Reverend Bayes, we have a problem - and a solution

Geoscientists often solve inverse problems to estimate values of parameters of interest given relevant data sets. Bayesian inference solves these problems by combining probability distributions that describe uncertainties in both observations and unknown parameters, and we require that the solution provides unbiased uncertainty estimates in order to inform evidence- or risk-based decisions. It has been known for over a century that employing different, but equivalent parametrisations of the same information can yield conditional probabilities that are mathematically inconsistent, a property referred to as the BK-inconsistency. Recently this inconsistency was shown to invalidate the solutions to physical problems found using several well-established methods of Bayesian inference. This talk explores the extent to which this inconsistency affects solutions to common geophysical problems. We demonstrate that changes in parametrisations result in inconsistent conditional prior probability densities, even though they represent exactly the same prior information. These inconsistent prior distributions can change Bayesian posterior solutions dramatically across various geoscientific problems including seismic impedance inversion, surface wave dispersion inversion, and travel time tomography, using real and synthetic data. Significantly different posterior statistics are obtained, including for maximum a posteriori (MAP) solutions, mean estimates, standard deviations, and full posterior distributions. Given that deterministic inversion is often equivalent to finding the MAP solution to specific Bayesian problems (the mathematical equations to be solved are identical), the BK-inconsistency also results in inconsistent solutions to deterministic inverse problems. Indeed, we show that solutions can potentially be designed, simply by changing the parametrisation. This study highlights that a careful rethinking of Bayesian inference and deterministic inversion may be required in physical problems, and we present one possible consistent method of solution.