We consider the problem of estimation in Hidden Markov models with finite state space and nonparametric emission distributions. Efficient estimators for the transition matrix are exhibited, and a semiparametric Bernstein-von Mises result is deduced. Following from this, we propose a modular approach using the cut posterior to jointly estimate the transition matrix and the emission densities. We derive a general theorem on contraction rates for this approach. We then show how this result may be applied to obtain a contraction rate result for the emission densities in our setting; a key intermediate step is an inversion inequality relating $L^1$ distance between the marginal densities to $L^1$ distance between the emissions. Finally, a contraction result for the smoothing probabilities is shown, which avoids the common approach of sample splitting. Simulations are provided which demonstrate both the theory and the ease of its implementation.

翻译：我们考虑具有有限状态空间和非参数发射分布的隐马尔可夫模型的估计问题。我们展示了转移矩阵的有效估计量，并推导出半参数伯恩斯坦-冯·米塞斯结果。在此基础上，我们提出了一种利用切割后验联合估计转移矩阵和发射密度的模块化方法。我们推导了该方法收缩率的一般性定理，并展示了如何将该结果应用于获取设定中发射密度的收缩率结论；其中关键中间步骤是建立边缘密度之间$L^1$距离与发射密度之间$L^1$距离的反演不等式。最后，我们证明了平滑概率的收缩结果，该方法避免了常用的样本分割策略。数值模拟结果既验证了理论的有效性，也展示了该方法的易实现性。