In high dimensional models where the posterior log-likelihood involves a determinant, traditional methods for evaluating the log-likelihood may fail due to massive memory requirements. Log-likelihoods containing determinants occur, for instance, if we assume Gaussian error- or prior distributions. We present a novel approach for evaluating such log-likelihoods when the matrix-vector product, Qv, with the matrix of interest is fast to compute. In this approach we utilise matrix functions (Higham (2008)), Krylov subspaces (Saad (2003)), and probing vectors (Tang and Saad (2010)) to construct an iterative method for computing the log-likelihood. Moreover, we apply the method for estimating range, scale and anisotropy hyper-parameters in a seismic inversion problem (see Buland and Omre (2003)) where we assume an SPDE prior (see Lindgren et al. (2010)).
Buland, A. and Omre, H. (2003). Bayesian linearized avo inversion. Geophysics, 68:185 - 198.
Higham, N. J. (2008). Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
Lindgren, F., Lindstrom, J., and Rue, H. (2010). An explicit link between Gaussian fields and Gaussian Markov random fields: The SPDE approach. Journal of the Royal Statistical Society, Series B, to appear.
Saad, Y. (2003). Iterative Methods for Sparse Linear Systems, 2nd Ed. SIAM.
Tang, J. and Saad, Y. (2010). A probing method for computing the diagonal of the matrix inverse. Technical report, Minnesota Supercomputing Institute for Advanced Computational Research.
Keywords: High dimensional Gaussian distribution; Log-likelihood evaluation; Hyper-parameter estimation; Iterative method
Biography: Erlend Aune is a PhD-student at NTNU working on iterative methods for inference in high dimensional seismic inversion.