I'm trying to prove the following whose context I will describe, but which is not strictly necessary to solving the problem. In natural language processing, specifically in part of speech tagging, given a sequence of words, $x_{1}...x_{n}$, one wishes to tag the words with their parts of speech, say $t_{1}...t_{n}$, and the tag sequence $t_{1}...t_{n}$ should be chosen so as to maximize $P(t_{1}...t_{n}|x_{1}...x_{n})$.
As an example, given the sentence, "John is good", the most likely tag sequence would be something like "NOUN VERB ADJECTIVE". For this particular sentence, most other possible tag sequences would have 0 probability, since the words themselves have few possibilities in what their parts of speech could be.
To find the tag sequence resulting in the maximal $P(t_{1}...t_{n}|x_{1}...x_{n})$, a few simplifying assumptions are made. First note that by Bayes' theorem, finding the tag sequence that maximizes $P(t_{1}...t_{n}|x_{1}...x_{n})$ is equivalent to finding the tag sequence that maximizes $P(x_{1}...x_{n}|t_{1}...t_{n})*P(t_{1}...t_{n})$.
My question concerns the following argument made in the textbook (Speech and LP) by Jurafsky): "The first assumption is that the probability of a word appearing is dependent only on its own part-of-speech tag; that it is independent of other words around it, and independent of the other tags around it."
Thus, $P(x_{1}...x_{n}|t_{1}...t_{n}) = \prod P(x_{n}|t_{n})$
I am triyng to follow why this final equation follows from the simplifying assumptions. It seems obvious, but how does it follow directly from the laws of probability?
The first step uses the Chain Rule:
\begin{align} P(x_1,\ldots,x_n\mid t_1,\ldots ,t_n) &= P(x_1\mid x_2,\ldots,x_n,t_1,\ldots ,t_n) \\ & \quad\times P(x_2\mid x_3,\ldots,x_n,t_1,\ldots ,t_n) \\ & \quad\times P(x_3\mid x_4,\ldots,x_n,t_1,\ldots ,t_n) \\ & \qquad\cdots \\ & \quad\times P(x_n\mid t_1,\ldots ,t_n) \\ & \\ &= P(x_1\mid t_1) \times P(x_2\mid t_2) \times \ldots \times P(x_n\mid t_n) \\ &\qquad\text{by the conditional independence assumption} \end{align}