By Bayes' rule, we have that $\hat t_{1:n} = \argmax_{t_1, ..., t_n} P(t_1...t_n|w_1...w_n) =$ $\argmax_{t_1, ..., t_n} \frac{P(w_1...w_n|t_1...t_n)P(t_1...t_n)}{P(w_1...w_n)} =$ $\argmax_{t_1, ..., t_n} P(w_1...w_n|t_1...t_n)P(t_1...t_n)$

HMM taggers make two simplifying assumptions:

• The probability of a word appearing depends only on its own tag and is independent of neighbouring words and tags: $P(w_1...w_n|t_1...t_n) \approx \prod_{i=1}^n P(w_i|t_i)$
• 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 which corresponds to the emission probability and the transition probability respectively.