I am self-studying Kevin Murphy's book, and trying to understand the math behind the HMM. I am struggling with the following derivation. Why can we in 17.46 cross out the $X_{1:t-1}$ term? my guess would be that $X_t$ is independent of $X_{1:t-1}$, but i am not 100% sure why in Markov Chain / HMM that would be the case. any hints much appreciated.
When $z_t$ is known the distribution of $x_t$ only depends on $z_t$. Hence $P(x_t|z_t, x_{(t-1)}) = P(x_t|z_t)$. This is one of the key features of any HMM.
This can be shown in the directed graph you can find in most HMM intro tutorials. I find the supplementary slides to Zucchini's book here (http://www2.imm.dtu.dk/courses/02433/) are particularly enlightening.
For instance slide 7 of week 2 states the following:
"Definition: If for a random variable A it holds that $Pr(A = a|B; C) = Pr(A = a|B)$, then A is said to be conditional independent of C given B. This is an important property influencing the random variables in a directed graph. For example in the figure on page 3 it holds that $X_t$ are conditional independent of $X_1$,...,$X_{t-1}$,$X_{t+1}$, ... given $C_t$ for all t. For the hidden state it holds that $C_{t+1}$ is conditional independent of $C_1$,...,$C_{t-1}$ given $C_t$. So, by conditioning on the parents of a random variable it is independent of everything else."
