Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators

作者：Kekkonen Hanne; Lassas Matti; Siltanen Samuli

来源：Inverse Problems, 2016, 32(8): 085005.

DOI：10.1088/0266-5611/32/8/085005

• Bayesian inverse problem
• posterior consistency
• white noise
• convergence rate
• microlocal analysis

The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable U(x, omega) is M-delta(y, omega)= A (U(x, omega))+ delta epsilon (y, omega), where A is a finitely many orders smoothing linear hypoelliptic operator and delta > 0 is the noise magnitude. The covariance operator C-U of U is smoothing of order 2r, self-adjoint, injective and elliptic pseudodifferential operator. If epsilon was taking values in L-2 then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem T-delta (m(delta)) = arg min(u epsilon H)(r) {vertical bar vertical bar Au - m(delta)vertical bar vertical bar(2)(L2) + delta(2)vertical bar vertical bar C-U(-1/2) u vertical bar vertical bar(2)(L2) However, Gaussian white noise does not take values in L-2 but in H-s where s > 0 is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of the conditional mean estimate to the correct solution as delta -> 0 is proven in appropriate function spaces using microlocal analysis. Also the frequentist posterior contractions rates are studied.

• 出版日期2016-8