Exercise 23.2

An HMM grammar is essentially a standard HMM whose state variable is $N$ (nonterminal, with values such as $Det$, $Adjective$, $Noun$ and so on) and whose evidence variable is $W$ (word, with values such as $is$, $duck$, and so on). The HMM model includes a prior ${\textbf{P}}(N_0)$, a transition model ${\textbf{P}}(N_{t+1}|N_t)$, and a sensor model ${\textbf{P}}(W_t|N_t)$. Show that every HMM grammar can be written as a PCFG. [Hint: start by thinking about how the HMM prior can be represented by PCFG rules for the sentence symbol. You may find it helpful to illustrate for the particular HMM with values $A$, $B$ for $N$ and values $x$, $y$ for $W$.]

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