# Noisy channel model

The noisy channel model is a framework used in spell checkers, question answering, speech recognition, and machine translation. In this model, the goal is to find the intended word given a word where the letters have been scrambled in some manner.

## Definition

Given an alphabet $\Sigma$, let $\Sigma^*$ be the set of all finite strings over $\Sigma$. Let the dictionary $D$ of valid words be some subset of $\Sigma^*$, i.e., $D\subseteq\Sigma^*$.

The noisy channel is the matrix

$\Gamma_{ws} = \Pr(s|w)$,

where $w\in D$ is the intended word and $s\in\Sigma^*$ is the scrambled word that was actually received.

## Example

Consider the English alphabet $\Sigma = \{a, b, c, ..., y, z, A, B, ..., Z, ...\}$. Some subset $D\subseteq\Sigma^*$ makes up the dictionary of valid English words.

There are several mistakes that may occur while typing, including:

1. Missing letters, e.g., leter instead of letter
4. Replacing letters, e.g., fimite instead of finite

To construct the noisy channel matrix $\Gamma$, we must consider the probability of each mistake, given the intended word ($\Pr(s|w)$ for all $w\in D$ and $s\in\Sigma^*$). These probabilities may be gathered, for example, by considering the Levenshtein distance between $s$ and $w$ or by comparing the draft of an essay with one that has been manually edited for spelling.

## Error-correction

The goal of the noisy channel model is to find the intended word given the scrambled word that was received. The decision function $\sigma : \Sigma^* \to D$ is a function that, given a scrambled word, returns the intended word.

Methods of constructing a decision function include the maximum likelihood rule, the maximum a posteriori rule, and the minimum distance rule.

In some cases, it may be better to accept the scrambled word as the intended word rather than attempt to find an intended word in the dictionary. For example, the word schönfinkeling may not be in the dictionary, but might in fact be the intended word.