# Inside–outside algorithm

{{ safesubst:#invoke:Unsubst||$N=Context |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In computer science, the inside–outside algorithm is a way of re-estimating production probabilities in a probabilistic context-free grammar. It was introduced James K. Baker in 1979 as a generalization of the forward–backward algorithm for parameter estimation on hidden Markov models to stochastic context-free grammars. It is used to compute expectations, for example as part of the expectation–maximization algorithm (an unsupervised learning algorithm).

## Inside and outside probabilities

The inside probability ${\displaystyle \beta _{j}(p,q)}$ is the total probability of generating words ${\displaystyle w_{p}\cdots w_{q}}$, given the root nonterminal ${\displaystyle N^{j}}$ and a grammar ${\displaystyle G}$:[1]

${\displaystyle \beta _{j}(p,q)=P(w_{pq}|N_{pq}^{j},G)}$

The outside probability ${\displaystyle \alpha _{j}(p,q)}$ is the total probability of beginning with the start symbol ${\displaystyle N^{1}}$ and generating the nonterminal ${\displaystyle N_{pq}^{j}}$ and all the words outside ${\displaystyle w_{p}\cdots w_{q}}$, given a grammar ${\displaystyle G}$:[1]

${\displaystyle \alpha _{j}(p,q)=P(w_{1(p-1)},N_{pq}^{j},w_{(q+1)m}|G)}$

## References

1. {{#invoke:citation/CS1|citation |CitationClass=book }}
• J. Baker (1979): Trainable grammars for speech recognition. In J. J. Wolf and D. H. Klatt, editors, Speech communication papers presented at the 97th meeting of the Acoustical Society of America, pages 547–550, Cambridge, MA, June 1979. MIT.
• Karim Lari, Steve J. Young (1990): The estimation of stochastic context-free grammars using the inside–outside algorithm. Computer Speech and Language, 4:35–56.
• Karim Lari, Steve J. Young (1991): Applications of stochastic context-free grammars using the Inside–Outside algorithm. Computer Speech and Language, 5:237–257.
• Fernando Pereira, Yves Schabes (1992): Inside–outside reestimation from partially bracketed corpora. Proceedings of the 30th annual meeting on Association for Computational Linguistics, Association for Computational Linguistics, 128–135.