This paper presents a research study focused on uncovering the hidden population distribution from the viewpoint of a variational non-Bayesian approach. It asserts that if the hidden probability density function (PDF) has continuous partial derivatives of at least half the dimension's order, it can be perfectly reconstructed from a stationary ergodic process: First, we establish that if the PDF belongs to the Wiener algebra, its canonical ensemble form is uniquely determined through the Fr\'echet differentiation of the Kullback-Leibler divergence, aiming to minimize their cross-entropy. Second, we utilize the result that the differentiability of the PDF implies its membership in the Wiener algebra. Third, as the energy function of the canonical ensemble is defined as a series, the problem transforms into finding solutions to the equations of analytic series for the coefficients in the energy function. Naturally, through the use of truncated polynomial series and by demonstrating the convergence of partial sums of the energy function, we ensure the efficiency of approximation with a finite number of data points. Finally, through numerical experiments, we approximate the PDF from a random sample obtained from a bivariate normal distribution and also provide approximations for the mean and covariance from the PDF. This study substantiates the excellence of its results and their practical applicability.

翻译：本文从变分非贝叶斯视角出发，针对隐藏总体分布的揭示问题展开研究。论证表明，若隐藏概率密度函数具有至少维度一半阶数的连续偏导数，则可通过平稳遍历过程完美重构该函数：首先，证明当概率密度函数属于维纳代数时，其正则系综形式可通过Kullback-Leibler散度的Fr\\'echet微分唯一确定，旨在最小化交叉熵；其次，利用概率密度函数的可微性蕴含其属于维纳代数的结论；第三，鉴于正则系综的能量函数定义为级数形式，问题转化为求解能量函数系数的解析级数方程。通过采用截断多项式级数并证明能量函数部分和的收敛性，保证了有限数据点下的逼近效率。最后，通过双变量正态分布随机样本的数值实验实施概率密度函数逼近，并基于该函数给出均值与协方差的估计值。本研究验证了方法的卓越性能及其实际可应用性。