Variational Prior Replacement in Bayesian Inference and Inversion

Many scientific investigations require that the values of a set of model parameters are estimated using recorded data. In Bayesian inference, information from both observed data and prior knowledge is combined to update model parameters probabilistically. Prior information represents our belief about the range of values that the variables can take, and their relative probabilities when considered independently of recorded data. Situations arise in which we wish to change prior information: (i) the subjective nature of prior information, (ii) cases in which we wish to test different states of prior information as hypothesis tests, and (iii) information from new studies may emerge so prior information may evolve over time. Estimating the solution to any single inference problem is usually computationally costly, as it typically requires thousands of model samples and their forward simulations. Therefore, recalculating the Bayesian solution every time prior information changes can be extremely expensive. We develop a mathematical formulation that allows prior information to be changed in a solution using variational methods, without performing Bayesian inference on each occasion. In this method, existing prior information is removed from a previously obtained posterior distribution and is replaced by new prior information. We therefore call the methodology variational prior replacement (VPR). We demonstrate VPR using a 2D seismic full waveform inversion example, where VPR provides almost identical posterior solutions compared to those obtained by solving independent inference problems using different priors. The former can be completed within minutes even on a laptop whereas the latter requires days of computations using high-performance computing resources. We demonstrate the value of the method by comparing the posterior solutions obtained using three different types of prior information.