There are many approaches to Bayesian computation with intractable likelihoods, including the exchange algorithm, approximate Bayesian computation (ABC), thermodynamic integration, and composite likelihood. These approaches vary in accuracy as well as scalability for datasets of significant size. The Potts model is an example where such methods are required, due to its intractable normalising constant. This model is a type of Markov random field, which is commonly used for image segmentation. The dimension of its parameter space increases linearly with the number of pixels in the image, making this a challenging application for scalable Bayesian computation. My talk will introduce various algorithms in the context of the Potts model and describe their implementation in C++, using OpenMP for parallelism. I will also discuss the process of releasing this software as an open source R package on the CRAN repository.

1. bayesImageS: an R package for Bayesian image analysis
Matt Moores
@MooresMt
https://mooresm.github.io/bayesImageS
NIASRA, University of Wollongong, NSW, Australia
useR! 2018
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2. Image Segmentation
Classifying the observed pixel intensities y := {yi }n
i=1 ∈ Rn
according to a set of latent labels z := {zi }n
i=1 ∈ [1, . . . , k]n
Assuming additive Gaussian noise:
yi | zi = j ∼ N µj, σ2
j
The resulting augmented likelihood is a mixture of Gaussians:
p(y, z | µ, σ2
, β) =
n
i=1
p yi | µzi , σ2
zi
p(zi | zi , β)
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Rydén & Titterington (1998) J. Comput. Graph. Stat. 7(2): 194–211.

3. Synthetic Data
z <- sample.int(3, 49, TRUE)
mu <- c(-1, 0, 1)
sigma <- rep(0.3, 3)
y <- rnorm(49, mu[z], sigma[z])
2
4
6
Y
−1
0
1
Z
0.0
0.2
0.4
0.6
density
Component
Means
−1
0
1
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4. Mixture Model
library(bayesImageS)
lPriors <- list(k=3, mu=c(-1,0,1), mu.sd=rep(0.25,3),
sigma=rep(0.5,3), sigma.nu=rep(5,3), lambda=rep(1,3))
mix.fit <- gibbsGMM(y, priors=lPriors)
mu[1] mu[2] mu[3]
0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000
0.4
0.8
1.2
1.6
−0.8
−0.4
0.0
0.4
−1.50
−1.25
−1.00
−0.75
−0.50
mu[1] mu[2] mu[3]
−1.4 −1.2 −1.0 −0.8 −0.6 −0.4 −0.8 −0.4 0.0 0.4 0.4 0.8 1.2 1.6
Estimated
True
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5. Pixel Classification
Ground Truth Classification
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6. Potts model
In many images, neighbouring pixels tend to share the same label.
This tendency can be represented using a hidden Markov random field:
p(zi | zi , β) =
exp{β i∼ δ(zi , z )}
k
j=1 exp{β i∼ δ(j, z )}
where:
β is the inverse temperature parameter,
i ∼ are the neighbours of pixel i,
and δ(x, y) is the Kronecker delta function.
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Potts (1952) Proceedings of the Cambridge Philosophical Society 48(1)

