Stochastic determination of matrix determinants.
Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations-matrices-acting on the data are often not accessible directly but are only represented indirectly in form of a computer routine. Such a routine implements the transformation a data vector undergoes under matrix multiplication. While efficient probing routines to estimate a matrix's diagonal or trace, based solely on such computationally affordable matrix-vector multiplications, are well known and frequently used in signal inference, there is no stochastic estimate for its determinant. We introduce a probing method for the logarithm of a determinant of a linear operator. Our method rests upon a reformulation of the log-determinant by an integral representation and the transformation of the involved terms into stochastic expressions. This stochastic determinant determination enables large-size applications in Bayesian inference, in particular evidence calculations, model comparison, and posterior determination.
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany and Ludwigs-Maximilians-Universität München, Geschwister-Scholl-Platz 1, D-80539 München, Germany.
Sebastian
S
Dorn
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany and Ludwigs-Maximilians-Universität München, Geschwister-Scholl-Platz 1, D-80539 München, Germany.
Torsten A
TA
Ensslin