The probability for a Katz backoff n-gram is given by
$$P_{BO}(w_n|w_{n-N+1:n-1}) = $$
$$
\begin{cases}
P^*(w_n|w_{n-N+1:n-1}), \text{ if } C(w_{n-N+1:n}) > 0 \\
\alpha(w_{n-N+1:n-1})P_{BO}(w_n|w_{n-N+2:n-1}), \text{ otherwise}
\end{cases}
$$
where $\alpha$ is a function that distributes higher-order n-grams to lower-order n-grams, to ensure that the backoff model gives a correct probability distribution.

Katz Backoff Representation

Katz Backoff is a kind of backoff where we rely on a discounted probability P* if we have seen this n-gram before. Otherwise, we recursively back off to the Katz probability for the (n-1)-gram.

University of Michigan - Ann Arbor

Backoff is the method of estimating probabilities using to a lower-order n-gram if we have zero evidence for a higher-order n-gram; if the n-gram we need has zero counts, we approximate it by backing off to the (n-1)-gram, and continue backing off until we reach a history that has some counts.

Backoff

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

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