How does a higher-order Markov model extend on first-order models?

A higher-order Markov model enhances first-order models by allowing the current state to depend not just on the immediate previous state, but on multiple preceding states. In a first-order Markov model, the assumption is that the future state is dependent only on the present state, which simplifies the modeling process but can limit accuracy in capturing the dynamics of certain systems.

In contrast, higher-order models capture longer-range dependencies by including a defined number of earlier states in their calculations. This means that the transitions can more accurately reflect the complexities of real-world processes where history has significant effects on future outcomes. This complexity can lead to improved predictions and a more nuanced understanding of the dynamics involved, particularly in time-series data or other contexts where past behaviors influence future behavior in a non-linear way.

Incorporating additional dependencies allows these models to better understand and predict sequences that exhibit patterns driven by longer histories rather than just immediate past states. Thus, the focus on these dependencies is what fundamentally differentiates higher-order Markov models from their first-order counterparts.