Log Gaussian Cox Processes - University Of Oxford

Log Gaussian Cox ProcessesChi Group MeetingFebruary 23, 2016

Outline Typical motivating application Introduction to LGCP model Brief overview of inference Applications in my work just getting started with this

MotivationImage: Lloyd, et al. 2015. ICML.

Outline Typical motivating application Introduction to LGCP model Brief overview of inference Applications in my work

ValueWhat are Gaussian processes?Time

What are Gaussian processes?ValuePosterior Likelihood * priorMarginal ussian

ValueWhat are Gaussian ussian

What is a Poisson process?Rate of point appearance: λ(t)Time

What is a Poisson process?Rate of point appearance: λ(t)TimeTime

What is a Log Gaussian Cox Process? doubly stochastic Poisson process Gaussian process modulated Poisson process sigmoidal Gaussian Cox processTime

What is a Log Gaussian Cox Process? doubly stochastic Poisson process Gaussian process modulated Poisson process sigmoidal Gaussian Cox processCox process inhomogeneous Poissonprocess with stochasticintensityTime

What is a Log Gaussian Cox Process? doubly stochastic Poisson process Gaussian process modulated Poisson process sigmoidal Gaussian Cox processCox process inhomogeneous Poissonprocess with stochasticintensityTime

What is a Log Gaussian Cox Process? doubly stochastic Poisson process Gaussian process modulated Poisson process sigmoidal Gaussian Cox processCox process inhomogeneous Poissonprocess with stochasticintensityAlways need a positive intensity, so - Take exponential- Sigmoid- SquareTime

Outline Typical motivating application Introduction to LGCP model Brief overview of inference Applications in my work

How can I do inference with this sort of model?IEBStats

More model specifications Number of points inside a given spacetime region:s[Flaxman, et al. 2015. “Fast Kronecker inference in Gaussian processes with non-Gaussian likelihoods.” ICML. ]

More model specifications Number of points inside a given spacetime region:iSimplify by introducing a spatial grid,yi count of points inside grid cell i[Flaxman, et al. 2015. “Fast Kronecker inference in Gaussian processes with non-Gaussian likelihoods.” ICML. ]

How can I do inference with this sort of model?Laplace approximationPosterior distribution,

How can I do inference with this sort of model?Laplace approximationPosterior distribution,Gaussianapproximationz0

How can I do inference with this sort of model?Laplace approximationPosterior distribution,Gaussianapproximationz0If assume a Normal centered at x0,and take Taylor series expansion around x0,math works out to show that Gaussian approximation of distribution is:posterior

How can I do inference with this sort of model?Laplace approximationPosterior distribution,GaussianapproximationNeed:- Find maximum of posterior- Hessian (-A) at maximumz0If assume a Normal centered at x0,and take Taylor series expansion around x0,math works out to show that Gaussian approximation of distribution is:posterior

How can I do inference with this sort of model?Laplace approximation Kronecker methodsCan decompose GP kernel as a product of covariance matrices(because on grid)Need to do lots of inversions and logdeterminants when doing GP regressionKronecker methods can speed this up quite abit

How can I do inference with this sort of model?Variational BayesNo grid required, but do need inducing points( which can best be set on a rectangular grid)[Lloyd, et al. 2015. “Variational inference for Gaussian process modulated Poissonprocesses” ICML. ]SamplingMetropolis Hastings x 2Hamiltonian Monte Carlo x 2[Adams, et al. 2009. “Tractable nonparametric Bayesian inference in Poisson processeswith Gaussian process intensities.” ICML. ]

Outline Typical motivating applications Introduction to LGCP model Brief overview of inference Applications in my work

References of interestTransformationInferenceNotesAdams, et al. 2009. “Tractablenonparametric Bayesian inferencein Poisson processes with Gaussianprocess intensities.” ICML.Sigmoid functionMultiple samplingschemesFlaxman, et al. 2015. “FastKronecker inference in Gaussianprocesses with non-Gaussianlikelihoods.” ICML.ExponentialLaplaceapproximationImplemented inGPML:http://www.cs.cmu.edu/ andrewgw/pattern/Gunter, et al. 2014. “EfficientBayesian nonarametric modelling ofstructured point processes.” UAI.Sigmoid functionMany samplingschemesMultiple realizationsfrom latent LCGPLloyd, et al. 2015. “Variationalinference for Gaussian processmodulated Poisson processes”ICML.SquareVariational Bayes

That’s all

Laplace approximation Posterior distribution, Gaussian approximation If assume a Normal centered at x 0, and take Taylor series expansion around x 0, math works out to show that Gaussian approximation of distribution is: posterior z 0 Ne