EM parameter estimation for a piecewise AR

We design a model meant to be the equivalent of Blake's weak string but in the probabilistic framework. Independent line sites delimit piecewise stationary Gaussian autoregressives AR(1) corrupted with Gaussian white noise. Thanks to the Bayesian interpretation, we define the joint probability which in turn yields the likelihood. We demonstrate how to make its computation possible in cubic time. This calculation allows the set of parameters to be tested but not estimated due to the complex form of the criterion. Yet the computations done so far provide the materials for an iterative maximization. Indeed, the Expectation Maximization algorithm happens to match the features of this model and is also easily calculable. When the likelihood is known, the cost of one step of the latter algorithm is negligible in comparison with the previous calculations.