# Understanding emission probability in HMM definition

This is rather basic question. I was going through Speech and Language Processing by Jurafsky and Martin. In the book, they define a Hidden Markov Model (HMM) as follows:

An HMM is specified by the following components:

• $$Q = q_1q_2 ...q_N$$ : a set of N states
• $$A = a_{11} ...a_{i j} ...a_{NN}$$ : a transition probability matrix $$A$$ , each > $$a_{ij}$$ representing the probability of moving from state $$i$$ to state $$j$$, s.t. $$\sum_{j=1}^Na_{ij}=1 \quad ∀i$$
• $$O = o_1o_2 ...o_T$$ : a sequence of $$T$$ observations, each one drawn from a vocabulary $$V = v_1, v_2,..., v_V$$
• $$B = b_i(o_t)$$ : a sequence of observation likelihoods, also called emission probabilities, each expressing the probability of an observation $$o_t$$ being generated from a state $$q_i$$
• $$π = π_1,π_2,...,π_N$$: an initial probability distribution over states. $$π_i$$ is the probability that the Markov chain will start in state $$i$$. Some states $$j$$ may have $$π_j = 0$$, meaning that they cannot be initial states. Also, $$\sum_{i=1}^n\pi_i=1$$

My doubt is shouldn't emission probabilities $$B$$ sum to 1? That is, shouldnt it be the case that $$\sum_{i=1}^n b_i(o_t)=1$$ (or maybe $$\sum_{t=1}^{n_t} b_i(o_t)=1$$). If not, why? If yes, why doesn't the book specify either of these?

In other words, $$\sum_{k=1}^{V} b_i\left(v_k\right) = 1$$. At every position in the sequence, the probability of emitting a given symbol given that you're in state $$i$$ is what's summed up to make a normalized distribution. This is true, whether you're at time $$t=1$$ or at time $$t=T$$ or any time in between.
For each possible state $$q_i$$, you'll have a different summing-to-one distribution, which is conditioned on you being in state $$q_i$$.