  We present the new Orthogonal Polynomials Approximation Algorithm (OPAA), a
parallelizable algorithm that estimates probability distributions using
functional analytic approach: first, it finds a smooth functional estimate of
the probability distribution, whether it is normalized or not; second, the
algorithm provides an estimate of the normalizing weight; and third, the
algorithm proposes a new computation scheme to compute such estimates.
  A core component of OPAA is a special transform of the square root of the
joint distribution into a special functional space of our construct. Through
this transform, the evidence is equated with the $L^2$ norm of the transformed
function, squared. Hence, the evidence can be estimated by the sum of squares
of the transform coefficients. Computations can be parallelized and completed
in one pass.
  OPAA can be applied broadly to the estimation of probability density
functions. In Bayesian problems, it can be applied to estimating the
normalizing weight of the posterior, which is also known as the evidence,
serving as an alternative to existing optimization-based methods.
