We introduce bandit importance sampling (BIS), a new class of importance sampling methods designed for settings where the target density is expensive to evaluate. In contrast to adaptive importance sampling, which optimises a proposal distribution, BIS directly designs the samples through a sequential strategy that combines space-filling designs with multi-armed bandits. Our method leverages Gaussian process surrogates to guide sample selection, enabling efficient exploration of the parameter space with minimal target evaluations. We establish theoretical guarantees on convergence and demonstrate the effectiveness of the method across a broad range of sampling tasks. BIS delivers accurate approximations with fewer target evaluations, outperforming competing approaches across multimodal, heavy-tailed distributions, and real-world applications to Bayesian inference of computationally expensive models.

A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes' rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and filtering with a nonparametric statespace model. A consistency rate for the posterior estimate is established.

Probability density function estimation with weighted samples is the main foundation of all adaptive importance sampling algorithms. Classically, a target distribution is approximated either by a non-parametric model or within a parametric family. However, these models suffer from the curse of dimensionality or from their lack of flexibility. In this contribution, we suggest to use as the approximating model a distribution parameterised by a variational autoencoder. We extend the existing framework to the case of weighted samples by introducing a new objective function. The flexibility of the obtained family of distributions makes it as expressive as a non-parametric model, and despite the very high number of parameters to estimate, this family is much more efficient in high dimension than the classical Gaussian or Gaussian mixture families. Moreover, in order to add flexibility to the model and to be able to learn multimodal distributions, we consider a learnable prior distribution for the variational autoencoder latent variables. We also introduce a new pre-training procedure for the variational autoencoder to find good starting weights of the neural networks to prevent as much as possible the posterior collapse phenomenon to happen. At last, we explicit how the resulting distribution can be combined with importance sampling, and we exploit the proposed procedure in existing adaptive importance sampling algorithms to draw points from a target distribution and to estimate a rare event probability in high dimension on two multimodal problems.

Wasserstein-Fisher-Rao (WFR) distance is a family of metrics to gauge the discrepancy of two Radon measures, which takes into account both transportation and weight change. Spherical WFR distance is a projected version of WFR distance for probability measures so that the space of Radon measures equipped with WFR can be viewed as metric cone over the space of probability measures with spherical WFR. Compared to the case for Wasserstein distance, the understanding of geodesics under the spherical WFR is less clear and still an ongoing research focus. In this paper, we develop a deep learning framework to compute the geodesics under the spherical WFR metric, and the learned geodesics can be adopted to generate weighted samples. Our approach is based on a Benamou-Brenier type dynamic formulation for spherical WFR. To overcome the difficulty in enforcing the boundary constraint brought by the weight change, a Kullback-Leibler (KL) divergence term based on the inverse map is introduced into the cost function. Moreover, a new regularization term using the particle velocity is introduced as a substitute for the Hamilton-Jacobi equation for the potential in dynamic formula. When used for sample generation, our framework can be beneficial for applications with given weighted samples, especially in the Bayesian inference, compared to sample generation with previous flow models.

Given a set of samples, the parameters of a GM are determined in such a way as to best fit the samples in a maximum likelihood way. Solutions for equally weighted samples are readily available, expectation-maximization (EM) based methods being the most prevalent because of low computational requirements and ease of implementation. So it comes as a surprise that GM estimation for weighted samples is hard to find in literature. It might be even more surprising that the standard reference [1] gives incorrect results, see Figure 1. 2. Context Applications for sample-to-density function approximation include clustering of unlabled data [2, 3], multi-target tracking [4, 5], group tracking [6], multilateration [7, 8], and arbitrary density representation in nonlinear filters [9, 10]. A popular basic solution to this is the k-means algorithm. It does not find a complete density representation, only the means of the individual clusters. The k-means algorithm uses hard sample-tomean associations, therefore yields merely approximate solutions but can be computationally optimized using k-d trees [11, 12]. Moreover, the global optimum can be found deterministically [13], therefore it can be used to provide an initial guess for more elaborate algorithms. A sample-to-density approximation that is optimal in a maximum likelihood sense can be searched with numerical optimization techniques such as the Newton algorithm that has quadratic convergence but high computational demand per iteration, quasi-Newton methods, the method of scoring, or the conjugate gradient method with slower convergence but less computational effort per iteration [14].

Boosting refers to a type of classification and regression algorithms that enjoy large popularity due to their excellent predictive accuracy on a wide range of datasets. The first boosting algorithms for classification, including the well known AdaBoost algorithm, were introduced by Schapire [1990], Freund and Schapire [1995], and Freund et al. [1996]. Later, several authors [Breiman, 1998, 1999, Friedman et al., 2000, Mason et al., 2000, Friedman, 2001] introduced the statistical view of boosting as a stagewise optimization approach. In particular, Friedman et al. [2000] first introduced boosting algorithms which iteratively optimize Bernoulli and multinomial likelihoods for binary and multiclass classification using Newton updates. Further, Friedman [2001] presented gradient descent based boosting algorithms for both regression and classification tasks with general loss functions.

Importance Sampling (IS) is a well-known Monte Carlo technique that approximates integrals involving a posterior distribution by means of weighted samples. In this work, we study the assignation of a single weighted sample which compresses the information contained in a population of weighted samples. Part of the theory that we present as Group Importance Sampling (GIS) has been employed implicitly in different works in the literature. The provided analysis yields several theoretical and practical consequences. For instance, we discuss the application of GIS into the Sequential Importance Resampling framework and show that Independent Multiple Try Metropolis schemes can be interpreted as a standard Metropolis-Hastings algorithm, following the GIS approach. We also introduce two novel Markov Chain Monte Carlo (MCMC) techniques based on GIS. The first one, named Group Metropolis Sampling method, produces a Markov chain of sets of weighted samples. All these sets are then employed for obtaining a unique global estimator. The second one is the Distributed Particle Metropolis-Hastings technique, where different parallel particle filters are jointly used to drive an MCMC algorithm. Different resampled trajectories are compared and then tested with a proper acceptance probability. The novel schemes are tested in different numerical experiments such as learning the hyperparameters of Gaussian Processes, the localization problem in a wireless sensor network and the tracking of vegetation parameters given satellite observations, where they are compared with several benchmark Monte Carlo techniques. Three illustrative Matlab demos are also provided.

A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes' rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and filtering with a nonparametric state-space model. A consistency rate for the posterior estimate is established.