7. Inverse Temperature
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8. Synthetic Data
Using the Swendsen-Wang algorithm to simulate from the Potts model with known β:
mask <- matrix(1, nrow=7, ncol=7)
neigh <- getNeighbors(mask, c(2,2,0,0))
blocks <- getBlocks(mask, 2)
k <- 3
beta <- 0.88
res.sw <- swNoData(beta, k, neigh, blocks, niter=200)
z <- matrix(max.col(res.sw$z)[1:nrow(neigh)], nrow=nrow(mask))
y <- rnorm(49, mu[z], sigma[z])
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9. Fitting the Model
lPriors <- list(k=3, mu=c(-1,0,1), mu.sd=rep(0.25,3),
sigma=rep(0.5,3), sigma.nu=rep(5,3), beta=c(0,2))
mh <- list(algorithm="ex",bandwidth=0.001,auxiliary=200)
mrf.fit <- mcmcPotts(y, neigh, blocks, lPriors, mh, niter=500, nburn=250)
0.0
0.5
1.0
0 100 200 300 400 500
beta
beta
0.0 0.5 1.0
Estimated
True
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10. Pixel Classification
Ground Truth Classification
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11. Intractable Likelihood
p(β|z) =
C−1(β)eβS(z)π(β)
β C−1(β)eβS(z)π(dβ)
(1)
The normalising constant has computational complexity O(nkn), since it involves a sum over all
possible combinations of the labels z ∈ Z:
C(β) =
z∈Z
eβS(z)
(2)
S(z) is the sufficient statistic of the Potts model:
S(z) =
i∼ ∈E
δ(zi , z ) (3)
where E is the set of all unique neighbour pairs.
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12. Algorithms for β
The argument mh$alg selects which algorithm to use for sampling from p(β|z):
pseudolikelihood
path sampling (thermodynamic integration)
approximate exchange algorithm (AEA)
approximate Bayesian computation (ABC-MCMC and ABC-SMC)
Bayesian indirect likelihood (BIL)
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Moores, Drovandi, Mengersen & Robert (2015) Stat. Comput. 25(1): 23–33.
Moores, Pettitt & Mengersen (2015; v2 2018) arXiv:1503.08066 [stat.CO]

13. Satellite Remote Sensing
−3050000
−3045000
−3040000
−3035000
−3030000
490000 495000 500000 505000 510000 515000
−1.0 −0.5 0.0 0.5 1.0
NDVI
Frequency −1.0 −0.5 0.0 0.5 1.0
050000100000150000
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14. Image segmentation with k = 5 labels
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15. Posterior Distributions
Pixel Intensity
Density
−1.0 −0.5 0.0 0.5 1.0
0.00.51.01.52.0
1.193 1.194 1.195 1.196
0200400600800
N = 450 Bandwidth = 0.0001203Density
AEA
BIL
ABC
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16. Segmentation of Anatomical Structures
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Radiography courtesy of Cathy Hargrave, Radiation Oncology Mater Centre, Queensland Health

17. External Field Prior
p(zi |zi∼ , β, µ, σ2
, yi ) =
exp {αi,zi + π(αi,zi )}
k
j=1 exp {αi,j + π(αi,j)}
π(zi |zi∼ , β) (4)
Isotropic translation:
π(αi,j) = log



1
nj h∈j
φ ∆(h, i)|µ∆ = 1.2, σ2
∆ = 7.32



(5)
where
nj is the number of voxels in object j
h ∈ j are the voxels in object j
∆(u, v) is the Euclidean distance between the coordinates of pixel u and pixel v
µ∆, σ2
∆ are parameters that describe the level of spatial variability of the object j
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18. Patient- and Organ-Specific Prior
αi (prostate) ∼ MVN






0.1
−0.5
0.2


 ,



4.12 0 0
0 2.92 0
0 0 0.92






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19. Seminal Vesicles
αi (SV) ∼ MVN






1.2
−0.7
−0.9


 ,



7.32 0 0
0 4.52 0
0 0 1.92






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20. Gaussian Random Field
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21. Results
−300 −250 −200 −150
150200250300
right−left (mm)
posterior−anterior(mm)
−300 −250 −200 −150
150200250300
right−left (mm)
posterior−anterior(mm)
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Moores, Hargrave, Deegan, Poulsen, Harden & Mengersen (2015) CSDA 86: 27–41.

22. Summary
bayesImageS can be used for segmentation of 2D and 3D images:
Update labels using chequerboard Gibbs sampling or Swendsen-Wang
Posterior for β using the exchange algorithm, ABC, pseudolikelihood, path sampling, or BIL
Fast implementation using RcppArmadillo
Parallelism using OpenMP
PkgDown documentation: https://mooresm.github.io/bayesImageS
Version 0.5-2 currently available on CRAN: https://CRAN.R-project.org/package=bayesImageS
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