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Posterior Exploration for Computationally Intensive Forward Models

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

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  1. 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

  2. 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

  3. 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.

  4. 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

  5. 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

  6. Single Site Metropolis • Doesn’t need very smart proposals • Computationally demanding (m updates/scan)

  7. Single site Metropolis: traces of 3 pixels Posterior mean image Computational effort (simulator evaluations xm)

  8. Multivariate Random Walk Metropolis • Highly multivariate – single update changes all pixels • Hard to choose Σz in high-dimensional settings

  9. Random Walk Metropolis: traces of 3 pixels

  10. Differential Evolution MCMC • Highly multivariate – single update changes all pixels • Like RWM but Σz info held in the P copies

  11. 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)

  12. 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

  13. 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

  14. MCMC Using Fast, Approximate Forward Models • Coarsened representations • Low-fidelity forward models FB Physical System Observations F2 F1 F0 FA

  15. Delayed Acceptance Metropolis • Pretests using the fast solver - from Christen & Fox (2004) • Can be used along with any previous mcmc scheme • Optimal acceptance rate?

  16. Augmented Posterior

  17. 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

  18. Alternative prior formulations

  19. Example: Electrical Impedance Tomography

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