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Pre-processing for Approximate Bayesian Computation in Image Analysis. Matt Moores Chris Drovandi Christian Robert Kerrie Mengersen. Context: Radiotherapy planning. Fast information synthesis. 1. Use a fan-beam CT to establish a treatment plan. - PowerPoint PPT Presentation
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Pre-processing for Approximate Bayesian Computation in Image
Analysis
Matt MooresChris Drovandi
Christian RobertKerrie Mengersen
Context: Radiotherapy planning
Fast informationsynthesis
1. Use a fan-beam CT to establish a treatment plan.
Less subject to artifacts induced by X-ray scatter or metal implants.
2. Use a cone-beam CT each day to determine any changes in the image and hence in the plan.
Less accurate but can be used in-situ.
• At present, all images are separately and manually inspected.• We want to automatically update the spatial prior information to
improve identification of features in images with low contrast-to-noise ratio.
• Represent the prior as an external field in a hidden Potts model of the image lattice: formulate the prior distribution of the latent pixel labels as a mixture of Gaussian fields, centred on the positions of the objects in a previous point in time.
Mixture of Gaussiansfan-beam cone-beam
Potts model
with informed external field
fan-beam external field
Image segmentationFan-beam CT Cone-beam CT
Simple Potts model External prior
Classification accuracy
Problem 1: imagescomputational challenges with big data
Problem 2: Potts modelDoubly intractable likelihood
analytically and com
putationally intractable
Problem 3: Potts modelEstimation of the inverse temperature
Option 1: PseudolikelihoodRyden and Titterington, 1998
• Approximate C( b ) using the product of the conditional densities p( xi | xi~l , b )
• Draw proposed values of b using a random walk and evaluate using a pseudolikelihood approximation to the Metropolis-Hastings ratio
Fast, but approximation error increases for large values of b
Option 2: Path samplingGelman and Meng, 1998
• Pre-compute E[S(x)] = Si~l d(xi ,xl) for fixed values of b• Use these values during model fitting to approximate E[S(x)|b
′ ]• Approximate the MH ratio using numerical integration
• Precomputation can be costly for large datasets, but output can be reused for multiple datasets of same size and number of mixture components.
0
]|)x([)}(/)(log{ 0
dSEZZ
Option 3: Exchange algorithmMurray, Gharamani and MacKay, 2006
• Use an auxiliary variable w drawn from p( xi | xi~l , b ) for proposed b’ so the partition function cancels out in the MH ratio:
• If b’ is drawn from a symmetric random walk then ratio of q’s = 1• If a uniform prior is used for b then ratio of p’s also cancels, so
log(r) = (b′ - b′′) S(x) + (b′′ - b′ ) S(w)
• Simulation of the auxiliary variable can be expensive (same dimension as latent labels x)
• Method is exact when perfect sampling is used to simulate w
)()}w(exp{)()}x(exp{)()|(
)()}w(exp{)()}x(exp{)()|(0000
000
zSZSq
zSZSq
Option 4: Approximations
• Cucala, Marin, Robert, Titterington (2009): approximate exchange algorithm using 500 iterations of Gibbs sampling for w, with reduced computational cost
• Fast [iterative, deterministic] approximations, eg:– Iterated conditional modes (ICM, Besag, 1986)– Variational Bayes (VB, McGrory, Titterington, Reeves, Pettitt, 2009)
but the problem of estimating the inverse temperature persists!– Path sampling and exchange algorithm estimate ratio of normalising
constants, ideal for MCMC, unsuitable for use in these algorithms– Pseudolikelihood directly approximates Z(b) so can be used in
combination with these algorithms
Software: BayesImageS
Comparison of methods
Comparison of no. pixels and elapsed time
Allocation of pixels was similar – Exchange algorithm was the most computationally expensive– Precomputation of E[S(z)| b ] for path sampling helped slightly– Pseudolikelihood was faster but produced a much higher
posterior estimate of b
Option 5: SMC-ABC
Still costly
Avoiding computational cost by pre-computation
The distribution of r(x)|q is independent of the data• By simulating pseudo-data for values of q, we can
create a mapping function f(q) to approximate E[r(x)| q ]
• This mapping function can be reused across multiple datasets, amortising its computational cost
• By mapping directly from qr(x), we avoid the need to simulate pseudo-data during model fitting
Sufficient statistics of the Potts model
Scalable SMC-ABC for the Potts model
Simulation study
Accuracy of posterior estimates of b
Pseudo-data Pre-computedb b
Algorithm
Post
erio
r esti
mat
e
Distribution of posterior sampling error for b
Pseudo-data Pre-computed
Erro
r
Algorithm
Improvement in runtime
Elapsed (wall clock) time CPU time
Pseudo-data Pre-computed Pseudo-data Pre-computed
Summary
Scalability of SMC-ABC can be improved by pre-computing an approximate mapping q r(x)
• Pre-computation tool 8 minutes on a 16 core Xeon server• Average runtime for SMC-ABC improved from 74.4 hours to
39 minutesThe mapping function represents the nonlinear, heteroskedastic
relationship between the parameter and the summary statistic.
The method handles noisy data. This is non-trivial.
Comments
1. Based on the pre-computation step, we draw the sufficient statistic from a Gaussian with mean and variance estimated from the pre-computations, then compare this with the representative observed sufficient statistic. This adds unnecessary noise. Instead, we could use the Gaussian density directly in the algorithm. This would avoid having to use ABC tolerances (hence removing that approximation effect entirely). We could still use SMC, but we would need to anneal the Gaussian likelihood in.
2. The method could be extended to multivariate applications, such as estimating both b and k for the hidden Potts model.
3. Most pixels don’t change allocation. 4. Careful consideration of spatial scale helps.
Finally, “so what”?
Further details