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July, 2000 Bayesian and MaxEnt Workshop 1

Tutorial on

Markov Chain Monte Carlo

Kenneth M. Hanson

Los Alamos National Laboratory

This presentation available at http://public.lanl.gov/kmh/talks/

- Revised 14/05/08

LA-UR-05-5680

Presented at the 29

th

International Workshop on Bayesian

Inference and Maximum Entropy Methods in Science and

Technology, Gif-sur-Yvette, France, July 8 – 13, 2000

July, 2000 Bayesian and MaxEnt Workshop 2

Acknowledgements

• MCMC experts

– Julian Besag, Jim Guberantus, John Skilling, Malvin Kalos

• General discussions

– Greg Cunningham, Richard Silver

July, 2000 Bayesian and MaxEnt Workshop 3

Problem statement

• Parameter space of n dimensions represented by vector x

• Given an “arbitrary” target probability density function

(pdf), q(x), draw a set of samples {x

k

} from it

• Only requirement typically is that, given x, one be able to

evaluate Cq(x), where C is an unknown constant

– MCMC algorithms do not typically require knowledge of the

normalization constant of the target pdf; from now on the

multiplicative constant C will not be made explicit

• Although focus here is on continuous variables, MCMC

can be applied to discrete variables as well

July, 2000 Bayesian and MaxEnt Workshop 4

Uses of MCMC

• Permits evaluation of the expectation values of functions

of x, e.g.,

〈 f(x)〉 = ∫ f(x) q(x) dx ≅ (1/K) Σ

k

f(x

k

)

– typical use is to calculate mean 〈x〉 and variance 〈(x - 〈x〉)

2

〉

• Also useful for evaluating integrals, such as the partition

function for properly normalizing the pdf

• Dynamic display of sequences provides visualization of

uncertainties in model and range of model variations

• Automatic marginalization; when considering any subset

of parameters of an MCMC sequence, the remaining

parameters are marginalized over

July, 2000 Bayesian and MaxEnt Workshop 5

Markov Chain Monte Carlo

x

2

Probability(x

1

, x

2

)

accepted step

rejected step

x

1

• Metropolis algorithm:

– draw trial step from

symmetric pdf, i.e.,

t(Δ x) = t(-Δ x)

– accept or reject trial step

– simple and generally

applicable

– relies only on calculation

of target pdf for any x

Generates sequence of random samples from an

arbitrary probability density function

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