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Adaptive Markov Chain Monte Carlo methods

Introduction

In recent years, a Bayesian statistical approach to inverse problems, has gained popularity (see e.g. book by Kaipio and Somersalo). In the Bayesian formulation, both the data and the quantity that is estimated are considered as random variables. Instead of point estimates, the goal is to find 'all' values (the distribution) of the unknown parameters that explain the noisy data equally well.

Since the distribution of the parameters is often analytically intractable, one needs methods for approximating the distributions. In the past few decades, MCMC has become a standard computational method for performing Bayesian statistical analyses. In MCMC, a markovian random walk is constructed in the parameter space in such a way that the resulting steps are samples from the parameter distribution. MCMC is demonstrated in the figure below for a simple nonlinear model fitting example.


Left: Parameter distribution (blue) and Maximum a Posteriori estimate (black). Right: distribution of the model fit and prediction.

Adaptive MCMC algorithms, initially developed in (Haario et al. 2001) have been forerunners of an emerging class of more effective MCMC algorithms, now an increasingly topical area of Bayesian statistics. Adaptive MCMC methods learn from the previous model simulations and tune the algorithm as the simulation proceeds. The theoretical properties of this approach have recently received considerable interest, and numerous applications have been reported. These methods are gradually gaining wide usage in various application areas, such as chemical engineering, ecological and hydrological modelling and climate research.

Adaptive MCMC Basics

Real-life modelling problems often provide specific challenges for the application of MCMC methods: the problems are high-dimensional, and the number of unknown parameters depends on the numerical discretization of the problem. The problems may include massive, time-dependent data sets that preclude the use of fixed or hand--tuned standard methods, necessitating the use of adaptive MCMC algorithms.

The basic method behing our adaptive methods is the Metropolis-algorithm developed already in early 50s by Nicholas Metropolis. In the metropolis algorithm, one randomly suggests a new point in the parameter space, after which the goodness of the point evaluated, compared to the current point and either accepted or rejected. The performance of the algorithm is strongly dependent on the proposal mechanism for the new candidate points. In adaptive methods, the history of sampled points is used to automatically adjust the proposal distribution.

The idea of adaptation was first introduced in the form of the Adaptive Metropolis (AM) algorithm in (Haario, 2001). The componentwise version (SCAM) followed in (Haario et al. 2005) and the Delayed Rejection Adaptive Metropolis (DRAM) that combines two different tuning strategies was introduced in (Haario et al. 2006). Below, the effect of adaptation is demonstrated.


Left: no adaptation - bad mixing, Middle: AM starts to adapt slowly, Right: DRAM, fast adaptation.

Applications

Adaptive MCMC methods have gained ground in many application areas. One of our focuses have been environmental and geophysical models, for example recovering ozone profiles from satellite measurements, see (Haario et al. 2004) and studying algae dynamics in lakes, see (Malve et al. 2005).

Lately, we have taken on a the challenge of studying uncertainties related to climate models. All climate models contain so called closure parameters, with which the experts tune their models. Our goal is to study the effect of the uncertainties in these parameters in the long term climate predictions made with these models. For an introduction to this project, see (Järvinen et al. 2010).


Left: GOMOS instument. Right: Ozone profile inverted from star occultation measurements.

MCMCRUN - Matlab package for adaptive MCMC, written by Marko Laine.

Heikki Haario, Eero Saksman and Johanna Tamminen: An Adaptive Metropolis Algorithm, Bernoulli 7(2), pages 223-242, 2001.

Heikki Haario, Eero Saksman and Johanna Tamminen: Componentwise adaptation for high dimensional MCMC, Computational Statistics 20 (2005) 2, pp 265-274.

Heikki Haario, Marko Laine, Antonietta Mira, and Eero Saksman: DRAM: Efficient adaptive MCMC, Statistics and Computing, 16(4), pages 339-354, 2006.

H. Haario, M. Laine, M. Lehtinen, E. Saksman and J. Tamminen: Markov chain Monte Carlo methods for high dimensional inversion in remote sensing (with discussion), J.R. Statist. Soc. B, 66, part 3, pages 591-607, 2004.

Olli Malve, Marko Laine and Heikki Haario: Estimation of winter respiration rates and prediction of oxygen regime in a lake using Bayesian inference, Ecological Modelling, 182(2), pages 183-197, 2005.

Heikki Järvinen, Petri Räisänen, Marko Laine, Johanna Tamminen, Alexander Ilin, Erkki Oja, Antti Solonen, and Heikki Haario: Estimation of ECHAM5 climate model closure parameters with adaptive MCMC, Atmospheric Chemistry and Physics Discussions, May 2010.

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