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I'm majoring in pure linguistics (not computational), and I don't have any basic knowledge regarding computational science or mathematics. But I happen to take "Automatic Speech Recognition" course in my graduate school and struggling with it.

I have a question regarding getting the formula for a component of the forward algorithm.

$$ \alpha_t(j) = \sum_{i=1}^{N} P(q_{t-1} = i, q_t=j, o_1^{t-1}, o^t|\lambda) $$

When $q$ is a hidden state, $o$ is a given observation, and $\lambda$ contains transition probability, emission probability and the start/end state.

Is the Markov assumption (the current state is only dependent upon the one right before it) assumed here? I thought so, because it contains $q_{t-1}=i$ and not $q_{t-2}=k$ or $q_{t-3}=l$.

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In general, to formally state that the Markov property holds, you need to have $P( x_t \mid x_{t-1:1}) = P(x_t \mid x_{t-1})$.

So, you cannot conclude only from $P(q_{t-1} = i, q_t=j, o_1^{t-1}, o^t|\lambda)$ that the Markov property holds, because $P(q_{t-1} = i, q_t=j, o_1^{t-1}, o^t|\lambda)$ is the joint probability of $q_{t-1} = i, q_t=j, o_1^{t-1}$ and $o^t$ given $\lambda$.

Nonetheless, the theory of hidden Markov model often assumes that a few properties (including the Markov property) hold

  1. the Markov property: $$P(q_{t+1} \mid q_{t}) = P(q_{t+1} \mid q_{t:1}),$$ where $q_t$ is the hidden state at time step $t$ and $q_{t:1} = q_t, q_{t-1}, \dots, q_1$.

  2. the stationarity property: $$P(q_{t_1 + 1} \mid q_{t_1}) = P(q_{t_2 + 1} \mid q_{t_2}),$$ for any time step $t_1$ and $t_2$. In other words, the state transition probabilities are independent of the actual time at which the transitions takes place.

  3. the output independence property: $$P(o^{T:1} \mid q_{T:1}) = \prod_{t=1}^T P(o^t \mid q_t , \lambda)$$ where $o^{T:1} = o^T, o^{T-1}, \dots, o^1$. In other words, this is the assumption that the output at time step $t$, $o^t$, is independent of the outputs at previous time steps.

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  • $\begingroup$ Oh, I think i should've put one more line for someone who's not familiar with the Forward Algorithm. Alphat(j)=P(o1,o2,...,oT-1,oT,qt=j|Lambda). Sorry that I don't know how to type mathematical symbols. $\endgroup$ – Jeeyoung Jeon Jun 17 at 0:18
  • $\begingroup$ @JeeyoungJeon The forward algorithm (FA) is used in context of HMMs, so all the assumptions that are made in the context of HMMs also hold in the context of FA. $\endgroup$ – nbro Jun 17 at 0:21
  • $\begingroup$ I edited my former comment, but I think I've found the answer from you already. Thanks! $\endgroup$ – Jeeyoung Jeon Jun 17 at 0:24

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