I am currently reading up on some hidden markov models [Source], and trying to understand some of the solution related to the evaluation problem, which is

Given observation sequence $\boldsymbol{O}$ = {$o_1$,$o_2$,$o_3$...,$o_t$}, and model $\lambda$ = {$\boldsymbol{A}$, $\boldsymbol{\pi}$ and $\boldsymbol{B}$}, how can $P(O|\lambda)$, or how well can a given model match a given observation sequence. In case one is to choose among several competing model, one would like to choose the model, which matches the observations best. [Evaluating problem]

On page 262 equation 15 it is stated: $$P(O,Q|\lambda) = P(O|Q,\lambda)P(Q,\lambda)$$

but isn't this incorrect? should it not be

$$P(O,Q|\lambda) = P(O|Q,\lambda)P(Q|\lambda)$$


Well, sure, it's just a typo. If you read a few lines before that, it says:

"[..] Thus we get

$$P(O|Q,\lambda) = b_{q_1}(O_1)\cdot b_{q_2}(O_2)\cdots b_{q_T}(O_T) \tag{13b}$$

The probability of such a state sequence $Q$ can be written as

$$P(Q|\lambda) = \pi_{q_1}a_{q_1q_2}a_{q_2q_3}\cdots a_{q_{T-1}q_T} \tag{14}$$

The joint probability of $O$ and $Q$, i.e., the probability that $O$ and $Q$ occur simultaneously, is simply the product of the above two terms, i.e.,

$$P(O,Q|\lambda) = P(O|Q,\lambda)P(Q,\lambda) \tag{15}\label{15}$$


So, you see that when equation $\ref{15}$ is described, reference is made to "the above two terms", which are precisely $P(O|Q,\lambda)$ and $P(Q|\lambda)$. The typo seems not to be repeated in the rest of the paper: for example, equation (16) for $P(O|\lambda)$ contains $P(O|Q,\lambda)P(Q|\lambda)$ and not $P(O|Q,\lambda)P(Q,\lambda)$.


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