Max-and-Smooth: a two-step approach for approximate Bayesian inference in latent Gaussian models

This is the final version. Available on open access from International Society for Bayesian Analysis (ISBA) via the DOI in this record. With modern high-dimensional data, complex statistical models are necessary, requiring computationally feasible inference schemes. We introduce Max-and-Smooth, an approximate Bayesian inference scheme for a flexible class of latent Gaussian models (LGMs) where one or more of the likelihood parameters are modeled by latent additive Gaussian processes. Max-and-Smooth consists of two-steps. In the first step (Max), the likelihood function is approximated by a Gaussian density with mean and covariance equal to either (a) the maximum likelihood estimate and the inverse observed information, respectively, or (b) the mean and covariance of the normalized likelihood function. In the second step (Smooth), the latent parameters and hyperparameters are inferred and smoothed with the approximated likelihood function. The proposed method ensures that the uncertainty from the first step is correctly propagated to the second step. Since the approximated likelihood function is Gaussian, the approximate posterior density of the latent parameters of the LGM (conditional on the hyperparameters) is also Gaussian, thus facilitating efficient posterior inference in high dimensions. Furthermore, the approximate marginal posterior distribution of the hyperparameters is tractable, and as a result, the hyperparameters can be sampled independently of the latent parameters. In the case of a large number of independent data replicates, sparse precision matrices, and high-dimensional latent vectors, the speedup is substantial in comparison to an MCMC scheme that infers the posterior density from the exact likelihood function. The proposed inference scheme is demonstrated on one spatially referenced real dataset and on simulated data mimicking spatial, temporal, and spatio-temporal inference problems. Our results show that Max-and-Smooth is accurate and fast.NER

Convergence of sparse variational inference in gaussian processes regression

Gaussian processes are distributions over functions that are versatile and mathematically convenient priors in Bayesian modelling. However, their use is often impeded for data with large numbers of observations, N, due to the cubic (in N) cost of matrix operations used in exact inference. Many solutions have been proposed that rely on M << N inducing variables to form an approximation at a cost of O(NM^2). While the computational cost appears linear in N, the true complexity depends on how M must scale with N to ensure a certain quality of the approximation. In this work, we investigate upper and lower bounds on how M needs to grow with N to ensure high quality approximations. We show that we can make the KL-divergence between the approximate model and the exact posterior arbitrarily small for a Gaussian-noise regression model with M<<N. Specifically, for the popular squared exponential kernel and D-dimensional Gaussian distributed covariates, M=O((log N)^D) suffice and a method with an overall computational cost of O(N(log N)^{2D}(\log\log N)^2) can be used to perform inference

Scalable Inference for Markov Processes with Intractable Likelihoods

Bayesian inference for Markov processes has become increasingly relevant in recent years. Problems of this type often have intractable likelihoods and prior knowledge about model rate parameters is often poor. Markov Chain Monte Carlo (MCMC) techniques can lead to exact inference in such models but in practice can suffer performance issues including long burn-in periods and poor mixing. On the other hand approximate Bayesian computation techniques can allow rapid exploration of a large parameter space but yield only approximate posterior distributions. Here we consider the combined use of approximate Bayesian computation (ABC) and MCMC techniques for improved computational efficiency while retaining exact inference on parallel hardware

Bayesian optimisation for likelihood-free cosmological inference

Many cosmological models have only a finite number of parameters of interest, but a very expensive data-generating process and an intractable likelihood function. We address the problem of performing likelihood-free Bayesian inference from such black-box simulation-based models, under the constraint of a very limited simulation budget (typically a few thousand). To do so, we adopt an approach based on the likelihood of an alternative parametric model. Conventional approaches to approximate Bayesian computation such as likelihood-free rejection sampling are impractical for the considered problem, due to the lack of knowledge about how the parameters affect the discrepancy between observed and simulated data. As a response, we make use of a strategy previously developed in the machine learning literature (Bayesian optimisation for likelihood-free inference, BOLFI), which combines Gaussian process regression of the discrepancy to build a surrogate surface with Bayesian optimisation to actively acquire training data. We extend the method by deriving an acquisition function tailored for the purpose of minimising the expected uncertainty in the approximate posterior density, in the parametric approach. The resulting algorithm is applied to the problems of summarising Gaussian signals and inferring cosmological parameters from the Joint Lightcurve Analysis supernovae data. We show that the number of required simulations is reduced by several orders of magnitude, and that the proposed acquisition function produces more accurate posterior approximations, as compared to common strategies.Comment: 16+9 pages, 12 figures. Matches PRD published version after minor modification