Sometimes, we need to obtain parse probabilities in cases where we don't already have a probabilistic parser. We solve this problem through a series of expectation and maximization steps by beginning with a parser with equal rule probabilities to incrementally improve our estimates, then parsing the sentence, computing a probability for each parse, and using these probabilities to weight the counts, re-estimate the rule probabilities, and so on, until we come up with a final probability. 

Inside-Outside Algorithm

A treebank is a corpus of already-parsed sentences, which helps to learn probabilities for the rules of a grammar. Using a treebank, we can use the below formula to compute the probability of each expansion of a non-terminal by counting the number of times that expansion occurs and then normalizing. Without a treebank, we would need to account for any ambiguities, and then parse.

University of California, Santa Cruz

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

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

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