Hidden Markov Model initial probability reestimate: Why $\pi^_i = \gamma_i(1)$ instead of $\pi^_i = \frac{\gamma_i(1)}{\sum_{j = 1}^N \gamma_j(1)}$

Question

In the sources I consulted it states that in the Baum Welch algorithm the reestimate of the initial probability of state $i$ of the HMM is $\pi^*_i = \gamma_i(1)$. But $\gamma_i(t)$ is the probability of being in state ${\displaystyle i}$ at time ${\displaystyle t}$ given the observed sequence ${\displaystyle Y}$ and the parameters ${\displaystyle \theta }$ (quote wiki)

So, then why does this probability not need to be normalised like so? :

$$\pi^*_i = \frac{\gamma_i(1)}{\sum_{j = 1}^N \gamma_j(1)}$$

After all normalizing is what is done for the reestimate of the transition probabilities and the emission probabilities too.

score 4 · Accepted Answer · answered Apr 03 '17 at 17:49

It is defined to be a probability. A probability is by definition already normalized. In particular, we are guaranteed that

$$\sum_{j=1}^N \gamma_j(1) = 1,$$

as there are only $N$ possibilities for the state that you're in at time $1$, and these $N$ cases have no overlap.

Hidden Markov Model initial probability reestimate: Why $\pi^*_i = \gamma_i(1)$ instead of $\pi^*_i = \frac{\gamma_i(1)}{\sum_{j = 1}^N \gamma_j(1)}$

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Hidden Markov Model initial probability reestimate: Why $\pi^_i = \gamma_i(1)$ instead of $\pi^_i = \frac{\gamma_i(1)}{\sum_{j = 1}^N \gamma_j(1)}$