22 research outputs found
Variational bayes for estimating the parameters of a hidden Potts model
Hidden Markov random field models provide an appealing representation of images and other spatial problems. The drawback is that inference is not straightforward for these models as the normalisation constant for the likelihood is generally intractable except for very small observation sets. Variational methods are an emerging tool for Bayesian inference and they have already been successfully applied in other contexts. Focusing on the particular case of a hidden Potts model with Gaussian noise, we show how variational Bayesian methods can be applied to hidden Markov random field inference. To tackle the obstacle of the intractable normalising constant for the likelihood, we explore alternative estimation approaches for incorporation into the variational Bayes algorithm. We consider a pseudo-likelihood approach as well as the more recent reduced dependence approximation of the normalisation constant. To illustrate the effectiveness of these approaches we present empirical results from the analysis of simulated datasets. We also analyse a real dataset and compare results with those of previous analyses as well as those obtained from the recently developed auxiliary variable MCMC method and the recursive MCMC method. Our results show that the variational Bayesian analyses can be carried out much faster than the MCMC analyses and produce good estimates of model parameters. We also found that the reduced dependence approximation of the normalisation constant outperformed the pseudo-likelihood approximation in our analysis of real and synthetic datasets
Computing the Cramer-Rao bound of Markov random field parameters: Application to the Ising and the Potts models
This report considers the problem of computing the Cramer-Rao bound for the
parameters of a Markov random field. Computation of the exact bound is not
feasible for most fields of interest because their likelihoods are intractable
and have intractable derivatives. We show here how it is possible to formulate
the computation of the bound as a statistical inference problem that can be
solve approximately, but with arbitrarily high accuracy, by using a Monte Carlo
method. The proposed methodology is successfully applied on the Ising and the
Potts models.% where it is used to assess the performance of three state-of-the
art estimators of the parameter of these Markov random fields
A cross-center smoothness prior for variational Bayesian brain tissue segmentation
Suppose one is faced with the challenge of tissue segmentation in MR images,
without annotators at their center to provide labeled training data. One option
is to go to another medical center for a trained classifier. Sadly, tissue
classifiers do not generalize well across centers due to voxel intensity shifts
caused by center-specific acquisition protocols. However, certain aspects of
segmentations, such as spatial smoothness, remain relatively consistent and can
be learned separately. Here we present a smoothness prior that is fit to
segmentations produced at another medical center. This informative prior is
presented to an unsupervised Bayesian model. The model clusters the voxel
intensities, such that it produces segmentations that are similarly smooth to
those of the other medical center. In addition, the unsupervised Bayesian model
is extended to a semi-supervised variant, which needs no visual interpretation
of clusters into tissues.Comment: 12 pages, 2 figures, 1 table. Accepted to the International
Conference on Information Processing in Medical Imaging (2019
Hidden Gibbs random fields model selection using Block Likelihood Information Criterion
Performing model selection between Gibbs random fields is a very challenging
task. Indeed, due to the Markovian dependence structure, the normalizing
constant of the fields cannot be computed using standard analytical or
numerical methods. Furthermore, such unobserved fields cannot be integrated out
and the likelihood evaluztion is a doubly intractable problem. This forms a
central issue to pick the model that best fits an observed data. We introduce a
new approximate version of the Bayesian Information Criterion. We partition the
lattice into continuous rectangular blocks and we approximate the probability
measure of the hidden Gibbs field by the product of some Gibbs distributions
over the blocks. On that basis, we estimate the likelihood and derive the Block
Likelihood Information Criterion (BLIC) that answers model choice questions
such as the selection of the dependency structure or the number of latent
states. We study the performances of BLIC for those questions. In addition, we
present a comparison with ABC algorithms to point out that the novel criterion
offers a better trade-off between time efficiency and reliable results
Pre-processing for approximate Bayesian computation in image analysis
Most of the existing algorithms for approximate Bayesian computation (ABC)
assume that it is feasible to simulate pseudo-data from the model at each
iteration. However, the computational cost of these simulations can be
prohibitive for high dimensional data. An important example is the Potts model,
which is commonly used in image analysis. Images encountered in real world
applications can have millions of pixels, therefore scalability is a major
concern. We apply ABC with a synthetic likelihood to the hidden Potts model
with additive Gaussian noise. Using a pre-processing step, we fit a binding
function to model the relationship between the model parameters and the
synthetic likelihood parameters. Our numerical experiments demonstrate that the
precomputed binding function dramatically improves the scalability of ABC,
reducing the average runtime required for model fitting from 71 hours to only 7
minutes. We also illustrate the method by estimating the smoothing parameter
for remotely sensed satellite imagery. Without precomputation, Bayesian
inference is impractical for datasets of that scale.Comment: 5th IMS-ISBA joint meeting (MCMSki IV