I am resitting CT4 and have a feeling that I have still failed to grasp the fundamental difference between forwards and backwards equations. At the moment I think that the forward equation is where you can move around as much as you like and then there is a small period of time where you can only make one transition and this is where you will end up, whereas the backwards equation represents the situation where the small period of time comes first and then after that you can make as many jumps as you like into the final state??? Is this right? Does anyone have a simpler way of explaining it? I find this quite difficult.
Say if it's a jump from state i to j from time s to t, and let k be any intermidiate states. The forward equation considers the last jump into state j (from k). This happens at time t-w, and we integrate over all w's. The backward equation considers the first jump out of state i (to k). This happens at time s+w, and we integrate over all w's.
I always get these mixed up too. It seems like "first jump" ought to be forwards. In a way, it's easier to understand in the non-time-homogeneous version. Then the forward equation tells you about d/dt (p_ij(s,t)) and the backward equation tells you about d/ds (p_ij(s,t)).
