A Bayesian approach to wavelet-based modelling of discontinuous functions applied to inverse problems

Abstract

Inverse problems are examples of regression with more unknowns than the amount of information in the data and hence constraints are imposed through prior information. The proposed method defines the underlying function as a wavelet approximation which is related to the data through a convolution. The wavelets provide a sparse and multi-resolution solution which can capture local behaviour in an adaptive way. Varied prior models are considered along with level-specific prior parameter estimation. Archaeological stratigraphy data are considered where vertical earth cores are analysed producing clear piecewise constant function estimates.

Keywords:

• Archaeological stratigraphy
• elastic-net
• Haar wavelet
• hierarchical models
• Laplace distribution
• Markov chain Monte Carlo
• sparsity

Acknowledgments

The authors thank the Editor and anonymous referees for their constructive comments which resulted in this improved version.