Ryan Prescott Adams
ABSTRACT
The inhomogeneous Poisson process is a point process that has varying intensity across its domain (usually time or space). For nonparametric Bayesian modeling, the Gaussian process is a useful way to place a prior distribution on this intensity. The combination of a Poisson process and GP is known as a Gaussian Cox process, or doubly-stochastic Poisson process. Likelihood-based inference in these models requires an intractable integral over an infinite-dimensional random function. In this paper we present the first approach to Gaussian Cox processes in which it is possible to perform inference without introducing approximations or finitedimensional proxy distributions. We call our method the Sigmoidal Gaussian Cox Process, which uses a generative model for Poisson data to enable tractable inference via Markov chain Monte Carlo. We compare our methods to competing methods on synthetic data and apply it to several real-world data sets.
- Adams, R. P., Murray, I., & MacKay, D. J. C. (2009). The Gaussian process density sampler. Advances in Neural Information Processing Systems 21 (pp. 9--16).Google Scholar
- Cox, D. R. (1955). Some statistical methods connected with series of events. Journal of the Royal Statistical Society, Series B, 17, 129--164.Google Scholar
- Cunningham, J., Shenoy, K., & Sahani, M. (2008a). Fast Gaussian process methods for point process intensity estimation. International Conference on Machine Learning 25 (pp. 192--199). Google ScholarDigital Library
- Cunningham, J., Yu, B., Shenoy, K., & Sahani, M. (2008b). Inferring neural firing rates from spike trains using Gaussian processes. Advances in Neural Information Processing Systems 20 (pp. 329--336).Google Scholar
- Diggle, P. (1985). A kernel method for smoothing point process data. Applied Statistics, 34, 138--147.Google ScholarCross Ref
- Duane, S., Kennedy, A. D., Pendleton, B. J., & Roweth, D. (1987). Hybrid Monte Carlo. Physics Letters B, 195, 216--222.Google ScholarCross Ref
- Gregory, P. C., & Loredo, T. J. (1992). A new method for the detection of a periodic signal of unknown shape and period. The Astrophysical Journal, 398, 146--168.Google ScholarCross Ref
- Heikkinen, J., & Arjas, E. (1999). Modeling a Poisson forest in variable elevations: a nonparametric Bayesian approach. Biometrics, 55, 738--745.Google ScholarCross Ref
- Jarrett, R. G. (1979). A note on the intervals between coal-mining disasters. Biometrika, 66, 191--193.Google ScholarCross Ref
- Kottas, A., & Sansóó, B. (2007). Bayesian mixture modeling for spatial Poisson process intensities, with applications to extreme value analysis. Journal of Statistical Planning and Inference, 137, 3151--3163.Google ScholarCross Ref
- Lewis, P. A. W., & Shedler, G. S. (1979). Simulation of a nonhomogeneous Poisson process by thinning. Naval Research Logistics Quarterly, 26, 403--413.Google ScholarCross Ref
- Møller, J., Syversveen, A. R., & Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scandinavian Journal of Statistics, 25, 451--482.Google ScholarCross Ref
- Murray, I., Ghahramani, Z., & MacKay, D. (2006). MCMC for doubly-intractable distributions. Uncertainty in Artificial Intelligence 22 (pp. 359--366).Google Scholar
- Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian processes for machine learning. Cambridge, MA: MIT Press. Google ScholarDigital Library
- Rathbun, S. L., & Cressie, N. (1994). Asymptotic properties of estimators for the parameters of spatial inhomogeneous Poisson point processes. Advances in Applied Probability, 26, 122--154.Google ScholarCross Ref
- Ripley, B. D. (1977). Modelling spatial patterns. Journal of the Royal Statistical Society, Series B, 39, 172--212.Google Scholar
Index Terms
- Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities
Recommendations
Efficient Bayesian inference of sigmoidal Gaussian Cox processes
We present an approximate Bayesian inference approach for estimating the intensity of a inhomogeneous Poisson process, where the intensity function is modelled using a Gaussian process (GP) prior via a sigmoid link function. Augmenting the model using a ...
Scalable nonparametric Bayesian inference on point processes with Gaussian processes
ICML'15: Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37In this paper we propose an efficient, scalable non-parametric Gaussian process model for inference on Poisson point processes. Our model does not resort to gridding the domain or to introducing latent thinning points. Unlike competing models that scale ...
Erlang mixture modeling for Poisson process intensities
AbstractWe develop a prior probability model for temporal Poisson process intensities through structured mixtures of Erlang densities with common scale parameter, mixing on the integer shape parameters. The mixture weights are constructed through ...
Comments