To resolve disambiguation, the PCFG will calculate a probability of each parse tree (derivation) $T$ of a given sentence $S$. 

The probability of tree $T$ given $S$ is defined as:

$P(T,S) = \prod_{i=1}^{n} P(RHS_i | LHS_i)$

From there, the PCFG will choose the parse tree with the greatest probability as the correct parse of $S$.

Carleton College

Applications of Probabilistic Parsing

Structural ambiguity occurs when the grammar can assign more than one parse
to a sentence. Two common kinds of ambiguity are attachment ambiguity and coordination ambiguity. A sentence has an attachment ambiguity if a particular constituent can be attached to the parse tree at more than one place. For instance, in the following example the gerundive-VP flying to Paris can be part of a gerundive sentence whose subject is the Eiffel Tower or it can be an adjunct modifying the VP headed by saw:
We saw the Eiffel Tower flying to Paris.
In coordination ambiguity phrases can be conjoined by a conjunction like and. For example, the phrase old men and women can be bracketed as [old [men and women]], referring to old men and old women, or as [old men] and [women], in
which case it is only the men who are old.
The fact that there are many grammatically correct but semantically unreasonable parses for naturally occurring sentences is an irksome problem that affects all parsers. Fortunately, the CKY algorithm is designed to efficiently handle structural ambiguities.

Ambiguity 

Probabilistic augmentation of context-free grammars in which each 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) 

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

CKY Parsing

One of the earliest recognition and parsing algorithms.
It is data-driven, bottom-up approach and it's a dynamic programming algorithm.
Standard version can only recognize CFGs in Chomsky Normal Form (CNF).

CKY Algorithm

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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