Appendix A. Model factorization. See Appendix H for references cited.
Let Xt denote the state of the system at time t, and let Yt denote data collected at time t, t=1,...,T. Let superscripts denote a sequence (i.e., XT = X1, …, XT). Let t denote time-varying vital rates, let denote process-model parameters that are constant over time and for all cutting cycles, and let denote parameters in the process model. Finally, is the parameter that sets the smoothness constraint. For a single cutting cycle, the joint probability distribution of all parameters and data factors as
A directed acyclic graph for this model is shown in Fig. A1.
|FIG. A1. Directed acyclic graph for the model. The series X1, …, XT is the trajectory of the hidden state, and the series Y1, …, YT is the data. Solid lines denote stochastic relationships, dotted lines denote deterministic relationships, boxes indicate observed data, and circles indicate model quantities (after Spiegelhalter et al. 1996).|