Pre-processing for approximate Bayesian computation in image analysis

Most of the existing algorithms for approximate Bayesian computation (ABC) assume that it is feasible to simulate pseudo-data from the model at each iteration. However, the computational cost of these simulations can be prohibitive for high dimensional data. An important example is the Potts model, which is commonly used in image analysis. Images encountered in real world applications can have millions of pixels, therefore scalability is a major concern. We apply ABC with a synthetic likelihood to the hidden Potts model with additive Gaussian noise. Using a pre-processing step, we fit a binding function to model the relationship between the model parameters and the synthetic likelihood parameters. Our numerical experiments demonstrate that the precomputed binding function dramatically improves the scalability of ABC, reducing the average runtime required for model fitting from 71 hours to only 7 minutes. We also illustrate the method by estimating the smoothing parameter for remotely sensed satellite imagery. Without precomputation, Bayesian inference is impractical for datasets of that scale.Comment: 5th IMS-ISBA joint meeting (MCMSki IV

Scalable Bayesian model averaging through local information propagation

We show that a probabilistic version of the classical forward-stepwise variable inclusion procedure can serve as a general data-augmentation scheme for model space distributions in (generalized) linear models. This latent variable representation takes the form of a Markov process, thereby allowing information propagation algorithms to be applied for sampling from model space posteriors. In particular, we propose a sequential Monte Carlo method for achieving effective unbiased Bayesian model averaging in high-dimensional problems, utilizing proposal distributions constructed using local information propagation. We illustrate our method---called LIPS for local information propagation based sampling---through real and simulated examples with dimensionality ranging from 15 to 1,000, and compare its performance in estimating posterior inclusion probabilities and in out-of-sample prediction to those of several other methods---namely, MCMC, BAS, iBMA, and LASSO. In addition, we show that the latent variable representation can also serve as a modeling tool for specifying model space priors that account for knowledge regarding model complexity and conditional inclusion relationships

Bayesian Inference for partially observed SDEs Driven by Fractional Brownian Motion

We consider continuous-time diffusion models driven by fractional Brownian motion. Observations are assumed to possess a non-trivial likelihood given the latent path. Due to the non-Markovianity and high-dimensionality of the latent paths, estimating posterior expectations is a computationally challenging undertaking. We present a reparameterization framework based on the Davies and Harte method for sampling stationary Gaussian processes and use this framework to construct a Markov chain Monte Carlo algorithm that allows computationally efficient Bayesian inference. The Markov chain Monte Carlo algorithm is based on a version of hybrid Monte Carlo that delivers increased efficiency when applied on the high-dimensional latent variables arising in this context. We specify the methodology on a stochastic volatility model allowing for memory in the volatility increments through a fractional specification. The methodology is illustrated on simulated data and on the S&P500/VIX time series and is shown to be effective. Contrary to a long range dependence attribute of such models often assumed in the literature, with Hurst parameter larger than 1/2, the posterior distribution favours values smaller than 1/2, pointing towards medium range dependence

Sequential Empirical Bayes method for filtering dynamic spatiotemporal processes

We consider online prediction of a latent dynamic spatiotemporal process and estimation of the associated model parameters based on noisy data. The problem is motivated by the analysis of spatial data arriving in real-time and the current parameter estimates and predictions are updated using the new data at a fixed computational cost. Estimation and prediction is performed within an empirical Bayes framework with the aid of Markov chain Monte Carlo samples. Samples for the latent spatial field are generated using a sampling importance resampling algorithm with a skewed-normal proposal and for the temporal parameters using Gibbs sampling with their full conditionals written in terms of sufficient quantities which are updated online. The spatial range parameter is estimated by a novel online implementation of an empirical Bayes method, called herein sequential empirical Bayes method. A simulation study shows that our method gives similar results as an offline Bayesian method. We also find that the skewed-normal proposal improves over the traditional Gaussian proposal. The application of our method is demonstrated for online monitoring of radiation after the Fukushima nuclear accident

Nested Sequential Monte Carlo Methods

We propose nested sequential Monte Carlo (NSMC), a methodology to sample from sequences of probability distributions, even where the random variables are high-dimensional. NSMC generalises the SMC framework by requiring only approximate, properly weighted, samples from the SMC proposal distribution, while still resulting in a correct SMC algorithm. Furthermore, NSMC can in itself be used to produce such properly weighted samples. Consequently, one NSMC sampler can be used to construct an efficient high-dimensional proposal distribution for another NSMC sampler, and this nesting of the algorithm can be done to an arbitrary degree. This allows us to consider complex and high-dimensional models using SMC. We show results that motivate the efficacy of our approach on several filtering problems with dimensions in the order of 100 to 1 000.Comment: Extended version of paper published in Proceedings of the 32nd International Conference on Machine Learning (ICML), Lille, France, 201