The underlying predictive model (UPM),\(p\left(y_{t}\middle|y_{t-1}^{\left(r_{t-1}\right)}\right)\), is the predictive density conditional on the run length and thus on the latest \(r_{t-1}\) observations, i.e.,\(y_{t-1}^{\left(r_{t-1}\right)}=y_{(t-r_{t-1}):(t-1)}\). The UPM is used to give more weight to those run lengths that better predict the new observation, and it is defined by the observation model,