In this paper, we introduce an innovative approach for addressing Bayesian inverse problems through the utilization of physics-informed invertible neural networks (PI-INN). The PI-INN framework encompasses two sub-networks: an invertible neural network (INN) and a neural basis network (NB-Net). The primary role of the NB-Net lies in modeling the spatial basis functions characterizing the solution to the forward problem dictated by the underlying partial differential equation. Simultaneously, the INN is designed to partition the parameter vector linked to the input physical field into two distinct components: the expansion coefficients representing the forward problem solution and the Gaussian latent noise. If the forward mapping is precisely estimated, and the statistical independence between expansion coefficients and latent noise is well-maintained, the PI-INN offers a precise and efficient generative model for Bayesian inverse problems, yielding tractable posterior density estimates. As a particular physics-informed deep learning model, the primary training challenge for PI-INN centers on enforcing the independence constraint, which we tackle by introducing a novel independence loss based on estimated density. We support the efficacy and precision of the proposed PI-INN through a series of numerical experiments, including inverse kinematics, 1-dimensional and 2-dimensional diffusion equations, and seismic traveltime tomography. Specifically, our experimental results showcase the superior performance of the proposed independence loss in comparison to the commonly used but computationally demanding kernel-based maximum mean discrepancy loss.

The fast growth of social networks and their data access limitations in recent years has led to increasing difficulty in obtaining the complete topology of these networks. However, diffusion information over these networks is available, and many algorithms have been proposed to infer the underlying networks using this information. The previously proposed algorithms only focus on inferring more links and ignore preserving the critical topological characteristics of the underlying social networks. In this paper, we propose a novel method called DANI to infer the underlying network while preserving its structural properties. It is based on the Markov transition matrix derived from time series cascades, as well as the node-node similarity that can be observed in the cascade behavior from a structural point of view. In addition, the presented method has linear time complexity (increases linearly with the number of nodes, number of cascades, and square of the average length of cascades), and its distributed version in the MapReduce framework is also scalable. We applied the proposed approach to both real and synthetic networks. The experimental results showed that DANI has higher accuracy and lower run time while maintaining structural properties, including modular structure, degree distribution, connected components, density, and clustering coefficients, than well-known network inference methods.

Large molecular representation models pre-trained on massive unlabeled data have shown great success in predicting molecular properties. However, these models may tend to overfit the fine-tuning data, resulting in over-confident predictions on test data that fall outside of the training distribution. To address this issue, uncertainty quantification (UQ) methods can be used to improve the models' calibration of predictions. Although many UQ approaches exist, not all of them lead to improved performance. While some studies have included UQ to improve molecular pre-trained models, the process of selecting suitable backbone and UQ methods for reliable molecular uncertainty estimation remains underexplored. To address this gap, we present MUBen, which evaluates different UQ methods for state-of-the-art backbone molecular representation models to investigate their capabilities. By fine-tuning various backbones using different molecular descriptors as inputs with UQ methods from different categories, we critically assess the influence of architectural decisions and training strategies. Our study offers insights for selecting UQ for backbone models, which can facilitate research on uncertainty-critical applications in fields such as materials science and drug discovery.

Context: Machine Learning (ML) is integrated into a growing number of systems for various applications. Because the performance of an ML model is highly dependent on the quality of the data it has been trained on, there is a growing interest in approaches to detect and repair data errors (i.e., data cleaning). Researchers are also exploring how ML can be used for data cleaning; hence creating a dual relationship between ML and data cleaning. To the best of our knowledge, there is no study that comprehensively reviews this relationship. Objective: This paper's objectives are twofold. First, it aims to summarize the latest approaches for data cleaning for ML and ML for data cleaning. Second, it provides future work recommendations. Method: We conduct a systematic literature review of the papers published between 2016 and 2022 inclusively. We identify different types of data cleaning activities with and for ML: feature cleaning, label cleaning, entity matching, outlier detection, imputation, and holistic data cleaning. Results: We summarize the content of 101 papers covering various data cleaning activities and provide 24 future work recommendations. Our review highlights many promising data cleaning techniques that can be further extended. Conclusion: We believe that our review of the literature will help the community develop better approaches to clean data.

Despite the remarkable advances in language modeling, current mainstream decoding methods still struggle to generate texts that align with human texts across different aspects. In particular, sampling-based methods produce less-repetitive texts which are often disjunctive in discourse, while search-based methods maintain topic coherence at the cost of increased repetition. Overall, these methods fall short in achieving holistic alignment across a broad range of aspects. In this work, we frame decoding from a language model as an optimization problem with the goal of strictly matching the expected performance with human texts measured by multiple metrics of desired aspects simultaneously. The resulting decoding distribution enjoys an analytical solution that scales the input language model distribution via a sequence-level energy function defined by these metrics. And most importantly, we prove that this induced distribution is guaranteed to improve the perplexity on human texts, which suggests a better approximation to the underlying distribution of human texts. To facilitate tractable sampling from this globally normalized distribution, we adopt the Sampling-Importance-Resampling technique. Experiments on various domains and model scales demonstrate the superiority of our method in metrics alignment with human texts and human evaluation over strong baselines.

We present a sequential Bayesian learning method for tracking non-stationary signal-to-noise ratios in LDPC codes using probabilistic graphical models. We represent the LDPC code as a cluster graph using a general purpose cluster graph construction algorithm called the layered trees running intersection property (LTRIP) algorithm. The channel noise estimator is a global Gamma cluster, which we extend to allow for Bayesian tracking of non-stationary noise variation. We evaluate our proposed model on real-world 5G drive test data. Our results show that our model is capable of tracking non-stationary channel noise, which outperforms an LDPC code with a fixed knowledge of the actual average channel noise.

