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LogGaussianCoxProcesses
ChiGroupMeetingFebruary23,2016
Outline
• Typicalmotivatingapplication
• IntroductiontoLGCPmodel
• Briefoverviewofinference
• Applicationsinmywork
…justgettingstartedwiththis…
Motivation
Image:Lloyd,etal.2015.ICML.
Outline
• Typicalmotivatingapplication
• IntroductiontoLGCPmodel
• Briefoverviewofinference
• Applicationsinmywork
WhatareGaussianprocesses?
Time
Value
WhatareGaussianprocesses?
Time
Value
likelihood prior
InfinitedimensionalGaussian
Posterior=Likelihood*priorMarginallikelihood
WhatareGaussianprocesses?
Time
Value
likelihood prior
InfinitedimensionalGaussian
WhatisaPoissonprocess?
Rateofpointappearance:λ(t)
Time
WhatisaPoissonprocess?
Rateofpointappearance:λ(t)
Time
Time
WhatisaLogGaussianCoxProcess?
…GaussianprocessmodulatedPoissonprocess
…sigmoidalGaussianCoxprocess
…doubly stochasticPoissonprocess
Time
WhatisaLogGaussianCoxProcess?
…GaussianprocessmodulatedPoissonprocess
…sigmoidalGaussianCoxprocess
…doubly stochasticPoissonprocess Coxprocess=inhomogeneous Poissonprocesswithstochasticintensity
Time
WhatisaLogGaussianCoxProcess?
…GaussianprocessmodulatedPoissonprocess
…sigmoidalGaussianCoxprocess
…doubly stochasticPoissonprocess Coxprocess=inhomogeneous Poissonprocesswithstochasticintensity
Time
WhatisaLogGaussianCoxProcess?
…GaussianprocessmodulatedPoissonprocess
…sigmoidalGaussianCoxprocess
…doubly stochasticPoissonprocess Coxprocess=inhomogeneous Poissonprocesswithstochasticintensity
Alwaysneedapositiveintensity,so…- Takeexponential- Sigmoid- Square
Time
Outline
• Typicalmotivatingapplication
• IntroductiontoLGCPmodel
• Briefoverviewofinference
• Applicationsinmywork
HowcanIdoinferencewiththissortofmodel?
IEB
Stats
Moremodelspecifications…
Numberofpointsinsideagivenspace-timeregion:
s
[Flaxman, etal.2015.“FastKronecker inferenceinGaussian processeswithnon-Gaussian likelihoods.” ICML.]
Moremodelspecifications…
Numberofpointsinsideagivenspace-timeregion:
Simplifybyintroducing aspatialgrid,
yi =countofpoints insidegridcelli
i
[Flaxman, etal.2015.“FastKronecker inferenceinGaussian processeswithnon-Gaussian likelihoods.” ICML.]
HowcanIdoinferencewiththissortofmodel?
Laplaceapproximation
Posteriordistribution,
HowcanIdoinferencewiththissortofmodel?
Laplaceapproximation
Posteriordistribution,
Gaussianapproximation
z0
HowcanIdoinferencewiththissortofmodel?
Laplaceapproximation
Posteriordistribution,
Gaussianapproximation
IfassumeaNormalcenteredatx0,andtakeTaylorseriesexpansionaroundx0,
mathworksouttoshowthatGaussianapproximationofdistribution is:posterior=
z0
HowcanIdoinferencewiththissortofmodel?
Laplaceapproximation
Posteriordistribution,
Gaussianapproximation
IfassumeaNormalcenteredatx0,andtakeTaylorseriesexpansionaroundx0,
mathworksouttoshowthatGaussianapproximationofdistribution is:posterior=
z0
Need:- Findmaximumofposterior- Hessian(-A)atmaximum
HowcanIdoinferencewiththissortofmodel?
Laplaceapproximation+ Kronecker methods
CandecomposeGPkernelasaproductofcovariancematrices(becauseongrid)
Needtodolotsofinversionsandlog-determinantswhendoingGPregression
Kronecker methodscanspeedthisupquiteabit
HowcanIdoinferencewiththissortofmodel?
Variational Bayes
MetropolisHastingsx2HamiltonianMonteCarlox2
[Adams, etal.2009.“TractablenonparametricBayesianinferenceinPoisson processeswithGaussianprocessintensities.” ICML.]
Nogrid required,butdoneedinducing points(…whichcanbestbesetonarectangulargrid)
[Lloyd, etal.2015.“Variational inferenceforGaussian processmodulatedPoissonprocesses” ICML.]
Sampling
Outline
• Typicalmotivatingapplications
• IntroductiontoLGCPmodel
• Briefoverviewofinference
• Applicationsinmywork
ReferencesofinterestTransformation Inference Notes
Adams,etal.2009.“TractablenonparametricBayesianinferenceinPoissonprocesseswithGaussianprocessintensities.”ICML.
Sigmoidfunction Multiplesamplingschemes
Flaxman,etal.2015.“FastKronecker inferenceinGaussianprocesseswithnon-Gaussianlikelihoods.”ICML.
Exponential Laplaceapproximation
Implemented inGPML:http://www.cs.cmu.edu/~andrewgw/pattern/
Gunter,etal.2014.“EfficientBayesiannonarametric modelling ofstructuredpointprocesses.”UAI.
Sigmoid function Manysamplingschemes
MultiplerealizationsfromlatentLCGP
Lloyd,etal.2015.“VariationalinferenceforGaussianprocessmodulatedPoissonprocesses”ICML.
Square Variational Bayes
That’sall…
