F. Llorente

F. Llorente, L. Martino, J. Read et al.

This survey gives an overview of Monte Carlo methodologies using surrogate models, for dealing with densities which are intractable, costly, and/or noisy. This type of problem can be found in numerous real-world scenarios, including stochastic optimization and reinforcement learning, where each evaluation of a density function may incur some computationally-expensive or even physical (real-world activity) cost, likely to give different results each time. The surrogate model does not incur this cost, but there are important trade-offs and considerations involved in the choice and design of such methodologies. We classify the different methodologies into three main classes and describe specific instances of algorithms under a unified notation. A modular scheme which encompasses the considered methods is also presented. A range of application scenarios is discussed, with special attention to the likelihood-free setting and reinforcement learning. Several numerical comparisons are also provided.

L. Martino, F. Llorente, E. Curbelo et al.

We propose a novel adaptive importance sampling scheme for Bayesian inversion problems where the inference of the variables of interest and the power of the data noise is split. More specifically, we consider a Bayesian analysis for the variables of interest (i.e., the parameters of the model to invert), whereas we employ a maximum likelihood approach for the estimation of the noise power. The whole technique is implemented by means of an iterative procedure, alternating sampling and optimization steps. Moreover, the noise power is also used as a tempered parameter for the posterior distribution of the the variables of interest. Therefore, a sequence of tempered posterior densities is generated, where the tempered parameter is automatically selected according to the actual estimation of the noise power. A complete Bayesian study over the model parameters and the scale parameter can be also performed. Numerical experiments show the benefits of the proposed approach.

F. Llorente, E. Curbelo, L. Martino et al.

Monte Carlo sampling methods are the standard procedure for approximating complicated integrals of multidimensional posterior distributions in Bayesian inference. In this work, we focus on the class of Layered Adaptive Importance Sampling (LAIS) scheme, which is a family of adaptive importance samplers where Markov chain Monte Carlo algorithms are employed to drive an underlying multiple importance sampling scheme. The modular nature of LAIS allows for different possible implementations, yielding a variety of different performance and computational costs. In this work, we propose different enhancements of the classical LAIS setting in order to increase the efficiency and reduce the computational cost, of both upper and lower layers. The different variants address computational challenges arising in real-world applications, for instance with highly concentrated posterior distributions. Furthermore, we introduce different strategies for designing cheaper schemes, for instance, recycling samples generated in the upper layer and using them in the final estimators in the lower layer. Different numerical experiments, considering several challenging scenarios, show the benefits of the proposed schemes comparing with benchmark methods presented in the literature.

F. Llorente, L. Martino, D. Delgado et al.

Understanding systems by forward and inverse modeling is a recurrent topic of research in many domains of science and engineering. In this context, Monte Carlo methods have been widely used as powerful tools for numerical inference and optimization. They require the choice of a suitable proposal density that is crucial for their performance. For this reason, several adaptive importance sampling (AIS) schemes have been proposed in the literature. We here present an AIS framework called Regression-based Adaptive Deep Importance Sampling (RADIS). In RADIS, the key idea is the adaptive construction via regression of a non-parametric proposal density (i.e., an emulator), which mimics the posterior distribution and hence minimizes the mismatch between proposal and target densities. RADIS is based on a deep architecture of two (or more) nested IS schemes, in order to draw samples from the constructed emulator. The algorithm is highly efficient since employs the posterior approximation as proposal density, which can be improved adding more support points. As a consequence, RADIS asymptotically converges to an exact sampler under mild conditions. Additionally, the emulator produced by RADIS can be in turn used as a cheap surrogate model for further studies. We introduce two specific RADIS implementations that use Gaussian Processes (GPs) and Nearest Neighbors (NN) for constructing the emulator. Several numerical experiments and comparisons show the benefits of the proposed schemes. A real-world application in remote sensing model inversion and emulation confirms the validity of the approach.

F. Llorente, L. Martino, V. Elvira et al.

Numerical integration and emulation are fundamental topics across scientific fields. We propose novel adaptive quadrature schemes based on an active learning procedure. We consider an interpolative approach for building a surrogate posterior density, combining it with Monte Carlo sampling methods and other quadrature rules. The nodes of the quadrature are sequentially chosen by maximizing a suitable acquisition function, which takes into account the current approximation of the posterior and the positions of the nodes. This maximization does not require additional evaluations of the true posterior. We introduce two specific schemes based on Gaussian and Nearest Neighbors (NN) bases. For the Gaussian case, we also provide a novel procedure for fitting the bandwidth parameter, in order to build a suitable emulator of a density function. With both techniques, we always obtain a positive estimation of the marginal likelihood (a.k.a., Bayesian evidence). An equivalent importance sampling interpretation is also described, which allows the design of extended schemes. Several theoretical results are provided and discussed. Numerical results show the advantage of the proposed approach, including a challenging inference problem in an astronomic dynamical model, with the goal of revealing the number of planets orbiting a star.