Rule changes and informative switches

Define function equivalence $r\sim r^{\prime }$ by $f_r=f_{r^{\prime }}$. A switch has three distinct descriptions:

1. $R_{t-1}\ne R_t$: rule-ID change;

2. $R_{t-1}\nsim R_t$: label-function change;

3. $f_{R_{t-1}}(X_t)\ne f_{R_t}(X_t)$: an informative disagreement at that point.