Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities

Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation, based on the asymptotic distribution of SAA solutions. The confidence regions are single ellipsoids with high probability. We also discuss the computation of simultaneous and individual confidence intervals

Symmetric Confidence Regions and Confidence Intervals for Normal Map Formulations of Stochastic Variational Inequalities

Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation, based on the asymptotic distribution of SAA solutions. The confidence regions are single ellipsoids with high probability. We also discuss the computation of simultaneous and individual confidence intervals

A new method to build confidence regions for solutions of stochastic variational inequalities

Stochastic variational inequalities model a large class of equilibrium problems subject to data uncertainty. The true solution to such a problem is usually estimated by a solution to its sample average approximation (SAA) problem. This paper proposed a new method to build asymptotically exact con dence regions for the true solution that are computable from the SAA solution

Variational Inequality Approach to Stochastic Nash Equilibrium Problems with an Application to Cournot Oligopoly

In this note we investigate stochastic Nash equilibrium problems by means of monotone variational inequalities in probabilistic Lebesgue spaces. We apply our approach to a class of oligopolistic market equilibrium problems where the data are known through their probability distributions.Comment: 19 pages, 2 table

Maximum-a-posteriori estimation with Bayesian confidence regions

Solutions to inverse problems that are ill-conditioned or ill-posed may have significant intrinsic uncertainty. Unfortunately, analysing and quantifying this uncertainty is very challenging, particularly in high-dimensional problems. As a result, while most modern mathematical imaging methods produce impressive point estimation results, they are generally unable to quantify the uncertainty in the solutions delivered. This paper presents a new general methodology for approximating Bayesian high-posterior-density credibility regions in inverse problems that are convex and potentially very high-dimensional. The approximations are derived by using recent concentration of measure results related to information theory for log-concave random vectors. A remarkable property of the approximations is that they can be computed very efficiently, even in large-scale problems, by using standard convex optimisation techniques. In particular, they are available as a by-product in problems solved by maximum-a-posteriori estimation. The approximations also have favourable theoretical properties, namely they outer-bound the true high-posterior-density credibility regions, and they are stable with respect to model dimension. The proposed methodology is illustrated on two high-dimensional imaging inverse problems related to tomographic reconstruction and sparse deconvolution, where the approximations are used to perform Bayesian hypothesis tests and explore the uncertainty about the solutions, and where proximal Markov chain Monte Carlo algorithms are used as benchmark to compute exact credible regions and measure the approximation error