Learning Objectives

$(1)$ calculate the probabilities of parse trees given a probabilistic context$-$free grammar $($PCFG$),(2)$ implement training and search algorithms for Hidden Markov Models $($HMMs$),$ and $(3)$ use neural networks to recognize the named entities $($NER$)$ in a sequence of tokens.

$0.$ A Probabilistic Context$-$Free Grammar $($PCFG$)$

Consider a PCFG with start symbol NP$,$ the following rules and probabilities, terminals cats, and, in