Bayesian Change Point Analysis of Linear Models on Graphs

Consider observations $y_1,\dots,y_n$ on nodes of a connected graph, where the $y_i$ independently come from $N(\theta_i, \sigma^2)$ distributions and an unknown partition divides the $n$ observations into blocks. One well-studied class of change point problems assumes the means $\theta_i$ are equal for all nodes within contiguous blocks of a simple graph of sequential observations; both frequentist and Bayesian approaches have been used to estimate the $\theta_i$ and the change points of the underlying partition. This paper examines a broad class of change point problems on general connected graphs in which a regression model is assumed to apply within each block of the partition of the graph. This general class also supports multivariate change point problems. We use Bayesian methods to estimate change points or block boundaries of the underlying partition. This paper presents the methodology for the general class of change point problems and develops new algorithms for implementation via Markov Chain Monte Carlo. The paper concludes with simulations and real data examples to demonstrate application of the methodology on a wide range of problems.