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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

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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(1θ)exp{−E(x;θ)}

θ)

θ)}:

Big energy implies small probability. Normalised byZ(θ) =ZXdxexp{−E(x; ICalled thepartition function. IDepends on the parametersθ. IIntractable in many interesting models.

Examples of Energy-based Models

Exponential Family Distributions

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E(x;θ) =−θTT(x) +h(x)

Gaussian, Gamma, Poisson, etc.

Typically easy.

Examples of Energy-based Models

Undirected Graphical Models

E(x;θ) =−xTV x−hTH h−xTJ h−xTα−hTβ

IIsing/Potts models, Boltzmann machines IPerhaps hidden unitsh. IOften with ﬁnite states. IHard!

Examples of Energy-based

Nonparametric Models

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E(x;θ) =g(x)

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

Models

Inference

GivenNdataD={xn}nN=1, what isθ?

p(θ| D)dθp(pD(θ|)θ)N =Rp(θ)nYp(xn|θ) =1

For interesting problems,θis often complex.

Use Markov chain Monte Carlo?

Quick Review of MCMC

We havep0(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 usingp0(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 aq(θˆ←θ)p0(θ) = q(θ←θ)p0(θ)

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

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