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:

University of Michigan - Ann Arbor

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.

Real word spelling correction

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

Speech and Language Processing (3rd ed. draft) 

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 $C(x)$:

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