In online randomized experiments or A/B tests, accurate predictions of participant inclusion rates are of paramount importance. These predictions not only guide experimenters in optimizing the experiment's duration but also enhance the precision of treatment effect estimates. In this paper we present a novel, straightforward, and scalable Bayesian nonparametric approach for predicting the rate at which individuals will be exposed to interventions within the realm of online A/B testing. Our approach stands out by offering dual prediction capabilities--it forecasts both the quantity of new customers expected in future time windows and, unlike available alternative methods, the number of times they will be observed. We derive closedform expressions for the posterior distributions of the quantities needed to form predictions about future user activity, thereby bypassing the need for numerical algorithms such as Markov chain Monte Carlo. After a comprehensive exposition of our model, we test its performance on experiments on real and simulated data, where we show its superior performance with respect to existing alternatives in the literature. 1 Introduction The problem of predicting the size of a population from which random samples are drawn has a long history in the statistics literature. Originally motivated by applications in ecology, where the goal is typically to determine the number of distinct species of animals within a population (Fisher et al., 1943; Good, 1953; Burnham and Overton, 1979), a variation of this problem has recently received considerable attention also in the genomics literature, where scientists are interested in predicting the number of future rare variants to be observed within a genomic study (Ionita-Laza et al., 2009; Zou et al., 2016; Chakraborty et al., 2019; Masoero et al., 2022).

In this paper, we introduce a Bayesian learning model to understand the behavior of Large Language Models (LLMs). We explore the optimization metric of LLMs, which is based on predicting the next token, and develop a novel model grounded in this principle. Our approach involves constructing an ideal generative text model represented by a multinomial transition probability matrix with a prior, and we examine how LLMs approximate this matrix. We discuss the continuity of the mapping between embeddings and multinomial distributions, and present the Dirichlet approximation theorem to approximate any prior. Additionally, we demonstrate how text generation by LLMs aligns with Bayesian learning principles and delve into the implications for in-context learning, specifically explaining why in-context learning emerges in larger models where prompts are considered as samples to be updated. Our findings indicate that the behavior of LLMs is consistent with Bayesian Learning, offering new insights into their functioning and potential applications.

The inadequate mixing of conventional Markov Chain Monte Carlo (MCMC) methods for multi-modal distributions presents a significant challenge in practical applications such as Bayesian inference and molecular dynamics. Addressing this, we propose Diffusive Gibbs Sampling (DiGS), an innovative family of sampling methods designed for effective sampling from distributions characterized by distant and disconnected modes. DiGS integrates recent developments in diffusion models, leveraging Gaussian convolution to create an auxiliary noisy distribution that bridges isolated modes in the original space and applying Gibbs sampling to alternately draw samples from both spaces. Our approach exhibits a better mixing property for sampling multi-modal distributions than state-of-the-art methods such as parallel tempering. We demonstrate that our sampler attains substantially improved results across various tasks, including mixtures of Gaussians, Bayesian neural networks and molecular dynamics.

In numerous healthcare and other contemporary applications, the variables of primary interest are obtained through indirect measurements, such as in the case of Magnetic Resonance Imaging (MRI) and Computed Tomography (CT). For some of these applications, the reliability of the results is of particular importance. The accuracy and trustworthiness of the outcomes obtained through indirect measurements are significantly influenced by two critical factors: the degree of uncertainty associated with the measuring instrument and the appropriateness of the (forward) model used for the reconstruction of the parameters of interest (measurand). In this paper, we consider Bayesian inversion to obtain the measurand from signals measured by the instrument and a noise model that mimics both the instrument noise and the error of the forward model.

In traditional studies on language evolution, scholars often emphasize the importance of sound laws and sound correspondences for phylogenetic inference of language family trees. However, to date, computational approaches have typically not taken this potential into account. Most computational studies still rely on lexical cognates as major data source for phylogenetic reconstruction in linguistics, although there do exist a few studies in which authors praise the benefits of comparing words at the level of sound sequences. Building on (a) ten diverse datasets from different language families, and (b) state-of-the-art methods for automated cognate and sound correspondence detection, we test, for the first time, the performance of sound-based versus cognate-based approaches to phylogenetic reconstruction. Our results show that phylogenies reconstructed from lexical cognates are topologically closer, by approximately one third with respect to the generalized quartet distance on average, to the gold standard phylogenies than phylogenies reconstructed from sound correspondences.

