The Bayesian Approach to Inverse Problems

These notes give various references to the literature that I will not, for reasons of brevity, give during the lectures themselves. 1. Gaussian Measures A comprehensive reference for Gaussian measures is [Bog98]. This book works in the generality of locally convex topological spaces, and has material on Gaussian measure on Banach and Hilbert spaces as particular cases. A similar book, focussed on the setting in Banach spaces, is [Lif95]. For a simpler introduction to the Hilbert space setting see the relevant material in [DZ92]. For a discussion of Radon measures, and statement that a probability measure on a seprable Hilbert space, with the Borel sigma algebra, is Radon, see [Bog98], Theorem A3.11. The statement that any two Radon measures with equal Fourier transform coincide is Theorem A3.18 in [Bog98]. Theorem 1.2 from our notes is Theorem 2.3.1 in [Bog98]. Regularity of functions drawn from Gaussian measures (see our Theorem 1.4) is discussed in some detail in [DZ92]. 2. Change of Measure from Gaussian Inverse problems in differential equations, and their Bayesian formulation, is covered in detail in [Stu10]. The Radon-Nikodym Theorem 2.1 may be found in [Dud02], as can Lemma 2.2, which we state here in complete detail, to enable a briefer (imprecise) statement in the lectures. The particular form of the statement adopted here may be found in [HSV07]. Lemma 2.2 Let μ, ν be probability measures on S × T where (S,A) and (T,B) are measurable spaces. Denote by (x, y) with x ∈ S and y ∈ T an element of S × T . Assume that μ ν and that μ has Radon-Nikodym derivative φ with respect to ν. Assume further that the conditional distribution of x|y under ν, denoted by ν(dx), exists. Then the conditional distribution of x|y under μ, denoted μ(dx), exists and μ ν. The Radon-Nikodym derivative is given by dμ dνy (x) = { 1 c(y)φ(x, y), if c(y) > 0, and 1 else (2.1)