Approximate Bayesian computation for the Ising/Potts model

Intro to ABC Simulation Study ABC Algorithms Ising/Potts model Image Analysis Conclusion
an introduction to
Approximate Bayesian Computation
Matt Moores
Research Fellow
Department of Statistics
University of Warwick
Warwick ML Club
June 12, 2017

Motivation
Inference for a parameter θ when it is:
impossible
or very expensive
to evaluate the likelihood p(y|θ)
ABC is a likelihood-free method for approximating
the posterior distribution
π(θ|y)
by generating pseudo-data from the model:
w ∼ f(·|θ)

Likelihood-free rejection sampler
Algorithm 1 Likelihood-free rejection sampler
1: Draw parameter value θ ∼ π(θ)
2: Generate w ∼ f(·|θ )
3: if w = y (the observed data) then
4: accept θ
5: end if
But if the observations y are continuous
(or the space y ∈ Y is enormous)
then P(w = y) ≈ 0
Tavar´e, Balding, Griﬃth & Donnelly (1997) Genetics 145(2)

ABC tolerance
accept θ if δ(w, y) <
where
> 0 is the tolerance level
δ(·, ·) is a distance function
(for an appropriate choice of norm)
Inference is more exact when is close to zero. but
more proposed θ are rejected
(tradeoﬀ between accuracy & computational cost)
Pritchard, Seielstad, Perez-Lezaun & Feldman (1999) Mol. Biol. Evol. 16(12)

Summary statistics
Computing δ(w, y) for w1, . . . , wn and y1, . . . , yn
can be very expensive for large n
Instead, compute summary statistics s(y)
e.g. suﬃcient statistics
(only available for exponential family)

Sufficient statistics
Fisher-Neyman factorisation theorem:
if s(y) is sufficient for θ
then p(y|θ) = f(y) g (s(y)|θ)
only applies to Potts, Ising, exponential random
graph models (ERGM)
otherwise, selection of suitable summary
statistics can be a very difficult problem

ABC rejection sampler
Algorithm 2 ABC rejection sampler
1: for all iterations t ∈ 1 . . . T do
2: Draw independent proposal θ ∼ π(θ)
4: if s(w) − s(y) < then
5: set θt ← θ
6: else
7: set θt ← θt−1
8: end if
9: end for
Approximates π(θ|y) by π (θ | s(w) − s(y) < )
Marin, Pudlo, Robert & Ryder (2012) Stat. Comput. 22(6)
Marin & Robert (2014) Bayesian Essentials with R §8.3

A trivial example
Gaussian with unknown variance:
y ∼ N(1, σ2
)
normalising constant:
Z(σ2
) = (2πσ2
)−n
2
natural conjugate prior:
π 1
σ2 ∼ Ga ν0
2
,
ν0ψ2
0
2
suﬃcient statistic:
s(y) = 1
n
n
i=1 (yi − 1)2
posterior is analytically tractable:
π 1
σ2 | y ∼ Ga ν0+n
2
,
ν0ψ2
0+ns(y)
2
∴ no need for ABC (nor MCMC) in practice

R code
π(τ|y)
Density
0 1 2 3 4 5 6 7
0.00.10.20.30.4
§
y ← rnorm (n=5, mean=1, sd=2/3)
n ← length ( y )
s sq ← sum(( y −1)ˆ2)/n
post nu ← nu0 + n
post ssd ← ( nu0 ∗ s0 ˆ2 + n∗ s sq )/2

now with ABC
π(τ)
0 1 2 3 4 5 6 7
0.00.51.01.5
πε(τ | δ(s(w), s(y)) < ε)
0 1 2 3 4 5 6 7
0.00.10.20.30.4
§
prop tau ← rgamma(10000 , nu0/2 , 0.5 ∗nu0∗ s0 ˆ2)
pseudo ← rnorm (n∗ 10000 , 1 , 1/ sqrt ( prop tau ))
pseudoMx ← matrix ( pseudo , nrow=10000, ncol=n)
pseudoSSD ← rowSums (( pseudoMx − 1)ˆ2)/n
ps norm ← abs ( pseudoSSD − s sq )
e p s i l o n ← sort ( ps norm ) [ 2 0 0 0 ]
prop keep ← prop tau [ ps norm <= e p s i l o n ]

choice of
0 1 2 3 4 5 6 7
0.00.20.40.6
(a) = 8.430
0 1 2 3 4 5 6 7
0.00.10.20.30.4
(b) = 1.427
0 1 2 3 4 5 6 7
0.00.20.40.6
(c) = 0.011
0 1 2 3 4 5 6 7
0.00.40.8
(d) = 0.001

concentration of measure
0 1 2 3 4
0.00.40.8
(a) n = 25, = 0.149
0 1 2 3 4
0.01.02.03.0
(b) n = 500, = 0.025
0 1 2 3 4
02468
(c) n = 104
, = 0.011
0 1 2 3 4
020406080100
(d) n = 106
, = 0.001

