Modeling with Bounded Partition Functions

Ryan Prescott Adams

Description

Modeling with BoundedPartition FunctionsRyan Prescott AdamsCavendish LaboratoryUniversity of Cambridgehttp://www.inference.phy.cam.ac.uk/rpa23/16 July 2008Overall Talk MessageI Energy functions are nice models for data.I Inference in energy models is often hard.I If you can draw exact samples, you can doMCMC inference.I I have a trick for generating exact data frommany energy models.I This trick is probably a bad idea.OutlineMotivationExamples of Energy ModelsInferenceQuick Review of MCMCDoubly-Intractable Posterior DistributionsExchange SamplingConceptAuxiliary VariablesBaby and ToyExact Sampling from Energy ModelsOutlineMotivationExamples of Energy ModelsInferenceQuick Review of MCMCDoubly-Intractable Posterior DistributionsExchange SamplingConceptAuxiliary VariablesBaby and ToyExact Sampling from Energy ModelsEnergy-based Models of DataFor some spaceX , write an energy: E(x; )Turn this into a probability distribution via:1p(xj) = expf E(x; )gZ()Big energy implies small probability.ZNormalised byZ() = dx expf E(x; )g:XI Called the partition function.I Depends on the parameters.I Intractable in many interesting models.Examples of Energy-based ModelsExponential Family DistributionsTE(x; ) = T(x)+h(x)I Gaussian, Gamma, Poisson, etc.I Typically easy.Examples of Energy-based ModelsUndirected Graphical ModelsT TT T TE(x; ) = x Vx h Hh x Jh x h I Ising/Potts models, Boltzmann machinesI Perhaps hidden ...

Informations

• Publié par: Imko
• Nombre de lectures: 17
• Langue: English

Extrait

Modeling with Bounded Partition Functions

Ryan Prescott Adams

Cavendish Laboratory University of Cambridge http://www.inference.phy.cam.ac.uk/rpa23/

16 July 2008

Overall Talk Message

I I I

I

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Energy functions are nice models for data. Inference in energy models is often hard. If you can draw exact samples, you can do MCMC inference. I have a trick for generating exact data from many energy models. This trick is probably a bad idea.

Outline

Motivation Examples of Energy Models Inference Quick Review of MCMC Doubly-Intractable Posterior Distributions

Exchange Sampling Concept Auxiliary Variables Baby and Toy

Exact Sampling from Energy Models

Outline

Motivation Examples of Energy Models Inference Quick Review of MCMC Doubly-Intractable Posterior Distributions

Exchange Sampling Concept Auxiliary Variables Baby and Toy

Exact Sampling from Energy Models

Energy-based Models of Data

For some spaceX, write an energy:E(x;

Turn this into a probability distribution via:

p(x|θ) = Z( 1 θ) exp{−E(x;θ)}

θ)

θ)}:

Big energy implies small probability. Normalised byZ(θ) = Z X dxexp{−E(x; I Called thepartition function. I Depends on the parametersθ. I Intractable in many interesting models.

Examples of Energy-based Models

Exponential Family Distributions

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E(x;θ) =−θ T T(x) +h(x)

Gaussian, Gamma, Poisson, etc.

Typically easy.

Examples of Energy-based Models

Undirected Graphical Models

E(x;θ) =−x T V x−h T H h−x T J h−x T α−h T β

I Ising/Potts models, Boltzmann machines I Perhaps hidden unitsh. I Often with finite states. I Hard!

Examples of Energy-based

Nonparametric Models

I I I

E(x;θ) =g(x)

g(x)a nonparametric function. Logistic Gaussian process. Hard!

Models

Inference

GivenNdataD={x n } n N =1 , what isθ?

p(θ| D) dθp( p D (θ | ) θ) N = R p(θ) n Y p(x n |θ) =1

For interesting problems,θis often complex.

Use Markov chain Monte Carlo?

Quick Review of MCMC

We havep 0 (θ)∝p(θ)and want to draw samples.

Markov chain (MC): a stochastic rule for wandering around in the space ofθ.

MC can have anequilibrium distribution.

Simulate a MC for a while andθis close to being a sample from the equilibrium distribution.

Write down a rule usingp 0 (θ)so that the MC hasp(θ)as its equilibrium distribution.

Metropolis–Hastings

Metropolis–Hastings is a popular MCMC variant.

MH Markov Transition Rule Current state isθ. ˆ ˆ 1.Make a proposalθ∼q(θ←θ). 2.Evaluate theacceptance ratio:

ˆ ˆ a q(θ ˆ ←θ)p 0 (θ) = q(θ←θ)p 0 (θ)

ˆ 3.Acceptθwith probability min(a,1), otherwise keepθ.