Bigrams or digrams are groups of two written letters, two syllables, or two words, and are very commonly used as the basis for simple statistical analysis of text. They are used in one of the most successful language models for speech recognition.[1] They are a special case of N-gram.

Gappy bigrams or skipping bigrams are word pairs which allow gaps (perhaps avoiding connecting words, or allowing some simulation of dependencies, as in a dependency grammar).

Head word bigrams are gappy bigrams with an explicit dependency relationship.

The term is also used in cryptography, where bigram frequency attacks have sometimes been used to attempt to solve cryptograms. See frequency analysis.

Bigrams help provide the conditional probability of a word given the preceding word, when the relation of the conditional probability is applied:

${\displaystyle P(W_n|W_{n-1}) = { P(W_{n-1},W_n) \over P(W_{n-1}) } }$

That is, the probability ${\displaystyle P() }$ of a word ${\displaystyle W_n}$ given the preceding word ${\displaystyle W_{n-1}}$ is equal to the probability of their bigram, or the co-occurrence of the two words ${\displaystyle P(W_{n-1},W_n)}$, divided by the probability of the preceding word.

## Bigram Frequency in the English language

The most common letter bigrams in the English language are listed below, according to Cornell University Math Explorer's Project[2] which measured over 40,000 words.

th 1.52%       en 0.55%       ng 0.18%
he 1.28%       ed 0.53%       of 0.16%
in 0.94%       to 0.52%       al 0.09%
er 0.94%       it 0.50%       de 0.09%
an 0.82%       ou 0.50%       se 0.08%
re 0.68%       ea 0.47%       le 0.08%
nd 0.63%       hi 0.46%       sa 0.06%
at 0.59%       is 0.46%       si 0.05%
on 0.57%       or 0.43%       ar 0.04%
nt 0.56%       ti 0.34%       ve 0.04%
ha 0.56%       as 0.33%       ra 0.04%
es 0.56%       te 0.27%       ld 0.02%
st 0.55%       et 0.19%       ur 0.02%


## References

1. Michael Collins. A new statistical parser based on bigram lexical dependencies. In Proceedings of the 34th Annual Meeting of the Association of Computational Linguistics, Santa Cruz, CA. 1996. pp.184-191.
2. Cornell Math Explorer's Project – Substitution Ciphers