Metropolis-Hastings proposals
Algorithm 3 ABC-MCMC
1: Initialise θ0 ∼ π(θ)
2: for all iterations t ∈ 1 . . . T do
3: Draw proposal θ ∼ q(· | θt−1)
5: Draw u ∼ Unif(0, 1)
6: if u < π(θ )q(θt−1|θ )
π(θt−1)q(θ |θt−1) and s(w) − s(y) < then
7: set θt ← θ
8: else
9: set θt ← θt−1
10: end if
11: end for
Unfortunately, this algorithm is prone to getting ”stuck”
Marjoram, Molitor, Plagnol & Tavar´e (2003) PNAS 100(26)
Lee & Latuszy´nski (2014) Biometrika 101(3)

Sequential Monte Carlo
Algorithm 4 ABC-SMC
1: Draw N particles θi ∼ π0(θ)
2: Draw N × M sets of pseudo-data wi,m ∼ f(·|θi)
3: repeat
4: Adaptively select ABC tolerance t
5: Update importance weights λi for each particle
6: if eﬀective sample size (ESS) < Nmin then
7: Resample particles according to their weights
8: end if
9: Update particles using Metropolis-Hastings step
(with adaptive proposal bandwidth σ2
t )
10: until
naccept
N < 0.015 or t < 10−9 or t ≥ 100
Targets a sequence of distributions π t (θ | s(w) − s(y) < t)
such that 1 > 2 > · · · > T
Drovandi & Pettitt (2011) Biometrics 67(1)
Del Moral, Doucet & Jasra (2012) Stat. Comput. 22(5)

hidden Markov random ﬁeld
Joint distribution of observed pixel intensities
y = (y1, . . . , yn) ∈ Rn and latent labels
z = (z1, . . . , zn) ∈ {1, . . . , k}n:
Pr(y, z | µ, σ2
, β) = L(y|µ, σ2
, z)π(z|β) (1)
Additive Gaussian noise:
yi | zi =j
iid
∼ N µj, σ2
j (2)
Potts model:
π(zi | zi, β) =
exp {β i∼ δ(zi, z )}
k
j=1 exp {β i∼ δ(j, z )}
(3)
Potts (1952) Proceedings of the Cambridge Philosophical Society 48(1)
Winkler (2003) Image Analysis, Random Fields and MCMC Methods, 2nd
ed.

Inverse Temperature
(e) β = 0.1 (f) β = 0.5 (g) β = 0.85
(h) β = 0.95 (i) β = 1.005 (j) β = 1.15

Doubly-intractable posterior
p(β|z) =
C−1(β)eβS(z)π(β)
β C−1(β)eβS(z)π(dβ)
(4)
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)
(5)
S(z) is the suﬃcient statistic of the Potts model:
S(z) =
i∼ ∈E
δ(zi, z ) (6)
where E is the set of all unique neighbour pairs.

bayesImageS
An R package for Bayesian image segmentation
using the hidden Potts model:
RcppArmadillo for fast computation in C++
OpenMP for parallelism
§
l i b r a r y ( bayesImageS )
p r i o r s ← l i s t ("k"=3,"mu"=rep (0 ,3) , "mu.sd"=sigma ,
"sigma"=sigma , "sigma.nu"=c (1 ,1 ,1) , "beta"=c (0 ,3))
mh ← l i s t ( algorithm="pseudo" , bandwidth =0.2)
r e s u l t ← mcmcPotts ( y , neigh , block ,NULL,
55000 ,5000 , p r i o r s ,mh)
Eddelbuettel & Sanderson (2014) RcppArmadillo: Accelerating R with
high-performance C++ linear algebra. CSDA 71

Bayesian computational methods
bayesImageS supports methods for updating the latent labels z:
Chequerboard updating (Winkler 2003)
Swendsen-Wang (1987)
and also methods for updating the inverse temperature β:
Pseudolikelihood (Ryd´en & Titterington 1998)
Path Sampling (Gelman & Meng 1998)
Exchange Algorithm (Murray, Ghahramani & MacKay 2006)
Approximate Bayesian Computation (Grelaud et al. 2009)
Sequential Monte Carlo (SMC-ABC)
(Del Moral, Doucet & Jasra 2012)

