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National Research Centerfor Environment & Health

Institute of Biomathematics& Biometry

Mathematical Modelling in Ecology and the Biosciences

Stochastic Simulation andBayesian Inference for Gibbs Fields:

The Software-Package ANTSInFields

Felix FriedrichLeipzig, 6.12.2002

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The Software Package ANTSInFields

ANTSInFields is a Software Package for Simulation and Statistical Inferenceon Gibbs Fields. It is intended for mainly two purposes: To support tea-ching by demonstrating well known sampling and estimation techniquesand for assistance in research.

ANTSInFieldsis available for download from http://www.antsinfields.de

contact: friedrich@antsinfields.de

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History

1995–1998:Development of Voyager

Portable and extensible System for simulation and data analysis

1998:Diploma thesis

Parameter estimation on Gibbs fields in the context of statistical ImageAnalysis

Need for implementation of• samplers for various Gibbs Fields (Ising Model and extensions)

• parameter estimators on simulated or external data.

today: ANTSInFields

Software for Simulation of and Statistical Inference on Gibbs Fields

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Aims / Requirements

Aims

• support teaching

• assistance for research

Requirements

• (really) easy to handle

• interactive visualization turned out to be efficient tool for teaching

• flexibility for research, testing new techniques etc.

• extensibility for implementing new samplers etc.

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Realization

• Strongly object oriented concept

allows implementation close to mathematical structureintuitive and self-explaining

easy implementation of interaction and consistent visualization

• modular design for extensibility, reusability

• command languagefor flexibility on intermediate level

ANTSInFields is written in OberonSystem 3 (ETH Zurich, N.Wirth, Gutknecht, H.Marais, E.Zelleret al.)

Oberon is also anOperating System. It runs on bare PC Hardware or as Emulation on Windows,MacOS, Linux,. . . (Portability)

ANTSInFields uses theVoyagerextension (University of Heidelberg, G.Sawitzki, M.Diller, F.Friedrich

et al.)

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Scope

ANTSInFields contains

• Handling and visualization of 1D, 2D and 3D data• Gibbs and Metropolis Hastings Algorithms, Simulated Annealing, Exact

Sampling (CFTP)• Bayesian image reconstruction methods• parameter estimators on Gibbs fields• . . .

ANTSInFields is attached to 2nd Edition of G. Winklers Book‘Image Analyis, Random Fields and Dynamic Monte Carlo Methods’, Sprin-ger Verlag

In progress: Meta compiler (alpha) for easy implementing of new models

Planned: Interface to R ( both command language and procedure calls).

Demonstration: Look and feel, Commands, Panels, Random Numbers

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Bayesian Image Restoration

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Random Fields

Notation

E finite space ofstates {•, •}

S ⊂ Zd finite index set

x = (xs)s∈S ∈ ES configuration

X := ES space of configurations { , . . . , , . . . , , . . . , }

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Gibbs Fields

We considerNeighbour-Gibbs fields

Π(x) =exp(−H(x))∑

y∈X exp(−H(y)),

whereH is of the form

H(x) =∑s∈S

f (xs, x∂(s))

with ∂(t) some neighbourhood oft, t ∈ S.

∂ =

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Ising Model

Easiest nontrivial case: E = {−1, 1}, nearest neighbours and isotropy.

An Ising Model with parameters β, h ∈ R is a Gibbs-Fieldwith energy

H(x) = −β∑s∼t

xsxt + h∑

s

xs

h: global tendency to take value1β: tendency of neighbours to be alike.

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Problems with sampling:

P(X = x) =exp(−H(x))∑

y∈X exp(−H(y))untractable,

but

P(Xt = xt|Xs = xs, s 6= t) =exp(xt(h + β

∑s∂t xs))

cosh(h + β∑

s∂t xs)easy to calculate

Solution→ MCMC techniques like

• Gibbs Sampler

• Metropolis Hastings Algorithm

• Exact Sampling

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Markov Chains

An irreducible Markov Chain with a stationary distribution µ is ergodicand fulfills

(a)P t(i, j)

t→∞−→ µ(j)

(b)fn

n→∞−→ Eµ(f (X))) in L2

ifEµ(f(X)) < ∞,

wherefn =

1

n

∑i=1...n

f(xi)

Demonstration: Reflected Markov Chain

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Algorithm for the Gibbs Sampler:

1. 0 7→ n

2. Samplex(0) from initial distribution, say uniform distribution on X

3. Apply Kt on x(n) for all t ∈ S

i.e. sample from local characteristicsΠ1 in each point

4. copyx(n) to x(n+1)

5. n + 1 7→ n

6. Return to step 3 until close enough toΠ.

Algorithm is a realization of a Markov chain with stationary distribution Π

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A sweep

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Visiting schedule

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Visiting schedule

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Visiting schedule:Whole sweep finished

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Demonstrations

• Ising Model, Gibbs Sampler

• Ising Model + Channel noise, MMSE

H(x, y) = −β∑s∼t

xsxt + h∑

s

xs +1

2ln(

p

1− p)∑

s

xsys

• Cooling Schemes, Simulated Annealing, ICM

• Grey-valued “Ising Model” (Potts and others)

H(x) = β∑s∼t

ϕ(xs, xt) + h∑

s

xs

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• Sampling from arbitrary Posterior Distributions

H(y, x) = β∑x∼t

ϕ(xs, xt) + h∑

s

xs +∑

s

ϑ(xs, ys),

• Φ-Model, Texture Synthesis

H(x) =∑

i

βi

∑s∼

it

ϕ(xs, xt) + h∑

s

xs

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Estimating (Hyper-)Parameters

Assume observations onΛ = Λ + ∂Λ

(Conditional) Maximum-Likelihood estimator (MLE)

θn := arg minθ− log(Pθ(Xt = xt, t ∈ Λ|Xs = xs, s ∈ ∂Λ))

= arg minθ

(log ZΛ(x∂Λ)−HΛ(xΛ | x∂Λ))

Problem: ZΛ not computable .Solution: Subsampling method (Younes(88), Winkler(01))

Alternative approach: Estimators regarding only the conditional distri-butions Pθ(Xt = xt|Xs, s ∈ ∂(t)) like:Coding, Maximum-Pseudolikelihood, Minimal least squares, MinimumChi Square estimator etc.

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Coding Estimator(CE)With

Λ+ = {t ∈ Λ | t1 + . . . + td even}the MLE on Λ+ becomes:

θn = arg minθ− log(P(Xt = xt, t ∈ Λ+|Xs = xs, s ∈ ∂Λ+))

= arg minθ− log

∑t∈Λ+

Pθ(Xt = xt|Xs, s ∈ ∂(t)).

Maximum-Pseudolikelihood Estimator (MPLE)Coding Estimator with replacement: Λ+ −→ Λ.

θn = arg minθ− log

∑t∈Λ

Pθ(Xt = xt|Xs, s ∈ ∂(t))

.

Demonstration: Estimating Parameters