Posterior Exploration for Computationally Intensive Forward Models

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Posterior Exploration for Computationally Intensive Forward Models. Shane Reese, Dept of Statistics, BYU Dave Higdon, Statistical Sciences, LANL Dave Moulton, Applied Math, LANL Jasper Vrugt , Hydrology, LANL Colin Fox, Physics, Otago. Computer Models at Los Alamos National Laboratory.

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### Posterior Exploration for Computationally Intensive Forward Models

Shane Reese, Dept of Statistics, BYU

Dave Higdon, Statistical Sciences, LANL

Dave Moulton, Applied Math, LANL

Jasper Vrugt, Hydrology, LANL

Colin Fox, Physics, Otago

Computer Models at Los Alamos National Laboratory

100

discovery

CFD

101

hydrology

102

response surface

103

number of simulations

agent-based models

104

sensitivity analysis

calibration & prediction

cosmology

forward Monte Carlo

105

106+

MCMC

extreme physics

utilization

ocean circulation

Example: Electrical Impedance Tomography

Current I injected into one electrode, -I/15 extracted from remaining 15 electrodes.

Conductivity: 24x24 lattice of pixels, each with a resistance between 2.5 and 4.5. Here =4, ☐=3.

Pairwise Difference Image Priors
• More commonly:
• u(d) = -d2 (GMRF)
• u(d) = -|d| (L1 MRF)
• scene-based, or template prior

Prior realization with (β,s)=(.5,.3)

m = 24x24 lattice with

1st order neighborhoods

Posterior Distribution

Posterior realizations for conductivity obtained using MCMC

Algorithms:

• Single-site Metropolis (ssm)
• Multivariate random walk Metropolis (rwm)
• Differential evolution (DE-MCMC)
• Using fast, approximate forward models
• Delayed rejection Metropolis
• Posterior augmentation
Single Site Metropolis
• Doesn’t need very smart proposals
Single site Metropolis: traces of 3 pixels

Posterior mean image

Computational effort (simulator evaluations xm)

Multivariate Random Walk Metropolis
• Highly multivariate – single update changes all pixels
• Hard to choose Σz in high-dimensional settings
Differential Evolution MCMC
• Highly multivariate – single update changes all pixels
• Like RWM but Σz info held in the P copies
Differential Evolution MCMC

x’p=xp+γ(xq-xr)+z

xp

xq

x’p

xr

• Proposal direction depends on randomly chosen pair
• Similar (in spirit?) to Christen & Fox’s proposal
• DE-MCMC ter-Braak (2006)
Differential Evolution MCMC: traces of 3 pixels

DE-MCMC for chain 1

DE-MCMC for chain 2

DE-MCMC for chain 3

SSM for equivalent computational effort

Differential Evolution MCMC: posterior for blue pixel in each of the 400 chains
• Mean and sd from each of the P=400 chains
• Initialized using a large SSM run
MCMC Using Fast, Approximate Forward Models
• Coarsened representations
• Low-fidelity forward models

FB

Physical System

Observations

F2

F1

F0

FA

Delayed Acceptance Metropolis
• Pretests using the fast solver - from Christen & Fox (2004)
• Can be used along with any previous mcmc scheme
• Optimal acceptance rate?
Final Remarks
• Hard to beat single site Metropolis for this type of problem
• Using fast, approximate solvers can speed things up by a factor of 2-5.
• Currently working on embedding an augmented MCMC scheme within a multigrid solver
• Use of template/scene priors
• Didn’t account for modeling error here