Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions that interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way.

Gaussian processes scale prohibitively with the size of the dataset. In response, many approximation methods have been developed, which inevitably introduce approximation error. This additional source of uncertainty, due to limited computation, is entirely ignored when using the approximate posterior. Therefore in practice, GP models are often as much about the approximation method as they are about the data. Here, we develop a new class of methods that provides consistent estimation of the combined uncertainty arising from both the finite number of data observed and the finite amount of computation expended. The most common GP approximations map to an instance in this class, such as methods based on the Cholesky factorization, conjugate gradients, and inducing points. For any method in this class, we prove (i) convergence of its posterior mean in the associated RKHS, (ii) decomposability of its combined posterior covariance into mathematical and computational covariances, and (iii) that the combined variance is a tight worst-case bound for the squared error between the method's posterior mean and the latent function. Finally, we empirically demonstrate the consequences of ignoring computational uncertainty and show how implicitly modeling it improves generalization performance on benchmark datasets.

Discovering governing equations from data is important to many scientific and engineering applications. Despite promising successes, existing methods are still challenged by data sparsity as well as noise issues, both of which are ubiquitous in practice. Moreover, state-of-the-art methods lack uncertainty quantification and/or are costly in training. To overcome these limitations, we propose a novel equation discovery method based on Kernel learning and BAyesian Spike-and-Slab priors (KBASS). We use kernel regression to estimate the target function, which is flexible, expressive, and more robust to data sparsity and noises. We combine it with a Bayesian spike-and-slab prior -- an ideal Bayesian sparse distribution -- for effective operator selection and uncertainty quantification. We develop an expectation propagation expectation-maximization (EP-EM) algorithm for efficient posterior inference and function estimation. To overcome the computational challenge of kernel regression, we place the function values on a mesh and induce a Kronecker product construction, and we use tensor algebra methods to enable efficient computation and optimization. We show the significant advantages of KBASS on a list of benchmark ODE and PDE discovery tasks.

In 1976, Lai constructed a nontrivial confidence sequence for the mean $\mu$ of a Gaussian distribution with unknown variance $\sigma$. Curiously, he employed both an improper (right Haar) mixture over $\sigma$ and an improper (flat) mixture over $\mu$. Here, we elaborate carefully on the details of his construction, which use generalized nonintegrable martingales and an extended Ville's inequality. While this does yield a sequential t-test, it does not yield an ``e-process'' (due to the nonintegrability of his martingale). In this paper, we develop two new e-processes and confidence sequences for the same setting: one is a test martingale in a reduced filtration, while the other is an e-process in the canonical data filtration. These are respectively obtained by swapping Lai's flat mixture for a Gaussian mixture, and swapping the right Haar mixture over $\sigma$ with the maximum likelihood estimate under the null, as done in universal inference. We also analyze the width of resulting confidence sequences, which have a curious dependence on the error probability $\alpha$. Numerical experiments are provided along the way to compare and contrast the various approaches.

The discovery of governing differential equations from data is an open frontier in machine learning. The sparse identification of nonlinear dynamics (SINDy) \citep{brunton_discovering_2016} framework enables data-driven discovery of interpretable models in the form of sparse, deterministic governing laws. Recent works have sought to adapt this approach to the stochastic setting, though these adaptations are severely hampered by the curse of dimensionality. On the other hand, Bayesian-inspired deep learning methods have achieved widespread success in high-dimensional probabilistic modeling via computationally efficient approximate inference techniques, suggesting the use of these techniques for efficient stochastic equation discovery. Here, we introduce HyperSINDy, a framework for modeling stochastic dynamics via a deep generative model of sparse governing equations whose parametric form is discovered from data. HyperSINDy employs a variational encoder to approximate the distribution of observed states and derivatives. A hypernetwork \citep{ha_hypernetworks_2016} transforms samples from this distribution into the coefficients of a differential equation whose sparse form is learned simultaneously using a trainable binary mask \citep{louizos_learning_2018}. Once trained, HyperSINDy generates stochastic dynamics via a differential equation whose coefficients are driven by a Gaussian white noise. In experiments, HyperSINDy accurately recovers ground truth stochastic governing equations, with learned stochasticity scaling to match that of the data. Finally, HyperSINDy provides uncertainty quantification that scales to high-dimensional systems. Taken together, HyperSINDy offers a promising framework for model discovery and uncertainty quantification in real-world systems, integrating sparse equation discovery methods with advances in statistical machine learning and deep generative modeling.

