In a Probabilistic Context-Free Grammar (PCFG), the probability of an ambiguous sentence is the sum of the probabilities of all valid parse trees for that sentence:

$$P(S) = \sum_{T \text{ s.t. } S=\text{yield}(T)} P(T)$$

Probability of an Ambiguous Sentence in a PCFG

In a Probabilistic Context-Free Grammar (PCFG), the probability of an unambiguous sentence is simply the probability of the single valid parse tree for that sentence: $$P(T,S) = P(T)$$

Probability of an Unambiguous Sentence in a PCFG

To predict the next word using a Probabilistic Context-Free Grammar (PCFG), the conditional probability of the next word $$w_i$$ given the preceding words is calculated as the ratio of their prefix probabilities:

$$P(w_i | w_1, w_2, ..., w_{i-1}) = \frac{P(w_1, w_2, ..., w_{i-1}, w_i)}{P(w_1, w_2, ..., w_{i-1})}$$

Formula for Next Word Prediction with PCFGs

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

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Applications of Probabilistic Parsing

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 for Language Modeling

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

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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