Adap'ski, posters abstracts

INFERENCE ON DIFFUSION PROCESSES BY POPULATION MONTE CARLO METHOD

Roberto Casarin
CEREMADE, University Paris IX Dauphine, France and  University of Brescia, Italy

E-mail: casarin@ceremade.dauphine.fr


ABSTRACT
In this work we apply Bayesian simulation
based inference to the estimation of the parameters in the drift
and diffusion terms of a stochastic differential equation (SDE),
from discretely observed data. The method, here proposed, requires
the discretisation of the stochastic differential equation and
some assumptions on the prior distribution of the drift and
diffusion parameters. Parameter estimation is then obtained by
averaging the values simulated from posterior distribution by
sequential importance sampling. In particular we use the
Population Monte Carlo (PMC) method, which allows to improve the
basic importance sampling by resampling values with higher
probability and adapting the importance distribution to the
sampling dynamic. Finally we provide a comparison on synthetic
data, between traditional Gibbs sampling and PMC method, also
combining them with the high frequency data augmentation
principle.


Keywords: Population Monte Carlo, Gibbs Sampling, Sequential Importance
Sampling, Monte Carlo Bridging, Stochastic Differential Equation, Ito
Process, CEV Process.

Genetic algorithms and Markov Chain Monte Carlo: Differential Evolution Markov Chain makes Bayesian computing easy

Cajo J. F. ter Braak,
Biometris, Wageningen University and Research Centre, Box 100, 6700 AC Wageningen, The Netherlands.

E-mail: Cajo.terbraak@wur.nl

Differential Evolution (DE) is a simple genetic algorithm for numerical optimization in real parameter spaces. In a statistical context one would not just want the optimum but also its uncertainty. The uncertainty distribution can be obtained by a Bayesian analysis (after specifying prior and likelihood) using Markov Chain Monte Carlo (MCMC) simulation. Here the essential ideas of DE and MCMC are integrated into Differential Evolution Markov Chain (DE-MC). DE-MC is a population MCMC algorithm, in which multiple chains are run in parallel. DE-MC solves an important problem in MCMC, namely that of choosing an appropriate scale and orientation for the jumping distribution. In DE-MC the jumps are simply a multiple of the differences of two random parameter vectors that are currently in the population. Simulations and examples illustrate the potential of DE-MC. The advantage of DE-MC over conventional MCMC are simplicity, speed of calculation and convergence, even for nearly collinear parameters and multimodal densities. For the expert, DE-MC is a form of parallel adaptive direction sampling in which the Gibbs step in the parallel direction is replaced by a Metropolis step with a near optimal step size. A pdf of the paper is available at http://www.biometris.nl/Markov%20Chain.pdf.


Time-Stability Condition and Ergodicity of Adaptive MC

Krzysztof Latuszynski, Warsaw School of Economics, Warsaw, Poland


Ergodicity results for adaptive Monte Carlo algorithms usually assume "time-stability" of transition kernels. On the other hand, a large class of time-inhomogenous Markov Chains is ergodic. This suggests existance of adaptive MC algorithms which fail to satisfy the "time-stability" condition but are still ergodic. I present a simple modification of Atchadé-Rosenthal ergodicity Theorems (3.1 and 3.2 in [1]) that does not assume "time-stability" of transition kernels and provide some examples.

[1] Atchade Y. F., Rosenthal J. S. "On Adaptive Markov Chain Monte Carlo Algorithms"

A DISTANCE-BASED DIAGNOSTIC FOR TRANS-DIMENSIONAL MARKOV CHAINS

Yanan Fan and S.A. Sisson

Over the last decade the use of trans-dimensional sampling
algorithms has become endemic in the statistical literature. In
spite of their application however, very few attempts have been
made to assess the whether the chain has reached its stationary
distribution. Those methods that have been proposed tend to
associate convergence of at most a small number of marginal
parametric functionals of subsets of the parameter vector with
that of the full chain. In this article we present a
distance-based method for the comparison of trans-dimensional
Markov chain sample-paths for a broad class of models, that
naturally incorporates the full set of parameters for each visited
model, and an arbitrary number of ``marginal'' functionals.
Illustration of the analysis of Markov chain sample-paths is
presented in two common modelling situations: finite mixture
analysis and a change-point problem.

BAYESIAN RECONSTRUCTION OF LOW RESOLUTION MAGNETIC RESONANCE IMAGING MODALITIES

John Kornak, Karl Young, Norbert Schuff and Michael Weiner
University of California, San Francisco, USA

Magnetic resonance imaging (MRI) data are available in a large variety
of modalities that individually can provide anatomical, metabolic,
physiological, and functional descriptions of the brain. Unfortunately
not all of these modalities are attainable with high signal-to-noise
ratio (SNR) and hence some modalities are necessarily acquired at low
spatial resolution. This research aims to utilise information from the
high SNR modality of structural MRI in order to boost the effective
resolution of lower SNR MRI modalities.

Structural MRI of the brain provides high resolution maps of the
spatial structure of tissue that can readily be segmented into gray
matter, white matter and cerebro-spinal fluid. These maps are here
employed as prior information to enable improved resolution
reconstruction of lower SNR MRI modalities such as: Magnetic Resonance
Spectroscopy Imaging for measuring metabolite levels; Perfusion
Imaging for measuring blood flow; Diffusion Tensor Imaging for
measuring diffusion in tissue; and functional Magnetic Resonance
Imaging for measuring blood oxygenation levels and hence functional
activation.

A Markov random field (MRF) prior distribution relates the segmented
tissue maps with the imaging modality to be reconstructed by
describing the expected behaviour of the signal within the different
homogeneous tissue regions, as well as the behaviour across tissue
boundaries. The likelihood relates the low-resolution imaging
modality's signal (observed in Fourier/$k$-space) to the map to be
reconstructed in standard image space. The ensuing posterior
distribution is then optimised via Markov chain Monte Carlo (MCMC)
methods in order to obtain reconstructed maps at the resolution of the
sMRI. However, the complex dependence structure relating every point
in Fourier-space to every point in image space leads to high computational
burden for MCMC sampling.



IMPROVED EFFICIENCY IN APPROXIMATE BAYESIAN COMPUTATION

Scott A. Sisson and Paola Borto and Stuart G. Coles

Recently a number of rejection-based algorithms have been proposed for
approximating the posterior distribution when it is impossible, or
computationally prohibitive to evaluate the likelihood
function. Observed data are instead compared with the parameter set
via a surrogate data set, simulated according to a known process.
The utility of these algorithms has been improved by the adoption of a
sampling scheme based on a Markov chain \cite{marjoram} (see
\cite{beamont} for an alternative approach), however the resulting
sampler can be inefficient.

Here we present an extension to the Markov chain
approach based on an augmented state space, in direct analogy with the
simulated tempering algorithm, that substantially improves sampler
efficiency \cite{bortot}. We demonstrate the proposed methodology on a stereological inclusions
analysis.


Marjoram, P., J. Molitor, V. Plagnol and S. Tavar\'e. Markov chain Monte Carlo without likelihoods. PNAS, 100: 15324--15328, 2003.
Beamont, M. A., W. Zhang and D. J. Balding. Approximate Bayesian computation in population genetics. Genetics, 162: 2025--2035, 2002.
Bortot, P., S. G. Coles, and S. A. Sisson. Inference for stereological extremes. Technical Report, 2004.