Most probabilistic parsers utilize Ney's 1991 probabilistic CKY algorithm, in which each word in a sentence is indexed and each constituent is encoded in a two-dimensional matrix.

Probabilistic CKY Algorithm

The goal of PCFGs is to propose the most likely parse *T̂* of a given sentence *S*.

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A **probabilistic context-free grammar (PCFG)** is a probabilistic augmentation of a context-free grammar in which each production rule is associated with a probability:

$$A \rightarrow B [p]$$

A PCFG is *consistent* if for all sentences, the probabilities for that sentence sum up to 1.

Probabilistic Context-Free Grammar (PCFG)

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

Speech and Language Processing (3rd ed. draft) 

To resolve ambiguity, a Probabilistic Context-Free Grammar (PCFG) calculates the probability of each possible parse tree (derivation) $$T$$ for a given sentence $$S$$. The joint probability of tree $$T$$ and sentence $$S$$ is defined as the product of the probabilities of all rules used in the derivation: $$P(T,S) = \prod_{i=1}^{n} P(RHS_i | LHS_i)$$. The PCFG then chooses the parse tree with the greatest probability as the correct parse of $$S$$.

PCFGs for Disambiguation

PCFGs are useful in language modeling as they can predict the probability of a sentence or part of a sentence

PCFGs for Language Modeling

PCFG Parsing Problem

The main problems of PCFGs are probability estimators are as follows:
- Poor independence assumptions
- Lack of lexical conditioning


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