MCMC-driven importance samplers

Abstract

Monte Carlo methods are the standard procedure for estimating complicated integrals of multidimensional posterior distributions in Bayesian inference. In this work, we focus on LAIS, a class of adaptive importance samplers where Markov chain Monte Carlo (MCMC) algorithms are employed to drive an underlying multiple importance sampling (IS) scheme. Its power lies in the simplicity of the layered framework: the upper layer locates proposal densities by means of MCMC algorithms, whereas the lower layer handles the multiple IS scheme, in order to compute the final estimators. The modular nature of LAIS allows for different possible choices in the upper and lower layers, yielding a variety of different performance and computational costs. In this work, we propose different enhancements in order to increase the efficiency and reduce the computational cost, of both upper and lower layers. The different variants are essential if we aim to address computational challenges arising in real-world applications, such as highly concentrated posterior distributions (due to large amounts of data, etc.). Hamiltonian-driven importance samplers are presented and tested. 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. Numerical experiments show the benefits of the proposed schemes as compared to vanilla version of LAIS and other benchmark methods.

Rare Events via Cross-Entropy Population Monte Carlo

“…Langevin based PMC (SL-PMC) [14] and Hamiltonian AIS (HAIS) [15]. Further MCMC-driven IS techniques are discussed by Llorent et al [16]. An important realization discussed by Llorent et al is that it is unknown what distribution the MCMC methods should target.…”

Convergence rates for optimised adaptive importance samplers

“…We note that the use of Langevin dynamics within the AIS is explored before, see, e.g., Fasiolo et al (2018), Elvira and Chouzenoux (2019), Mousavi et al (2021), also see, Martino et al (2017b,a), Llorente et al (2021) for the use of Markov chain Monte Carlo (MCMC) based proposals. However, these ideas are distinct from our work, in the sense that they explore driving the parameters (or samples) w.r.t.…”

Iterative importance sampling with Markov chain Monte Carlo sampling in robust Bayesian analysis

“…The efficient importance sampling consists of selecting a proposal distribution (given a density kernel) using a least squares problem and then using the proposed distribution in an independent Metropolis Hasting sampling. Moreover, in (Llorente et al, 2021), a layered adaptive importance sampling algorithm is presented which combined MCMC algorithms with importance sampling and different strategies to re-use generated samples. The layered adaptive importance sampling algorithm generates samples using two layers.…”

“…The layered adaptive importance sampling algorithm generates samples using two layers. The upper layer generates samples using a MCMC algorithm which are later used in a multiple importance sampling scheme (lower layer) (Llorente et al, 2021). A recycling layered adaptive importance sampling scheme is presented which re-uses the samples from the upper layer in the lower layer (Llorente et al, 2021).…”

“…This scheme is similar to the method proposed in this paper. However, a key difference between the (Liesenfeld and Richard, 2008) and (Llorente et al, 2021) papers and this paper is that we re-use the MCMC samples from one run by weighting with a different prior, and that we use an efficient sample size for determining when to re-run the MCMC sampler, which leads to fewer MCMC runs to cover the hyperparameter space.…”