Inside–outside algorithm

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]

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

Base Case:

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

General case:

Suppose there is a rule ${\displaystyle N_{j}\rightarrow N_{r}N_{s}}$ in the grammar, then the probability of generating ${\displaystyle w_{p}\cdots w_{q}}$ starting with a subtree rooted at ${\displaystyle N_{j}}$ is:

${\displaystyle \sum _{k=p}^{k=q-1}P(N_{j}\rightarrow N_{r}N_{s})\beta _{r}(p,k)\beta _{s}(k+1,q)}$

The inside probability ${\displaystyle \beta _{j}(p,q)}$ is just the sum over all such possible rules:

${\displaystyle \beta _{j}(p,q)=\sum _{N_{r},N_{s}}\sum _{k=p}^{k=q-1}P(N_{j}\rightarrow N_{r}N_{s})\beta _{r}(p,k)\beta _{s}(k+1,q)}$

Base Case:

${\displaystyle \alpha _{j}(1,n)={\begin{cases}1&{\mbox{if }}j=1\\0&{\mbox{otherwise}}\end{cases}}}$

Here the start symbol is ${\displaystyle N_{1}}$.

General case:

Suppose there is a rule ${\displaystyle N_{r}\rightarrow N_{j}N_{s}}$ in the grammar that generates ${\displaystyle N_{j}}$. Then the left contribution of that rule to the outside probability ${\displaystyle \alpha _{j}(p,q)}$ is:

${\displaystyle \sum _{k=q+1}^{k=n}P(N_{r}\rightarrow N_{j}N_{s})\alpha _{r}(p,k)\beta _{s}(q+1,k)}$

Now suppose there is a rule ${\displaystyle N_{r}\rightarrow N_{s}N_{j}}$ in the grammar. Then the right contribution of that rule to the outside probability ${\displaystyle \alpha _{j}(p,q)}$ is:

${\displaystyle \sum _{k=1}^{k=p-1}P(N_{r}\rightarrow N_{s}N_{j})\alpha _{r}(k,q)\beta _{s}(k,p-1)}$

The outside probability ${\displaystyle \alpha _{j}(p,q)}$ is the sum of the left and right contributions over all such rules:

${\displaystyle \alpha _{j}(p,q)=\sum _{N_{r},N_{s}}\sum _{k=q+1}^{k=n}P(N_{r}\rightarrow N_{j}N_{s})\alpha _{r}(p,k)\beta _{s}(q+1,k)+\sum _{N_{r},N_{s}}\sum _{k=1}^{k=p-1}P(N_{r}\rightarrow N_{s}N_{j})\alpha _{r}(k,q)\beta _{s}(k,p-1)}$

• 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.

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