The cold posterior effect (CPE) (Wenzel et al., 2020) in Bayesian deep learning shows that, for posteriors with a temperature T < 1, the resulting posterior predictive could have better performances than the Bayesian posterior (T = 1). As the Bayesian posterior is known to be optimal under perfect model specification, many recent works have studied the presence of CPE as a model misspecification problem, arising from the prior and/or from the likelihood function. In this work, we provide a more nuanced understanding of the CPE as we show that misspecification leads to CPE only when the resulting Bayesian posterior underfits. In fact, we theoretically show that if there is no underfitting, there is no CPE. In Bayesian deep learning, the cold posterior effect (CPE) (Wenzel et al., 2020) refers to the phenomenon in which if we artificially "temper" the posterior by either p(θ|D) (p(D|θ)p(θ)) The discovery of the CPE has sparked debates in the community about its potential contributing factors. If the prior and likelihood are properly specified, the Bayesian solution (i.e., T = 1) should be optimal (Gelman et al., 2013), assuming approximate inference is properly working. Hence, the presence of the CPE implies either the prior (Wenzel et al., 2020; Fortuin et al., 2022), the likelihood (Aitchison, 2021; Kapoor et al., 2022), or both are misspecified. This has been, so far, the main argument of many works trying to explain the CPE. One line of research examines the impact of the prior misspecification on the CPE (Wenzel et al., 2020; Fortuin et al., 2022). The priors of modern Bayesian neural networks are often selected for tractability. Consequently, the quality of the selected priors in relation to the CPE is a natural concern.

We present VBPI-Mixtures, an algorithm designed to enhance the accuracy of phylogenetic posterior distributions, particularly for tree-topology and branchlength approximations. Despite the Variational Bayesian Phylogenetic Inference (VBPI), a leading-edge black-box variational inference (BBVI) framework, achieving remarkable approximations of these distributions, the multimodality of the tree-topology posterior presents a formidable challenge to sampling-based learning techniques such as BBVI. Advanced deep learning methodologies such as normalizing flows and graph neural networks have been explored to refine the branch-length posterior approximation, yet efforts to ameliorate the posterior approximation over tree topologies have been lacking. As a result, VBPI-Mixtures is capable of capturing distributions over tree-topologies that VBPI fails to model. We deliver state-ofthe-art performance on difficult density estimation tasks across numerous real phylogenetic datasets. Phylogenetic inference has a wide range of applications in various fields, such as molecular evolution, epidemiology, ecology, and tumor progression, making it an essential tool for modern evolutionary research. Bayesian phylogenetics allows researchers to reason about uncertainty in their findings about the evolutionary relationship between species. The posterior distribution over phylogenetic trees given the species data is, however, challenging to infer, since the latent space is a Cartesian product of the discrete tree-topology space and the continuous branch-length space. Furthermore, the cardinality of the tree-topology space grows as a double factorial of the number of species (taxa), making the marginal likelihood computationally intractable in most interesting problem settings.

Sparse linear regression methods for high-dimensional data often assume that residuals have constant variance. When this assumption is violated, it can lead to bias in estimated coefficients, prediction intervals (PI) with improper length, and increased type I errors. We propose a heteroscedastic high-dimensional linear regression model through a partitioned empirical Bayes Expectation Conditional Maximization (H-PROBE) algorithm. H-PROBE is a computationally efficient maximum a posteriori estimation approach based on a Parameter-Expanded Expectation-Conditional-Maximization algorithm. It requires minimal prior assumptions on the regression parameters through plug-in empirical Bayes estimates of hyperparameters. The variance model uses a multivariate log-Gamma prior on coefficients that can incorporate covariates hypothesized to impact heterogeneity. The motivation of our approach is a study relating Aphasia Quotient (AQ) to high-resolution T2 neuroimages of brain damage in stroke patients. AQ is a vital measure of language impairment and informs treatment decisions, but it is challenging to measure and subject to heteroscedastic errors. It is, therefore, of clinical importance -- and the goal of this paper -- to use high-dimensional neuroimages to predict and provide PIs for AQ that accurately reflect the heterogeneity in residual variance. Our analysis demonstrates that H-PROBE can use markers of heterogeneity to provide narrower PI widths than standard methods without sacrificing coverage. Through extensive simulation studies, we exhibit that H-PROBE results in superior prediction, variable selection, and predictive inference than competing methods.

Driven by the enormous success of neural networks in a broad spectrum of machine learning applications, see Goodfellow et al. [16] and Schmidhuber [29] for an introduction, the theoretical understanding of network based methods is a dynamic and flourishing research area at the intersection of mathematical statistics, optimization and approximation theory. In addition to theoretical guarantees, uncertainty quantification is an important and challenging problem for neural networks and has motivated the introduction of Bayesian neural networks, where for each network weight a distribution is learned, see Graves [17] and Blundell et al. [8] and numerous subsequent articles. In this work we study the Gibbs posterior distribution for a stochastic neural network. In a nonparametric regression problem, we show that the corresponding estimator achieves a minimax-optimal prediction risk bound up to a logarithmic factor. Moreover, the method is adaptive with respect to the unknown regularity and structure of the regression function. While early theoretical foundations for neural nets are summarized by Anthony & Bartlett [4], the excellent approximation properties of deep neural nets, especially with the ReLU activation function, have been discovered in the last years, see e.g.