We propose a deep importance sampling method that is suitable for estimating rare event probabilities in high-dimensional problems. We approximate the optimal importance distribution in a general importance sampling problem as the pushforward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a squared tensor-train decomposition. The squared tensor-train decomposition provides a scalable ansatz for building order-preserving high-dimensional transformations via density approximations. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. To compute expectations over unnormalized probability distributions, we design a ratio estimator that estimates the normalizing constant using a separate importance distribution, again constructed via a composition of transformations in tensor-train format. This offers better theoretical variance reduction compared with self-normalized importance sampling, and thus opens the door to efficient computation of rare event probabilities in Bayesian inference problems. Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.

Detecting anomalies in multivariate time series(MTS) data plays an important role in many domains. The abnormal values could indicate events, medical abnormalities,cyber-attacks, or faulty devices which if left undetected could lead to significant loss of resources, capital, or human lives. In this paper, we propose a novel and innovative approach to anomaly detection called Bayesian State-Space Anomaly Detection(BSSAD). The BSSAD consists of two modules: the neural network module and the Bayesian state-space module. The design of our approach combines the strength of Bayesian state-space algorithms in predicting the next state and the effectiveness of recurrent neural networks and autoencoders in understanding the relationship between the data to achieve high accuracy in detecting anomalies. The modular design of our approach allows flexibility in implementation with the option of changing the parameters of the Bayesian state-space models or swap-ping neural network algorithms to achieve different levels of performance. In particular, we focus on using Bayesian state-space models of particle filters and ensemble Kalman filters. We conducted extensive experiments on five different datasets. The experimental results show the superior performance of our model over baselines, achieving an F1-score greater than 0.95. In addition, we also propose using a metric called MatthewCorrelation Coefficient (MCC) to obtain more comprehensive information about the accuracy of anomaly detection.