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I was reading a tutorial paper on Hidden Markov Models. The paper describes

Given a set of states $S$ we can observe a series over time $\vec{z} \in S^T$, where $\vec{z}={z_t,z_{t-1}\ldots,z_1}$. As a convention, we will assume that there is an initial state and initial observation $z_0 \equiv s_0$, where $s_0$ represents the initial probability distribution over states at time 0.

thus we have $P(z_t|z_{t-1},\ldots,z_1)=P(z_t|z_{t-1},\ldots,z_1,z_0)$ for any state sequence. Why?

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The markov assumption is that the probability of being in any particular state only depends on the last $k$ states. That may be what the equation that you are asking about is trying to show. When you advance forward one state then you one term from the history because it is assumed to be independent from the current state given the more recent history.

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