Qualitative Robustness in Bayesian Inference

The practical implementation of Bayesian inference requires numerical approximation when closed-form expressions are not available. What types of accuracy (convergence) of the numerical approximations guarantee robustness and what types do not? In particular, is the recursive application of Bayes' rule robust when subsequent data or posteriors are approximated? When the prior is the push forward of a distribution by the map induced by the solution of a PDE, in which norm should that solution be approximated? Motivated by such questions, we investigate the sensitivity of the distribution of posterior distributions (i.e. posterior distribution-valued random variables, randomized through the data) with respect to perturbations of the prior and data generating distributions in the limit when the number of data points grows towards infinity

Brittleness of Bayesian inference and new Selberg formulas

The incorporation of priors in the Optimal Uncertainty Quantification (OUQ) framework \cite{OSSMO:2011} reveals brittleness in Bayesian inference; a model may share an arbitrarily large number of finite-dimensional marginals with, or be arbitrarily close (in Prokhorov or total variation metrics) to, the data-generating distribution and still make the largest possible prediction error after conditioning on an arbitrarily large number of samples. The initial purpose of this paper is to unwrap this brittleness mechanism by providing (i) a quantitative version of the Brittleness Theorem of \cite{BayesOUQ} and (ii) a detailed and comprehensive analysis of its application to the revealing example of estimating the mean of a random variable on the unit interval $[0,1]$ using priors that exactly capture the distribution of an arbitrarily large number of Hausdorff moments. However, in doing so, we discovered that the free parameter associated with Markov and Kre\u{\i}n's canonical representations of truncated Hausdorff moments generates reproducing kernel identities corresponding to reproducing kernel Hilbert spaces of polynomials. Furthermore, these reproducing identities lead to biorthogonal systems of Selberg integral formulas. This process of discovery appears to be generic: whereas Karlin and Shapley used Selberg's integral formula to first compute the volume of the Hausdorff moment space (the polytope defined by the first $n$ moments of a probability measure on the interval $[0,1]$), we observe that the computation of that volume along with higher order moments of the uniform measure on the moment space, using different finite-dimensional representations of subsets of the infinite-dimensional set of probability measures on $[0,1]$ representing the first $n$ moments, leads to families of equalities corresponding to classical and new Selberg identities.Comment: 73 pages. Keywords: Bayesian inference, misspecification, robustness, uncertainty quantification, optimal uncertainty quantification, reproducing kernel Hilbert spaces (RKHS), Selberg integral formula

Fast rates for support vector machines using Gaussian kernels

For binary classification we establish learning rates up to the order of $n^{-1}$ for support vector machines (SVMs) with hinge loss and Gaussian RBF kernels. These rates are in terms of two assumptions on the considered distributions: Tsybakov's noise assumption to establish a small estimation error, and a new geometric noise condition which is used to bound the approximation error. Unlike previously proposed concepts for bounding the approximation error, the geometric noise assumption does not employ any smoothness assumption.Comment: Published at http://dx.doi.org/10.1214/009053606000001226 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Conditioning Gaussian measure on Hilbert space

For a Gaussian measure on a separable Hilbert space with covariance operator $C$, we show that the family of conditional measures associated with conditioning on a closed subspace $S^{\perp}$ are Gaussian with covariance operator the short $\mathcal{S}(C)$ of the operator $C$ to $S$. We provide two proofs. The first uses the theory of Gaussian Hilbert spaces and a characterization of the shorted operator by Andersen and Trapp. The second uses recent developments by Corach, Maestripieri and Stojanoff on the relationship between the shorted operator and $C$-symmetric oblique projections onto $S^{\perp}$. To obtain the assertion when such projections do not exist, we develop an approximation result for the shorted operator by showing, for any positive operator $A$, how to construct a sequence of approximating operators $A^{n}$ which possess $A^{n}$-symmetric oblique projections onto $S^{\perp}$ such that the sequence of shorted operators $\mathcal{S}(A^{n})$ converges to $\mathcal{S}(A)$ in the weak operator topology. This result combined with the martingale convergence of random variables associated with the corresponding approximations $C^{n}$ establishes the main assertion in general. Moreover, it in turn strengthens the approximation theorem for shorted operator when the operator is trace class; then the sequence of shorted operators $\mathcal{S}(A^{n})$ converges to $\mathcal{S}(A)$ in trace norm

Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis

We show how the discovery of robust scalable numerical solvers for arbitrary bounded linear operators can be automated as a Game Theory problem by reformulating the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games. When the solution space is a Banach space $B$ endowed with a quadratic norm $\|\cdot\|$, the optimal measure (mixed strategy) for such games (e.g. the adversarial recovery of $u\in B$, given partial measurements $[\phi_i, u]$ with $\phi_i\in B^*$, using relative error in $\|\cdot\|$-norm as a loss) is a centered Gaussian field $\xi$ solely determined by the norm $\|\cdot\|$, whose conditioning (on measurements) produces optimal bets. When measurements are hierarchical, the process of conditioning this Gaussian field produces a hierarchy of elementary bets (gamblets). These gamblets generalize the notion of Wavelets and Wannier functions in the sense that they are adapted to the norm $\|\cdot\|$ and induce a multi-resolution decomposition of $B$ that is adapted to the eigensubspaces of the operator defining the norm $\|\cdot\|$. When the operator is localized, we show that the resulting gamblets are localized both in space and frequency and introduce the Fast Gamblet Transform (FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT can be used to solve and diagonalize arbitrary PDEs with constant coefficients, the FGT can be used to decompose a wide range of continuous linear operators (including arbitrary continuous linear bijections from $H^s_0$ to $H^{-s}$ or to $L^2$) into a sequence of independent linear systems with uniformly bounded condition numbers and leads to $\mathcal{O}(N \operatorname{polylog} N)$ solvers and eigenspace adapted Multiresolution Analysis (resulting in near linear complexity approximation of all eigensubspaces).Comment: 142 pages. 14 Figures. Presented at AFOSR (Aug 2016), DARPA (Sep 2016), IPAM (Apr 3, 2017), Hausdorff (April 13, 2017) and ICERM (June 5, 2017

Extreme points of a ball about a measure with finite support

We show that, for the space of Borel probability measures on a Borel subset of a Polish metric space, the extreme points of the Prokhorov, Monge-Wasserstein and Kantorovich metric balls about a measure whose support has at most n points, consist of measures whose supports have at most n+2 points. Moreover, we use the Strassen and Kantorovich-Rubinstein duality theorems to develop representations of supersets of the extreme points based on linear programming, and then develop these representations towards the goal of their efficient computation