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Tractable Nonparametric Bayesian Inference in Poisson Processes with Gaussian Process Intensity

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Tractable Nonparametric Bayesian Inference in Poisson Processes with Gaussian Process Intensity

by

Ryan P. Adams, Iain Murray, and David J.C. MacKay

(ICML 2009)

Presented by Lihan He

ECE, Duke University

July 31, 2009

Outline

- Introduction
- The model
- Poisson distribution
- Poisson process
- Gaussian process
- Gaussian Cox process
- Generating data from Gaussian Cox process

- Inference by MCMC
- Experimental results
- Conclusion

2/19

Introduction

Inhomogeneous Poisson process

- A counting process
- Rate of arrivals varies in time or space
- Intensity function (s)
- Astronomy, forestry, birth model, etc.

3/19

Introduction

How to model the intensity function (s)

- Using Gaussian process
- Nonparametrical approach
- Called Gaussian Cox process

Difficulty: intractable in inference

- Double-stochastic process
- Some approximation methods in previous research
- This paper: tractable inference
- Introducing latent variables
- MCMC inference – Metropolis-Hastings method
- No approximation

4/19

Model: Poisson distribution

Discrete random variable X has p.m.f.

for k = 0, 1, 2, …

- Number of event arrivals
- Parameter
- E[X] =
- Conjugate prior: Gamma distribution

5/19

Model: Poisson process

The Poisson process is parameterized by an intensity function

such that the random number of event within a subregion

is Poisson distributed with parameter

for k = 0, 1, 2, …

- N(0)=0
- The number of events in disjoint subregions are independent
- No events happen simultaneously
- Likelihood function

6/19

Model: Poisson process

Two-dimensional spatial Poisson process

One-dimensional temporal Poisson process

7/19

Model: Gaussian Cox process

Using Gaussian process prior for intensity function (s)

*: upper bound on (s)

σ : logistic function

g(s): random scalar function, drawn from a Gaussian process prior

8/19

Model: Gaussian process

Definition: Let g=(g(x1), g(x2), …, g(xN)) be an N-dimensional vector of function values evaluated at N points x1:N. P(g) is a Gaussian process if for any finite subset {x1, …, xN} the marginal distribution over that finite subset g has a multivariate Gaussian distribution.

- Nonparametric prior (without parameterizing g, as g=wTx)
- Infinite dimension prior (dimension N is flexible), but only need to work with finite dimensional problem
- Fully specified by the mean function and the covariance function

Mean function is usually defined to be zero

Example covariance function

9/19

Model: Generating data from Gaussian Cox process

Objective: generate a set of event {sk}k=1:K on some subregion T which are drawn from Poisson process with intensity function

10/19

Inference

Given a set of K event {sk}k=1:Kon some subregion T as observed data, what is the posterior distribution over (s)?

Poisson process likelihood function

Posterior

11/19

Inference

Augment the posterior distribution by introducing latent variables to make the MCMC-based inference tractable.

Observed data:

Introduced latent variables:

Total number of thinned events M

Locations of thinned events

Values of the function g(s) at the thinned events

Values of the function g(s) at the observed events

Complete likelihood

12/19

Inference

MCMC inference: sample

Sample M and: Metropolis-Hasting method

Metropolis-Hasting method: draw a new sample xt+1based on the last sample xt and a proposal distribution q(x’;xt)

Sample x’ from proposal q(x’; xt)

2.Compute acceptance ratio

3.Sample r~U(0,1)

4.If r<a, accept x’ as new sample, i.e., xt+1=x’; otherwise, reject x’, let xt+1=xt.

13/19

Inference

Sample M: Metropolis-Hasting method

Proposal distribution for inserting one thinned event

Proposal distribution for deleting one thinned event

Acceptance ratio for inserting one thinned event

Acceptance ratio for deleting one thinned event

14/19

m

Inference

Sample : Metropolis-Hasting method

Acceptance ratio for sampling a thinned event

Sample gM+K: Hamiltonian Monte Carlo method (Duane et al, 1987)

Sample *: place Gamma prior on *

Conjugate prior, the posterior can be derived analytically.

15/19

Experimental results

Synthetic data

53 events

29 events

235 events

16/19

Experimental results

Coal mining disaster data

191 coal mine explosions in British from year 1875 to 1962

17/19

Experimental results

Redwoods data

195 redwood locations

18/19

Conclusion

- Proposed a novel method of inference for the Gaussian Cox process that avoids the intractability of such model;
- Using a generative prior that allows exact Poisson data to be generated from a random intensity function drawn from a transformed Gaussian process;
- Using MCMC method to infer the posterior distribution of the intensity function;
- Compared to other method, having better result;
- Having significant computational demands: infeasible for data sets that have more than several thousand event.

19/19