Lake Menteith, Scotland
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0

ABC-SMC for the hidden Potts model
§
l i b r a r y ( bayess )
data ( Menteith )
y ← as . matrix ( Menteith )
l i b r a r y ( bayesImageS )
mask ← matrix (1 , nrow=nrow( y ) , ncol=ncol ( y ))
neigh ← getNeighbors (mask , c (2 ,2 ,0 ,0))
block ← getBlocks (mask , 2)
p r i o r s ← l i s t ( k=6, mu=rep (256/2 , 6) ,
mu. sd=rep (256/ 6 ,6) , sigma=rep (256/ 6 ,6) ,
sigma . nu=rep (6 , 6) , beta ← c (0 ,2))
r e s ← smcPotts ( as . vector ( y ) , neigh , block ,
param=l i s t ( npart =2000, nstat =5) , p r i o r s=p r i o r s )
6h 15min for 100 SMC iterations (N=2000, M=5)

ABC tolerance
SMC iteration
εt
0 20 40 60 80 100
02000400060008000100001200014000

approximate posterior ( t = 37.5)
β
Density
1.280 1.285 1.290 1.295 1.300 1.305 1.310 1.315
0204060
Equivalent to 120,801 iterations of the ABC rejection sampler

Eﬀective Sample Size
SMC iteration
ESS
0 20 40 60 80 100
800100012001400160018002000

image segmentation
(a) Original image (b) Potts labels

Summary
ABC is a method for likelihood-free inference
It enables inference for models that are
otherwise computationally intractable
Main components of ABC:
π(θ) proposal density for θ
f(·|θ) generative model for w
tolerance level
δ(·, ·) distance function
s(y) summary statistics

References I
M. Moores, A. N. Pettitt & K. Mengersen
Scalable Bayesian inference for the inverse temperature of a hidden Potts
model.
arXiv:1503.08066 [stat.CO], 2015.
M. Moores, C. C. Drovandi, K. Mengersen & C. P. Robert
Pre-processing for approximate Bayesian computation in image analysis.
Statistics & Computing 25(1): 23–33, 2015.
J.-M. Marin, P. Pudlo, C. P. Robert & R. Ryder
Approximate Bayesian computational methods.
Statistics & Computing, 22(6): 1167–80, 2012.
A. Grelaud, C. P. Robert, J.-M. Marin, F. Rodolphe & J.-F. Taly
ABC likelihood-free methods for model choice in Gibbs random ﬁelds.
Bayesian Analysis, 4(2): 317–36, 2009.
J.-M. Marin & C. P. Robert
Bayesian Essentials with R
Springer-Verlag, 2014.

References II
A. Lee & K. Latuszy´nski
Variance bounding and geometric ergodicity of Markov chain Monte Carlo
kernels for approximate Bayesian computation
Biometrika 101(3): 655–671, 2014.
P. Del Moral, A. Doucet & A. Jasra
An adaptive sequential Monte Carlo method for approximate Bayesian
computation.
Statistics & Computing, 22(5): 1009–20, 2012.
C. C. Drovandi & A. N. Pettitt
Estimation of Parameters for Macroparasite Population Evolution Using
Approximate Bayesian Computation
Biometrics 67(1): 225–233, 2011.
P. Marjoram, J. Molitor, V. Plagnol & S. Tavar´e
Markov chain Monte Carlo without likelihoods.
Proc. Natl Acad. Sci. USA, 100(26): 15324–15328, 2003.

References III
J. Pritchard, M. Seielstad, A. Perez-Lezaun & M. Feldman
Population Growth of Human Y Chromosomes: A Study of Y
Chromosome Microsatellites.
Mol. Biol. Evol. 16(12): 1791–98, 1999.
S. Tavaré, D. Balding, R, Griffiths & P. Donnelly
Inferring coalescence times from DNA sequence data.
Genetics, 145(2): 505–18, 1997.
R. B. Potts
Some generalized order-disorder transformations.
Proc. Cambridge Philosophical Society, 48(1): 106–109, 1952.
G. Winkler
Image analysis, random fields and Markov chain Monte Carlo methods
2nd
ed., Springer-Verlag, 2003.
D. Eddelbuettel
Seamless R and C++ integration with Rcpp
Springer-Verlag, 2013.

Approximate Bayesian computation for the Ising/Potts model

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