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Importance sampling (IS) methods are broadly used to approximate posterior distributions or their moments. In the standard IS approach, samples are drawn from a single proposal distribution and weighted adequately. However, since the performance in IS depends on the mismatch between the targeted and the proposal distributions, several proposal densities are often employed for the generation of samples. Under this multiple importance sampling (MIS) scenario, extensive literature has addressed the selection and adaptation of the proposal distributions, interpreting the sampling and weighting steps in different ways. In this paper, we establish a novel general framework with sampling and weighting procedures when more than one proposal is available. The new framework encompasses most relevant MIS schemes in the literature, and novel valid schemes appear naturally. All the MIS schemes are compared and ranked in terms of the variance of the associated estimators. Finally, we provide illustrative examples revealing that, even with a good choice of the proposal densities, a careful interpretation of the sampling and weighting procedures can make a significant difference in the performance of the method.

We propose a Monte Carlo algorithm to sample from high dimensional probability distributions that combines Markov chain Monte Carlo and importance sampling. We provide a careful theoretical analysis, including guarantees on robustness to high dimensionality, explicit comparison with standard Markov chain Monte Carlo methods and illustrations of the potential improvements in efficiency. Simple and concrete intuition is provided for when the novel scheme is expected to outperform standard schemes. When applied to Bayesian variable-selection problems, the novel algorithm is orders of magnitude more efficient than available alternative sampling schemes and enables fast and reliable fully Bayesian inferences with tens of thousand regressors.

This article describes a new adaptive Kriging method combined with adaptive importance sampling approximating the optimal auxiliary by iteratively building a Gaussian mixture distribution. The aim is to iteratively reduce both the modeling and sampling errors simultaneously, thus avoiding limitations in cases of very rare failure events. At each iteration, a near optimal auxiliary Gaussian distribution is defined and new samples are drawn from it following the scheme of adaptive multiple importance sampling (MIS). The corresponding estimator is provided as well as its variance. A new learning function is developed as a generalization of the U learning function for MIS populations. A stopping criterion is proposed based on both the modeling error and the variance of the estimator. Results on benchmark problems show that the method exhibits very good performances on both efficiency and accuracy.

Off-policy learning exhibits greater instability when compared to on-policy learning in reinforcement learning (RL). The difference in probability distribution between the target policy ( ) and the behavior policy (b) is a major cause of instability. High variance also originates from distributional mismatch. The variation between the target policy’s distribution and the behavior policy’s distribution can be reduced using importance sampling (IS). However, importance sampling has high variance, which is exacerbated in sequential scenarios. We propose a smooth form of importance sampling, specifically relative importance sampling (RIS), which mitigates variance and stabilizes learning. To control variance, we alter the value of the smoothness parameter in RIS. We develop the first model-free relative importance sampling off-policy actor-critic (RIS-off-PAC) algorithms in RL using this strategy. Our method uses a network to generate the target policy (actor) and evaluate the current policy ( ) using a value function (critic) based on behavior policy samples. Our algorithms are trained using behavior policy action values in the reward function, not target policy ones. Both the actor and critic are trained using deep neural networks. Our methods performed better than or equal to several state-of-the-art RL benchmarks on OpenAI Gym challenges and synthetic datasets.

An innovative adaptive neural network multisphere importance sampling (ANNM-IS) is proposed and integrated with symbiotic organism search (SOS) to form a framework for finding an engineering optimal design. Building a single sphere in IS to enhance the computational efficiency has been used for decades, ANNM-IS provides a pioneering idea, in which multi-spheres are built. “Adaptive point”, found by neural network (NN), is proposed to help for generating multiple spheres. ANNM-IS is further integrated with SOS to update NN for next iteration. As optimization iterations increase, adaptive NN provides more accurate reliability estimates. A two-step SOS, considering exploration and exploitation, is designed to enhance the search performance. Four reliability problem are first solved to confirm the correctness and effectiveness of ANNM-IS, then another four structural optimization problem including a building controller design and a 25-bar truss design are solved. Results shown that the proposed method drastically reduces the amount of function evaluation and computation time without sacrificing accuracy in reliability compared to those of other sampling methods. The developed framework can solve a complex structural optimization problem of accurate reliability with affordable price. The supporting source codes are available for download at https://github.com/johnthedy/RBDO-using-MIS-NN-SOS .

