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.

翻译：本文提出了一种新的正交多项式逼近算法（OPAA），该算法具有并行化特性，通过泛函分析方法估计概率分布：首先，它给出概率分布的平滑泛函估计，无论该分布是否归一化；其次，该算法提供归一化权重的估计；第三，算法提出了一种新的计算方案来实现此类估计。OPAA的核心组成部分是将联合分布的平方根变换至我们构建的特定泛函空间。通过该变换，证据被等同于变换后函数的$L^2$范数的平方。因此，证据可通过变换系数的平方和进行估计。计算过程可并行化且一次完成。OPAA可广泛用于概率密度函数的估计。在贝叶斯问题中，它可应用于估计后验分布的归一化权重（即证据），作为现有基于优化方法的一种替代方案。