By Bayes' rule, the most probable tag sequence $\hat t_{1:n}$ for an observation sequence of words $w_1...w_n$ is:

$\hat t_{1:n} = \argmax_{t_1, ..., t_n} P(t_1...t_n|w_1...w_n)$ $= \argmax_{t_1, ..., t_n} \frac{P(w_1...w_n|t_1...t_n)P(t_1...t_n)}{P(w_1...w_n)}$ = argmax_{t_1, ..., t_n} P(w_1...w_n|t_1...t_n)P(t_1...t_n)

HMM taggers make two simplifying assumptions:

1. The probability of a word appearing depends only on its own tag and is independent of neighboring words and tags: $P(w_1...w_n|t_1...t_n) \approx \prod_{i=1}^n P(w_i|t_i)$
2. The probability of a tag is dependent only on the previous tag, rather than the entire tag sequence: $P(t_1...t_n) \approx \prod_{i=1}^n P(t_i|t_{i-1})$

Hence, it follows that: $\hat t_{1:n} \approx \argmax_{t_1, ..., t_n} \prod_{i=1}^n P(w_i|t_i)P(t_i|t_{i-1})$

The two parts of this equation correspond to the emission probability and the transition probability, respectively.