Inference via low-dimensional couplings

Abstract

We investigate the low-dimensional structure of deterministic transformations between random variables, i.e., transport maps between probability measures. In the context of statistics and machine learning, these transformations can be used to couple a tractable "reference" measure (e.g., a standard Gaussian) with a target measure of interest. Direct simulation from the desired measure can then be achieved by pushing forward reference samples through the map. Yet characterizing such a map-e.g., representing and evaluating it-grows challenging in high dimensions. The central contribution of this paper is to establish a link between the Markov properties of the target measure and the existence of low-dimensional couplings, induced by transport maps that are sparse and/or decomposable. Our analysis not only facilitates the construction of transformations in high-dimensional settings, but also suggests new inference methodologies for continuous non-Gaussian graphical models. For instance, in the context of nonlinear state-space models, we describe new variational algorithms for filtering, smoothing, and sequential parameter inference. These algorithms can be understood as the natural generalization-to the non-Gaussian case-of the square-root Rauch-Tung-Striebel Gaussian smoother.

Localization for MCMC: sampling high-dimensional posterior distributions with local structure

“…More importantly, we have not rigorously defined localization for non-Gaussian problems, where enforcing bandedness of the covariance matrices may not be sufficient because higher moments become important. Taking inspiration instead from the sparsity of the precision matrix, a useful route may be to consider the conditional independence structure of more general non-Gaussian Markov random fields [56]. We hope to address these issues in future work.…”

Scalable Approximate Inference and Some Applications

“…There has been a collection of recent works, (such as Rezende & Mohamed, 2015;Kingma et al, 2016;Marzouk et al, 2016;Spantini et al, 2017), that approximate the target distributions with complex proposals obtained by iterative variable transforms in a similar way to our proposals q in (2.7). The key difference, however, is that these methods explicitly parameterize the transforms T and optimize the parameters by back-propagation, while our method, by leveraging the nonparametric nature of SVGD, constructs the transforms T sequentially in closed forms, requiring no back-propagation.…”

Multilevel Monte Carlo for Smoothing via Transport Methods

“…For instance, to obtain a MSE of O( 2) for some > 0 when approximating filtering distributions associated with Euler-discretized diffusions with constant diffusion coefficients, the cost of the PF is O( −3 ) while the cost of the MLPF is O( −2 log( ) 2 ). In this article we consider a new approach to replace the particle filter, using transport methods in [27]. In the context of filtering, one expects that the proposed method improves upon the MLPF by yielding, under assumptions, a MSE of O( 2 ) for a cost of O( −2 ).…”

“…The main idea in this article is to adopt an alternative method to the PF. The approach is to use transport methods [27]. Transport maps have been used for Bayesian inference [10,19] and more specifically for parameter estimation in [24] based on a related multi-scale idea.…”

“…The basic idea is to obtain a map such that the image of samples from an easy-to-sample distribution through this map has exactly the type which one desires. In [27] it is shown how to develop numerical approximations of maps, associated exactly to the distributions of interest in this article. These approximations often induce i.i.d.…”