README.md

In Freguglia/mrf2dbayes: Bayesian Analysis of Markov Random Fields on 2d Lattices

mrf2dbayes

The goal of mrf2dbayes is to provide out-of-the-box implementations of Bayesian inference algorithms for Markov Random Fields on 2-dimensional lattices with pairwise interactions. It can be viewed as a Bayesian extension of the mrf2d package.

Installation

The development version from GitHub can be installed with:

# install.packages("devtools")
devtools::install_github("Freguglia/mrf2dbayes")

Functionalities

Likelihood approximations

Because the likelihood function for Markov Random Fields is unavailable due to an intractable normalizing constant, mrf2dbayes uses an approximate-posterior distribution approach, which substitutes the likelihood function for an approximation in the acceptance ratio of the Metropolis-Hastings algorithm steps.

log-Likelihood approximations are represented by llapprox object. These can be created with the homonym function llapprox() and passing a reference field, an interaction structure (mrfi object from mrf2d), an interaction family (also introduced in mrf2d), a method and its related additional arguments.

Approximation methods currently implemented are:

• "pseudo": Pseudolikelihood approximation.
• "gt": Monte-Carlo Likelihood approximation from Geyer & Thompson (1992).
• TODO: "wl": Wang-Landau algorithm.
• TODO: "cpseudo": Corrected Pseudolikelihood approach from …

library(mrf2dbayes)
# Example MRF from mrf2d
z <- mrf2d::Z_potts

lla <- llapprox(z, mrfi(1), "oneeach", method = "pseudo")

MRF parameters posterior sampling

mrfbayes() runs a Metropolis-Hastings algorithm to sample from the posterior distribution of the MRF parameters considering centered Gaussian priors and Gaussian transition kernels. It takes an observed (discrete) random field, llapprox object and the size of the chain (nsamples) as arguments, as well as other arguments to control parameters such as the prior distribution variance and the transition kernel variance.

metrop <- mrfbayes(z, lla, nsamples = 10000)

A mrfbayes_out object is returned. plot() and summary() methods are available.

plot(metrop)

summary(metrop)
#> # A tibble: 2 x 6
#> # Groups:   position [2]
#>   position interaction   q025   mean   q975     sd
#>   <chr>    <chr>        <dbl>  <dbl>  <dbl>  <dbl>
#> 1 (0,1)    different   -1.05  -1.02  -0.987 0.0153
#> 2 (1,0)    different   -0.993 -0.963 -0.934 0.0151

Hidden MRF Bayesian analysis

Considering a Gaussian mixture driven by a Hidden MRF, a Metropolis-Within-Gibbs approach is used to sample the hidden (latent) field, the parameters of the hidden MRF and the parameters of the emission distribution simultaneously.

A Gaussian prior is used for the means of the emission distribution and an Inverse-Gamma for the variances.

bold5000 <- mrf2d::bold5000
# Dummy discrete field as reference
dummy <- matrix(sample(0:4, 
                       prod(dim(bold5000)), replace = TRUE),
                nrow = nrow(bold5000), ncol = ncol(bold5000))

lla_bold <- llapprox(dummy, mrfi(1), "onepar", method = "pseudo")

hmrf_metrop <- hmrfbayes(bold5000, lla_bold, nsamples = 20000)

mrf2d::cplot(bold5000)

plot(hmrf_metrop, "pars")

plot(hmrf_metrop, "theta")

plot(hmrf_metrop, "zprobs")

Freguglia/mrf2dbayes documentation built on Jan. 19, 2024, 11:21 p.m.