Orthogonal Polynomials Quadrature Algorithm (OPQA): A Functional Analytical Approach to Bayesian Inference

In this paper, we present the new Orthogonal Polynomials-Quadrature Algorithm (OPQA), a parallelizable algorithm that estimates both the posterior and the evidence in a Bayesian analysis in one pass by means of a functional analytic approach. First, OPQA relates the evidence to an orthogonal projection onto a special basis of our construct. Second, it lays out a fast and accurate computational scheme to compute the transform coefficients. OPQA can be summarized as follows. First, we consider the $L^2$ space associated with a measure with exponential weights. Then we constuct a multivariate orthogonal basis which is dense in this space, such density being guaranteed by the Riesz's Theorem. As we project the square root of the joint distribution onto this basis of our choice, the density of the basis allows us to invoke the Parseval Identity, which equates the evidence with the sum of squares of the transform coefficients of this orthogonal projection. To compute those transform coefficients, we propose a computational scheme using Gauss-Hermite quadrature in higher dimensions. Not only does this approach avoids the potential high variance problem associated with random sampling methods, it significantly reduces the complexity of the computation and enables one to speed up the computational speed by parallelization. This new algorithm does not make any assumption about the independence of the latent variable, nor do we assume any knowledge of the prior. It solves for both the evidence and the posterior in one pass. An outline of the theoretical proof of the supporting algorithm will be provided.