9 research outputs found
Approximate Inference for Nonstationary Heteroscedastic Gaussian process Regression
This paper presents a novel approach for approximate integration over the
uncertainty of noise and signal variances in Gaussian process (GP) regression.
Our efficient and straightforward approach can also be applied to integration
over input dependent noise variance (heteroscedasticity) and input dependent
signal variance (nonstationarity) by setting independent GP priors for the
noise and signal variances. We use expectation propagation (EP) for inference
and compare results to Markov chain Monte Carlo in two simulated data sets and
three empirical examples. The results show that EP produces comparable results
with less computational burden
Non-Gaussian Process Regression
Standard GPs offer a flexible modelling tool for well-behaved processes.
However, deviations from Gaussianity are expected to appear in real world
datasets, with structural outliers and shocks routinely observed. In these
cases GPs can fail to model uncertainty adequately and may over-smooth
inferences. Here we extend the GP framework into a new class of time-changed
GPs that allow for straightforward modelling of heavy-tailed non-Gaussian
behaviours, while retaining a tractable conditional GP structure through an
infinite mixture of non-homogeneous GPs representation. The conditional GP
structure is obtained by conditioning the observations on a latent transformed
input space and the random evolution of the latent transformation is modelled
using a L\'{e}vy process which allows Bayesian inference in both the posterior
predictive density and the latent transformation function. We present Markov
chain Monte Carlo inference procedures for this model and demonstrate the
potential benefits compared to a standard GP
String and Membrane Gaussian Processes
In this paper we introduce a novel framework for making exact nonparametric
Bayesian inference on latent functions, that is particularly suitable for Big
Data tasks. Firstly, we introduce a class of stochastic processes we refer to
as string Gaussian processes (string GPs), which are not to be mistaken for
Gaussian processes operating on text. We construct string GPs so that their
finite-dimensional marginals exhibit suitable local conditional independence
structures, which allow for scalable, distributed, and flexible nonparametric
Bayesian inference, without resorting to approximations, and while ensuring
some mild global regularity constraints. Furthermore, string GP priors
naturally cope with heterogeneous input data, and the gradient of the learned
latent function is readily available for explanatory analysis. Secondly, we
provide some theoretical results relating our approach to the standard GP
paradigm. In particular, we prove that some string GPs are Gaussian processes,
which provides a complementary global perspective on our framework. Finally, we
derive a scalable and distributed MCMC scheme for supervised learning tasks
under string GP priors. The proposed MCMC scheme has computational time
complexity and memory requirement , where
is the data size and the dimension of the input space. We illustrate the
efficacy of the proposed approach on several synthetic and real-world datasets,
including a dataset with millions input points and attributes.Comment: To appear in the Journal of Machine Learning Research (JMLR), Volume
1
Gaussian process product models for nonparametric nonstationarity
Stationarity is often an unrealistic prior as-sumption for Gaussian process regression. One solution is to predefine an explicit non-stationary covariance function, but such co-variance functions can be difficult to spec-ify and require detailed prior knowledge of the nonstationarity. We propose the Gaus-sian process product model (GPPM) which models data as the pointwise product of two latent Gaussian processes to nonparametri-cally infer nonstationary variations of ampli-tude. This approach differs from other non-parametric approaches to covariance function inference in that it operates on the outputs rather than the inputs, resulting in a signifi-cant reduction in computational cost and re-quired data for inference. We present an ap-proximate inference scheme using Expecta-tion Propagation. This variational approx-imation yields convenient GP hyperparame-ter selection and compact approximate pre-dictive distributions