Recent transfer learning (TL) approaches in industrial intelligent fault diagnosis (FD) mostly follow the "pre-train and fine-tuning" paradigm to address data drift, which emerges from variable working conditions. However, we find that this approach is prone to the phenomenon known as catastrophic forgetting. Furthermore, performing frequent models fine-tuning on the resource-constrained edge nodes can be computationally expensive and unnecessary, given the excellent transferability demonstrated by existing models. In this work, we propose the Drift-Aware Weight Consolidation (DAWC), a method optimized for edge deployments, mitigating the challenges posed by frequent data drift in the industrial Internet of Things (IIoT). DAWC efficiently manages multiple data drift scenarios, minimizing the need for constant model fine-tuning on edge devices, thereby conserving computational resources. By detecting drift using classifier confidence and estimating parameter importance with the Fisher Information Matrix, a tool that measures parameter sensitivity in probabilistic models, we introduce a drift detection module and a continual learning module to gradually equip the FD model with powerful generalization capabilities. Experimental results demonstrate that our proposed DAWC achieves superior performance compared to existing techniques while also ensuring compatibility with edge computing constraints. Additionally, we have developed a comprehensive diagnosis and visualization platform.

Industrial artificial intelligence (IAI) stresses the application of artificial intelligence techniques to industries, with some inherent challenges, such as uncertainties in sensory signals, real-time data processing, high modelling accuracy, and the interpretability of predictive models and results [1-7]. Recently, the IAI concept has received considerable attention worldwide due to the availability of cheaper sensors for data acquisition, powerful computing facilities and advanced algorithms that perform speedily at lower computational cost, larger storage devices and cloud computing technology for data management, and faster communication systems for sharing and delivering data. Although the IAI concept is not well-defined so far, the development of advanced machine learning algorithms is strongly expected so that they can meet these requirements of IAI. Machine learning has been a very active research area in AI over the past decades, and significant efforts in building predictive learner models have been made [8]. Among these approaches, the most popular and widely used ones include multilayer perceptrons with error-backpropagation algorithms (MLPs) [9], support vector machines (SVMs) [10], Bayesian networks (BNs) [11], and adaptive neuro-fuzzy inference systems (ANFIS) [12].

Quality of Service (QoS) prediction is an essential task in recommendation systems, where accurately predicting unknown QoS values can improve user satisfaction. However, existing QoS prediction techniques may perform poorly in the presence of noise data, such as fake location information or virtual gateways. In this paper, we propose the Probabilistic Deep Supervision Network (PDS-Net), a novel framework for QoS prediction that addresses this issue. PDS-Net utilizes a Gaussian-based probabilistic space to supervise intermediate layers and learns probability spaces for both known features and true labels. Moreover, PDS-Net employs a condition-based multitasking loss function to identify objects with noise data and applies supervision directly to deep features sampled from the probability space by optimizing the Kullback-Leibler distance between the probability space of these objects and the real-label probability space. Thus, PDS-Net effectively reduces errors resulting from the propagation of corrupted data, leading to more accurate QoS predictions. Experimental evaluations on two real-world QoS datasets demonstrate that the proposed PDS-Net outperforms state-of-the-art baselines, validating the effectiveness of our approach.

Bayesian variable selection methods are powerful techniques for fitting and inferring on sparse high-dimensional linear regression models. However, many are computationally intensive or require restrictive prior distributions on model parameters. In this paper, we proposed a computationally efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are required through the use of plug-in empirical Bayes estimates of hyperparameters. Efficient maximum a posteriori (MAP) estimation is completed through a Parameter-Expanded Expectation-Conditional-Maximization (PX-ECM) algorithm. The PX-ECM results in a robust computationally efficient coordinate-wise optimization which -- when updating the coefficient for a particular predictor -- adjusts for the impact of other predictor variables. The completion of the E-step uses an approach motivated by the popular two-group approach to multiple testing. The result is a PaRtitiOned empirical Bayes Ecm (PROBE) algorithm applied to sparse high-dimensional linear regression, which can be completed using one-at-a-time or all-at-once type optimization. We compare the empirical properties of PROBE to comparable approaches with numerous simulation studies and analyses of cancer cell drug responses. The proposed approach is implemented in the R package probe.

Autoregressive large language models (LLMs) compress knowledge from their training data through next-token conditional distributions. This limits tractable querying of this knowledge to start-to-end autoregressive sampling. However, many tasks of interest -- including sequence continuation, infilling, and other forms of constrained generation -- involve sampling from intractable posterior distributions. We address this limitation by using amortized Bayesian inference to sample from these intractable posteriors. Such amortization is algorithmically achieved by fine-tuning LLMs via diversity-seeking reinforcement learning algorithms: generative flow networks (GFlowNets). We empirically demonstrate that this distribution-matching paradigm of LLM fine-tuning can serve as an effective alternative to maximum-likelihood training and reward-maximizing policy optimization. As an important application, we interpret chain-of-thought reasoning as a latent variable modeling problem and demonstrate that our approach enables data-efficient adaptation of LLMs to tasks that require multi-step rationalization and tool use.