The Effect of Prior Parameters in a Bayesian Approach to Inferring Material Properties from Experimental Measurements

Abstract

A Bayesian formulation for inference of material properties from experimental signals is proposed with a normal likelihood quantifying noise and an inverse gamma prior expressing uncertainty about noise variance. Asymptotic analysis shows that, for large samples, the posterior distribution is maximized at the database signal that minimizes an L2 norm between the experimental data and database entries. The largest posterior probability tends to 1 as the number of data points on the signals tends to ∞. There exists a critical ratio between the mode of the inverse gamma prior and the true variance of the normally distributed noise. This ratio determines how the posterior probability varies with the shape parameter of the inverse gamma prior. Numerical results based on an analytical expression and simulations confirm the conclusions from the asymptotic analysis. These results show that posterior probabilities are insensitive to the choice of prior mode when the prior’s shape parameter is taken to be small. In spite of the use of interpolation in database construction, the approach is limited to cases where a small number of parameters is to be identified. When multiple signal types are analyzed simultaneously, a method using a weighted average of posterior distributions from the different signal types is proposed. This method is compared with a classical Bayesian approach that uses a joint likelihood. Effective choices of prior parameters are recommended.