A review of statistical modelling and inference for electrical capacitance tomography

Bayesian inference applied to electrical capacitance tomography, or other inverse problems, provides a framework for quantified model fitting. Estimation of unknown quantities of interest is based on the posterior distribution over the unknown permittivity and unobserved data, conditioned on measured data. Key components in this framework are a prior model requiring a parametrization of the permittivity and a normalizable prior density, the likelihood function that follows from a decomposition of measurements into deterministic and random parts, and numerical simulation of noise-free measurements. Uncertainty in recovered permittivities arises from measurement noise, measurement sensitivities, model inaccuracy, discretization error and a priori uncertainty; each of these sources may be accounted for and in some cases taken advantage of. Estimates or properties of the permittivity can be calculated as summary statistics over the posterior distribution using Markov chain Monte Carlo sampling. Several modified Metropolis-Hastings algorithms are available to speed up this computationally expensive step. The bias in estimates that is induced by the representation of unknowns may be avoided by design of a prior density. The differing purpose of applications means that there is no single 'Bayesian' analysis. Further, differing solutions will use different modelling choices, perhaps influenced by the need for computational efficiency. We solve a reference problem of recovering the unknown shape of a constant permittivity inclusion in an otherwise uniform background. Statistics calculated in the reference problem give accurate estimates of inclusion area, and other properties, when using measured data. The alternatives available for structuring inferential solutions in other applications are clarified by contrasting them against the choice we made in our reference solution.