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：我们考虑了使用有限状态空间和非参数排放分布的隐藏Markov模型的估计问题。 展示了过渡矩阵的有效估计数据,并推断出Bernstein- von Mises的半参数结果。 在此之后,我们提议采用模块化方法,使用切割后框来共同估计过渡矩阵和排放密度。 我们为这一方法得出了一个关于收缩率的一般理论。 然后,我们展示了如何应用这一结果来获得我们所设定排放密度的收缩率结果; 关键的中间步骤是边缘密度之间距离为1美元到排放距离1美元之间的倾斜不平等。 最后,显示了平滑概率的收缩结果,这避免了样本分离的共同方法。 提供了模拟数据,既表明了理论,也表明了其实施容易。</s>