The algorithm takes the input sentence $X={x_{1}, x_{2},..., x_{k},..., x_{n}}$ and generates a set of candidate words $C(x_{i})$ for each word $x_{i}$. The candidates include every word with a small edit distance from $x_{i}$. Then we pick the sentence with the highest language model probability.

Each sentence is scored by the noisy channel:

Noisy channel algorithm for real-word correction

Assume the probability of writing a word correctly is $\alpha$, and distribute $1-\alpha$ evenly over all other candidate corrections $C(x)$:

Channel model of the noisy channel

Detecting and correcting spelling errors even if they accidentally result in an actual
real word. 

This can happen from typographical errors (insertion, deletion, transposition) that accidentally produce a real word, or cognitive errors where the writer substituted the wrong spelling of a homophone or near-homophone. It is a much more difficult task than non-word errors correction since any word in the input text could be an error.

University of Michigan - Ann Arbor

Spelling correction is often considered from two perspectives, non-word spelling correction and real word spelling correction.

Spelling correction and the noisy channel

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

Speech and Language Processing (3rd ed. draft) 

Real word spelling correction

- Is used for treating the misspelled word as if a correctly spelled word had been “distorted” by being passed through a noisy communication channel.
- Is a kind of Bayesian inference.

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