The von Mises–Fisher distribution is one of the most widely used probability distributions to describe directional data. Finite mixtures of von Mises–Fisher distributions have found numerous applications. However, the likelihood function for the finite mixture of von Mises–Fisher distributions is unbounded and consequently the maximum likelihood estimation is not well defined. To address the problem of likelihood degeneracy, we consider a penalized maximum likelihood approach whereby a penalty function is incorporated. We prove strong consistency of the resulting estimator. An Expectation–Maximization algorithm for the penalized likelihood function is developed and experiments are performed to examine its performance.

von Mises-Fisher 分布是描述方向数据的最广泛使用的概率分布之一。von Mises-Fisher 分布的有限混合已找到许多应用。然而，von Mises-Fisher 分布的有限混合的似然函数是无界的，因此最大似然估计没有很好的定义。为了解决似然退化的问题，我们考虑了一种惩罚最大似然方法，其中结合了惩罚函数。我们证明了结果估计量的强一致性。开发了惩罚似然函数的期望最大化算法，并进行了实验以检查其性能。