Grammar theory to model symbol strings originated from work in computational linguistics aiming to understand the structure of natural languages. Probabilistic context free grammars (PCFGs) have been applied in probabilistic modeling of RNA structures almost 40 years after they were introduced in computational linguistics.
PCFGs extend context-free grammars similar to how hidden Markov models extend regular grammars. Each production is assigned a probability. The probability of a derivation (parse) is the product of the probabilities of the productions used in that derivation. These probabilities can be viewed as parameters of the model, and for large problems it is convenient to learn these parameters via machine learning. A probabilistic grammar's validity is constrained by context of its training dataset.
PCFGs have application in areas as diverse as natural language processing to the study the structure of RNA molecules and design of programming languages. Designing efficient PCFGs has to weigh factors of scalability and generality. Issues such as grammar ambiguity must be resolved. The grammar design affects results accuracy. Grammar parsing algorithms have various time and memory requirements.
Derivation: The process of recursive generation of strings from a grammar.
Parsing: Finding a valid derivation using an automaton.
Parse Tree: The alignment of the grammar to a sequence.
An example of a parser for PCFG grammars is the pushdown automaton. The algorithm parses grammar nonterminals from left to right in a stack-like manner. This brute-force approach is not very efficient. In RNA secondary structure prediction variants of the Cocke–Younger–Kasami (CYK) algorithm provide more efficient alternatives to grammar parsing than pushdown automata. Another example of a PCFG parser is the Stanford Statistical Parser which has been trained using Treebank.
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A stochastic grammar (statistical grammar) is a grammar framework with a probabilistic notion of grammaticality: Stochastic context-free grammar Statistical parsing Data-oriented parsing Hidden Markov model Estimation theory The grammar is realized as a language model. Allowed sentences are stored in a database together with the frequency how common a sentence is. Statistical natural language processing uses stochastic, probabilistic and statistical methods, especially to resolve difficulties that arise because longer sentences are highly ambiguous when processed with realistic grammars, yielding thousands or millions of possible analyses.
The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events, especially in the context of Markov information sources and hidden Markov models (HMM). The algorithm has found universal application in decoding the convolutional codes used in both CDMA and GSM digital cellular, dial-up modems, satellite, deep-space communications, and 802.
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