For uncertain structures with the coexisting random and interval inputs, effectively estimating the lower and upper bounds of failure probability is always a challenge. To address this issue, this paper first proposes a two-stage adaptive radial-based importance sampling (TARBIS) method, where two optimal spheres are sought successively in two stages to estimate the bounds of failure probability. Then, by replacing the true limit state function using the Kriging model, a Kriging-assisted TARBIS (K-TARBIS) is further developed to improve the computational efficiency. In the first stage, the training points mostly contributing to the estimation of two bounds of failure probability are identified by a system reliability theory-based U ( SYSU ) learning function to update the Kriging model. In the second stage, the Kriging model is updated only on sample points contributing to the estimation of the upper bound of failure probability. Throughout the active learning process, the Kriging model is sequentially updated in a series of small sub-candidate sample pools of TARBIS, which greatly reduces the computational cost. The accuracy and efficiency of the proposed method are demonstrated through four representative examples.

Time-dependent reliability and reliability sensitivity analysis in presence of random uncertainty is widespread in equipment structures. To this end, this paper establishes a sequentially quadratic surrogate method. Firstly, the global reliability sensitivity analysis (GRS) is transformed into the classification problem of the time-dependent performance function outputs by means of conditional probability formula. Secondly, referring to the strategy of the Meta-IS method, the Kriging model of time-dependent performance function is employed to construct the importance sampling function to generate the importance sampling (IS) samples of failure domain efficiently. Furthermore, the Kriging model is updated in the IS samples set through the single-loop adaptive Kriging method to realize the accurate identification of the failure indicator function of IS samples, as well as simulation of time-dependent failure probability. Finally, utilize the information of the failure samples obtained by the estimation of time-dependent reliability to evaluate GRS. The proposed algorithm has excellent computational efficiency and applicability due to the conversion of the conditional probability formula, which enables the computational consumption of the time-dependent reliability and GRS analysis independent of the dimensions of the inputs, as well as the Meta-IS method, which improves the sampling efficiency and is applicable to the case of complex implicit performance function. The given examples fully verify the conclusions.

This paper investigates adaptive importance sampling algorithms for which the policy, the sequence of distributions used to generate the particles, is a mixture distribution between a flexible kernel density estimate (based on the previous particles), and a “safe” heavy-tailed density. When the share of samples generated according to the safe density goes to zero but not too quickly, two results are established: (i) uniform convergence rates are derived for the policy toward the target density; (ii) a central limit theorem is obtained for the resulting integral estimates. The fact that the asymptotic variance is the same as the variance of an “oracle” procedure with variance-optimal policy, illustrates the benefits of the approach. In addition, a subsampling step (among the particles) can be conducted before constructing the kernel estimate in order to decrease the computational effort without altering the performance of the method. The practical behavior of the algorithms is illustrated in a simulation study.

Multiple importance sampling combines the probability density functions of several sampling techniques into an importance function. The combination weights are the proportion of samples used for the particular techniques. This paper addresses the determination of the optimal combination weights from a few initial samples. Instead of the numerically unstable optimization of the variance, in our solution the quasi-optimal weights are obtained by solving a linear equation, which leads to simpler computations and more robust estimations. The proposed method is validated with 1D numerical examples and with the direct lighting problem of computer graphics.

To ensure the reliability of complex structures, a novel reliability assessment method combining an active learning kriging (ALK) model with adaptive weighted importance sampling (AWIS), the ALK–AIWS, was proposed in this work. The initial design of experiment (DoE) points were first generated using a modified Metropolis algorithm to construct a kriging metamodel. The Markov chain state seeds were then used as the centers for the importance sampling density function to simulate the training data in a given important region. Thus, the kriging surrogate model was updated using the revised DoE produced by the active learning function, and the failure probability can be evaluated using the entire training data set. An AWIS method was also introduced considering the contribution of the design point to the structural failure probability. Finally, the failure probabilities of several numerical examples and a complex engineering design case were evaluated verifying the efficiency, accuracy, and applicability of the proposed ALK–AWIS method, which provides an alternative approach to reliability evaluation in practical engineering applications.