Fast and scalable non-parametric Bayesian inference for Poisson point processes

We study the problem of non-parametric Bayesian estimation of the intensity function of a Poisson point process. The observations are $n$ independent realisations of a Poisson point process on the interval $[0,T]$. We propose two related approaches. In both approaches we model the intensity function as piecewise constant on $N$ bins forming a partition of the interval $[0,T]$. In the first approach the coefficients of the intensity function are assigned independent gamma priors, leading to a closed form posterior distribution. On the theoretical side, we prove that as $n\rightarrow\infty,$ the posterior asymptotically concentrates around the "true", data-generating intensity function at an optimal rate for $h$-H\"older regular intensity functions ($0 < h\leq 1$). In the second approach we employ a gamma Markov chain prior on the coefficients of the intensity function. The posterior distribution is no longer available in closed form, but inference can be performed using a straightforward version of the Gibbs sampler. Both approaches scale well with sample size, but the second is much less sensitive to the choice of $N$. Practical performance of our methods is first demonstrated via synthetic data examples. We compare our second method with other existing approaches on the UK coal mining disasters data. Furthermore, we apply it to the US mass shootings data and Donald Trump's Twitter data.Comment: 45 pages, 22 figure

Efficient Non-parametric Bayesian Hawkes Processes

In this paper, we develop an efficient nonparametric Bayesian estimation of the kernel function of Hawkes processes. The non-parametric Bayesian approach is important because it provides flexible Hawkes kernels and quantifies their uncertainty. Our method is based on the cluster representation of Hawkes processes. Utilizing the stationarity of the Hawkes process, we efficiently sample random branching structures and thus, we split the Hawkes process into clusters of Poisson processes. We derive two algorithms -- a block Gibbs sampler and a maximum a posteriori estimator based on expectation maximization -- and we show that our methods have a linear time complexity, both theoretically and empirically. On synthetic data, we show our methods to be able to infer flexible Hawkes triggering kernels. On two large-scale Twitter diffusion datasets, we show that our methods outperform the current state-of-the-art in goodness-of-fit and that the time complexity is linear in the size of the dataset. We also observe that on diffusions related to online videos, the learned kernels reflect the perceived longevity for different content types such as music or pets videos

Object Edge Contour Localisation Based on HexBinary Feature Matching

This paper addresses the issue of localising object edge contours in cluttered backgrounds to support robotics tasks such as grasping and manipulation and also to improve the potential perceptual capabilities of robot vision systems. Our approach is based on coarse-to-fine matching of a new recursively constructed hierarchical, dense, edge-localised descriptor, the HexBinary, based on the HexHog descriptor structure first proposed in [1]. Since Binary String image descriptors [2]– [5] require much lower computational resources, but provide similar or even better matching performance than Histogram of Orientated Gradient (HoG) descriptors, we have replaced the HoG base descriptor fields used in HexHog with Binary Strings generated from first and second order polar derivative approximations. The ALOI [6] dataset is used to evaluate the HexBinary descriptors which we demonstrate to achieve a superior performance to that of HexHoG [1] for pose refinement. The validation of our object contour localisation system shows promising results with correctly labelling ~86% of edgel positions and mis-labelling ~3%