Non-Linearities in Gaussian Processes with Integral Observations

Gaussian processes (GP) can be used for inferring latent continuous functions also based on aggregate observations corresponding to integrals of the function, for example to learn daily rate of new infections in a population based on cumulative observations collected only weekly. We extend these approaches to cases where the observations correspond to aggregates of arbitrary non-linear transformations of a GP. Such models are needed, for example, when the latent function of interest is known to be non-negative or bounded. We present a solution based on Markov chain Monte Carlo with numerical integration for aggregation, and demonstrate it in binned Poisson regression and in non-invasive detection of fouling using ultrasound waves.

1. Introduction

Gaussian processes (GP) provide a flexible basis for learning e.g. latent spatiotemporal functions based on noisy observations. In this work we concentrate in use of GPs to infer latent functions based on aggregated observations that relate to integrals of the function [1]–[3], also sometimes referred to as binned data [4]. Instead of directly observing a noisy realisation of the function itself, we observe a noisy average or sum of it over some, typically temporal or spatial, region. A prototypical application would be modeling a daily rate of incidences of a disease based on records of total number of new cases per week or month. In this context, GPs have been used for example to model malaria incidences and poverty rates on a finer scale based on data aggregated by administrative districts [1], [5], and computed tomography for reconstructing a 3D object based on signal attenuation along linear paths through the object [2], [6].