We introduce a density basis of the trigonometric polynomials that is suitable to mixture modelling. Statistical and geometric properties are derived, suggesting it as a circular analogue to the Bernstein polynomial densities. Nonparametric priors are constructed using this basis and a simulation study shows that the use of the resulting Bayes estimator may provide gains over comparable circular density estimators previously suggested in the literature.

From a theoretical point of view, we propose a general prior specification framework for density estimation on compact metric space using sieve priors. This is tailored to density bases such as the one considered herein and may also be used to exploit their particular shape-preserving properties. Furthermore, strong posterior consistency is shown to hold under notably weak regularity assumptions and adaptive convergence rates are obtained in terms of the approximation properties of positive linear operators generating our models.

我们介绍了适用于混合建模的三角多项式的密度基。导出了统计和几何特性，表明它是伯恩斯坦多项式密度的圆形模拟。非参数先验是使用此基础构建的，并且模拟研究表明，使用所得的贝叶斯估计器可能比之前文献中建议的可比圆形密度估计器提供增益。

从理论的角度来看，我们提出了一个通用的先验规范框架，用于使用筛分先验在紧凑度量空间上进行密度估计。这适用于密度基础，例如本文考虑的密度基础，也可用于开发它们特定的形状保持特性。此外，强大的后验一致性显示出在明显弱的规律性假设下成立，并且根据生成我们模型的正线性算子的近似特性获得了自适应收敛率。