There has been a long-standing and widespread belief that Bayesian Optimization (BO) with standard Gaussian process (GP), referred to as standard BO, is ineffective in high-dimensional optimization problems. This perception may partly stem from the intuition that GPs struggle with high-dimensional inputs for covariance modeling and function estimation. While these concerns seem reasonable, empirical evidence supporting this belief is lacking. In this paper, we systematically investigated BO with standard GP regression across a variety of synthetic and real-world benchmark problems for high-dimensional optimization. Surprisingly, the performance with standard GP consistently ranks among the best, often outperforming existing BO methods specifically designed for high-dimensional optimization by a large margin. Contrary to the stereotype, we found that standard GP can serve as a capable surrogate for learning high-dimensional target functions. Without strong structural assumptions, BO with standard GP not only excels in high-dimensional optimization but also proves robust in accommodating various structures within the target functions. Furthermore, with standard GP, achieving promising optimization performance is possible by only using maximum likelihood estimation, eliminating the need for expensive Markov-Chain Monte Carlo (MCMC) sampling that might be required by more complex surrogate models. We thus advocate for a re-evaluation and in-depth study of the potential of standard BO in addressing high-dimensional problems.

Out-of-distribution (OOD) detection aims to detect testing samples far away from the in-distribution (ID) training data, which is crucial for the safe deployment of machine learning models in the real world. Distance-based OOD detection methods have emerged with enhanced deep representation learning. They identify unseen OOD samples by measuring their distances from ID class centroids or prototypes. However, existing approaches learn the representation relying on oversimplified data assumptions, e.g., modeling ID data of each class with one centroid class prototype or using loss functions not designed for OOD detection, which overlook the natural diversities within the data. Naively enforcing data samples of each class to be compact around only one prototype leads to inadequate modeling of realistic data and limited performance. To tackle these issues, we propose PrototypicAl Learning with a Mixture of prototypes (PALM) which models each class with multiple prototypes to capture the sample diversities, and learns more faithful and compact samples embeddings to enhance OOD detection. Our method automatically identifies and dynamically updates prototypes, assigning each sample to a subset of prototypes via reciprocal neighbor soft assignment weights. To learn embeddings with multiple prototypes, PALM optimizes a maximum likelihood estimation (MLE) loss to encourage the sample embeddings to be compact around the associated prototypes, as well as a contrastive loss on all prototypes to enhance intra-class compactness and inter-class discrimination at the prototype level. Compared to previous methods with prototypes, the proposed mixture prototype modeling of PALM promotes the representations of each ID class to be more compact and separable from others and the unseen OOD samples, resulting in more reliable OOD detection. Moreover, the automatic estimation of prototypes enables our approach to be extended to the challenging OOD detection task with unlabelled ID data. Extensive experiments demonstrate the superiority of PALM over previous methods, achieving state-of-the-art average AUROC performance of 93.82 on the challenging CIFAR-100 benchmark. Code is available at https://github.com/jeff024/PALM. Deep learning (DL) plays a crucial role in many real-world applications, such as autonomous driving (Huang et al., 2020), medical diagnosis (Zimmerer et al., 2022), and cyber-security (Nguyen et al., 2022).

Estimating the structure of a Bayesian network, in the form of a directed acyclic graph (DAG), from observational data is a statistically and computationally hard problem with essential applications in areas such as causal discovery. Bayesian approaches are a promising direction for solving this task, as they allow for uncertainty quantification and deal with well-known identifiability issues. From a probabilistic inference perspective, the main challenges are (i) representing distributions over graphs that satisfy the DAG constraint and (ii) estimating a posterior over the underlying combinatorial space. We propose an approach that addresses these challenges by formulating a joint distribution on an augmented space of DAGs and permutations. We carry out posterior estimation via variational inference, where we exploit continuous relaxations of discrete distributions. We show that our approach can outperform competitive Bayesian and non-Bayesian benchmarks on a range of synthetic and real datasets.

Sparse mixture of experts (SMoE) offers an appealing solution to scale up the model complexity beyond the mean of increasing the network's depth or width. However, effective training of SMoE has proven to be challenging due to the representation collapse issue, which causes parameter redundancy and limited representation potentials. In this work, we propose a competition mechanism to address this fundamental challenge of representation collapse. By routing inputs only to experts with the highest neural response, we show that, under mild assumptions, competition enjoys the same convergence rate as the optimal estimator. We further propose CompeteSMoE, an effective and efficient algorithm to train large language models by deploying a simple router that predicts the competition outcomes. Consequently, CompeteSMoE enjoys strong performance gains from the competition routing policy while having low computation overheads. Our extensive empirical evaluations on two transformer architectures and a wide range of tasks demonstrate the efficacy, robustness, and scalability of CompeteSMoE compared to state-of-the-art SMoE strategies.

Supervised fairness-aware machine learning under distribution shifts is an emerging field that addresses the challenge of maintaining equitable and unbiased predictions when faced with changes in data distributions from source to target domains. In real-world applications, machine learning models are often trained on a specific dataset but deployed in environments where the data distribution may shift over time due to various factors. This shift can lead to unfair predictions, disproportionately affecting certain groups characterized by sensitive attributes, such as race and gender. In this survey, we provide a summary of various types of distribution shifts and comprehensively investigate existing methods based on these shifts, highlighting six commonly used approaches in the literature. Additionally, this survey lists publicly available datasets and evaluation metrics for empirical studies. We further explore the interconnection with related research fields, discuss the significant challenges, and identify potential directions for future studies.