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.