In some control problems, we are given a model for the dynamical system so we can predict the future of the system if we are able to observe the current state or accurately estimate the current state from a sequence of observations. For example, in financial prediction, the observations might correspond to stock prices and the noise might be due to small errors in recording the time of stock transactions. Presumably, there are other factors besides the instantaneous price of stocks that are important in predicting the behavior of the stock market, e.g., news of a change in the interest rate. Techniques depend on the hypothesis space and the performance criterion. At the risk of introducing terms before they are defined, we might search the space of hidden Markov models using gradient search methods looking for the maximum likely model given the data.
Some practitioners decompose the task of inference into two stages.
In the second stage, the conditioned data is used to search for a model. In some cases, the sources of noise should be considered as an integral part of the dynamical system and therefore deserving of a more central role in inference. In cases where the sources of noise are well understood, it makes sense to use signal processing techniques to filter the noise and simplify subsequent inference. There are a variety of inference problems involving dynamical systems. Here we mention a few of the more important problems that we will revisit often in subsequent sections of the tutorial.