I am studying the the forward-backward algorithm used in Hidden Markov Models. I understand that you are trying to propagate through a sequence (and the available states) to find the most probable state in the model at any given time.
I am however struggling with the following notation (for a forward variable) where the probability of being in state $i$ at time $t$ having observed letters $X^{1} ...X^{t}$ in a model $M(\omega)$ is:
$\alpha_{i}(t) = P(S^t = I,X^{1} ...X^{t}|\omega)$
where the start state is initialized as:
$\alpha_{start}(0) = 1$
which is fine. However where I am struggling is how to compute $\alpha_{i}(t)$ recursively by propagations i.e.
$\alpha_{i}(t+1) = \sum\limits_{j \in S} \alpha_{j}(t)t_{ij}e_{iX^{t+1}} = \sum\limits_{j \in N^{-}(i)} \alpha_{j}(t)t_{ij}e_{iX^{t+1}}$
where $e_{iX^{t}}$ are emissions and $t_{ij}$ are transitions, and $\sum\limits_{j \in N^{-}(i)}$ is the neighbourhood notation which I interpret as instead of state $j$ being in all states, it is found in from the previous state (although correct me if I'm wrong!).
To my eyes this reads as "I can calculate the probability of being in the next state at $t+1$ given the observed sequence by calculating the probability of being in state $j$ at time $t$ having observed the letters $X^{1} ...X^{t}$.
But what I don't understand is........how is $\alpha_{j}(t)$ any different to $\alpha_{i}(t+1)$?
Even more confusing for me is when you propagate through a state where there is no emission observed (imagine flipping a coin but it landing on a small table where if it falls off the emission is not recorded, but the fall is) the equation is modified to:
$\alpha(t+1) = \sum\limits_{j \in N^{-}(i)} \alpha_{j}(t+1)t_{ij}$
where the only explanation for $t+1$ appearing both sides of the equation is because iterating the equation leads to a stable set of values $\alpha_{i}(t+1)$
I know that the Markov property is that the probability of being in a certain state is only dependent on the previous state, but this notation is confusing me.
If anyone could explain this in simple terms (perhaps on an abstract example) then that would be great.
Thanks.