The problem of reducing a Hidden Markov Model (HMM) to one of smaller dimension that exactly reproduces the same marginals is tackled by using a system-theoretic approach. Realization theory tools are extended to HMMs by leveraging suitable algebraic representations of probability spaces. We propose two algorithms that return coarse-grained equivalent HMMs obtained by stochastic projection operators: the first returns models that exactly reproduce the single-time distribution of a given output process, while in the second the full (multi-time) distribution is preserved. The reduction method exploits not only the structure of the observed output, but also its initial condition, whenever the latter is known or belongs to a given subclass. Optimal algorithms are derived for a class of HMM, namely observable ones.

翻译：本文采用系统理论方法，研究将隐马尔可夫模型约简至更低维度，使其精确再现相同边际分布的问题。通过利用概率空间合适的代数表示，将实现理论工具推广至隐马尔可夫模型。我们提出两种算法，通过随机投影算子返回粗粒度化的等价隐马尔可夫模型：第一种算法返回精确再现给定输出过程单时间分布的模型，而第二种算法则保留完整的（多时间）分布。该约简方法不仅利用了观测输出结构，还利用了初始条件（当已知或属于特定子类时）。针对一类可观测隐马尔可夫模型，推导出了最优算法。