Stochastic Methods for NLP Probabilistic Context-Free Parsers Probabilistic Lexicalized Context-Free Parsers Hidden Markov Models – Viterbi Algorithm Statistical Decision-Tree Models

Probabilistic CFG 1. sent <- np, vp. p(sent) = p(r 1 ) * p(np) * p(vp). 2. np <- noun. p(np) = p(r 2 ) * p(noun) noun <- dog. p(noun) = p(dog). The probabilities are taken from a particular corpus of text.

Probabilistic Lexicalized CFG 1. sent <- np(noun), vp(verb). p(sent) = p(r 1 ) * p(np) * p(vp) * p(verb|noun). 2. np <- noun. p(np) = p(r 2 ) * p(noun) noun <- dog. p(noun) = p(dog). Note that we've introduced the probability of a particular verb given a particular noun.

Markov Chain Discrete random process: The system is in various states and we move from state to state. The probability of moving to a particular next state (a transition) depends solely on the current state and not previous states (the Markov property). May be modeled by a finite state machine with probabilities on the edges.

Hidden Markov Model Each state (or transition) may produce an output. The outputs are visible to the viewer, but the underlying Markov model is not. The problem is often to infer the path through the model given a sequence of outputs. The probabilities associated with the transitions are known a priori. There may be more than one start state. The probability of each start state may also be known.

Uses of HMM Parts of speech (POS) tagging Speech recognition Handwriting recognition Machine Translation Cryptanalysis Many other non-NLP applications

Viterbi Algorithm Used to find the mostly likely sequence of states (the Viterbi path) in a HMM that leads to a given sequence of observed events. Runs in time proportional to (number of observations) * (number of states) 2. Can be modified if the state depends on the last n states (instead of just the last state). Take time (number of observations) * (number of states) n

Viterbi Algorithm - Assumptions The system at any given time is in one particular state. There are a finite number of states. Transitions have an associated incremental metric. Events are cumulative over a path, i.e., additive in some sense.

Viterbi Algorithm - Code See the

Uses in NLP Parts of speech (POS) tagging: The observations are the words of the sentences. The HMM nodes are the parts of speech. Speech recognition: The observations are the sound waves (after some processing). The HMM may contain the words in the text sentence, or the phonemes.

Statistical Decision-Tree Model SPATTER (Magerman) Alternative approach to CFGs. Uses statistical measures generated by hand annotation of large corpus of text. Automatically discovers disambiguation criteria for parsing Uses a stack decoding algorithm Finds one tree then uses branch-and-bound

Stack Decoding Algorithm Similar to beam search, but claims to use a stack, instead of a priority queue. The n best nodes (partial solutions) are kept. The best node is expanded and its children are put on the stack. The stack is then trimmed back to n nodes.