If none of the existing runs explains the new observation well enough, the prior predictive density,\(p\left(y_{t}\middle|y_{t-1}^{\left(0\right)}\right)\), will dominate, resulting in a high posterior probability for a regime shift.
As we are mostly interested in retrospective analysis of regime shifts, we use the smoothed run length probabilities to find the most likely segmentation of the data or the most likely set of regimes by maximizing the product of run length probabilities over the whole time series (Perälä et al. 2016). We can find the maximum among all possible combinations of regimes or we can focus on some subset of regimes by setting certain constraints for the regimes. We have decided to set a constraint for the minimum length of the regimes (\(M\)). This constraint is not used for the first and the last regimes, though, since their start and end points can be outside the time frame of our data. We use uniform priors for the mean and variance parameters in each of the time series analyzed. The lower and upper limits for the uniform priors were assigned so that all plausible parameter values were contained in the intervals. The posterior inference of the observation model parameters is carried out by a sequential Monte Carlo algorithm (Perälä et al. 2016), using 100,000 particles.