When a sequence satisfies a Markov condition with $au = 1$, the data is characterized by a first-order Markov model. In this case, the factorization of the joint probability simplifies to a product of probabilities for each element given only the immediately preceding element: $P(x_1, \ldots, x_T) = P(x_1) \prod_{t=2}^T P(x_t \mid x_{t-1})$. When $au = k$, the data is characterized by a $k^{ extrm{th}}$-order Markov model, which conditions on the $k$ previous time steps. For discrete data like language, a true Markov model estimates $P(x_t \mid x_{t-1})$ by counting the relative frequency of each word occurring in each context, allowing the most likely sequence to be computed efficiently using dynamic programming.