive similarity. In its broadest form, the universal law states that the level of perceived similarity between a pair of stimuli should decay as a concave function of their distance when embedded in an appropriate psychological space. While extensively studied, evidence in support of the universal law has relied on low-dimensional stimuli and small stimulus sets that are very different from their real-world counterparts. This is largely because pairwise comparisons - as required for similarity judgments - scale quadratically in the number of stimuli. We provide direct evidence for the universal law in a naturalistic high-dimensional regime by analyzing an existing dataset of 214, 200 human similarity judgments and a newly collected dataset of 390, 819 human generalization judgments (N = 2406 US participants) across three sets of natural images. The Universal Law of Generalization Holds for Naturalistic Stimuli Statement of Relevance Humans constantly form generalizations, whether when trying to identify the color of an object or reasoning about which action to take based on past experiences.