The free energy principle is based on Bayesian probability theory, which is a branch of mathematics that deals with the calculation of probabilities. In Bayesian probability theory, probabilities are represented by numbers between 0 and 1, where 0 indicates that an event is impossible and 1 indicates that it is certain to occur.

The free energy principle proposes that the brain uses Bayesian inference to constantly update its beliefs about the world based on sensory information. This process can be formalized mathematically using Bayesian equations, which are used to calculate the probability of different events occurring. For example, the probability of event A occurring given that event B has occurred can be calculated using the following equation:

P(A|B) = P(B|A) * P(A) / P(B)

where P(A|B) is the probability of event A occurring given that event B has occurred, P(B|A) is the probability of event B occurring given that event A has occurred, P(A) is the probability of event A occurring, and P(B) is the probability of event B occurring.

The free energy principle also proposes that the brain uses these probabilities to minimize surprise or uncertainty, which can be measured using the concept of "free energy". The free energy of a system is a measure of the difference between the supposed brain's predictions and the sensory information it receives. Minimizing free energy corresponds to supposing that the brain is making the most accurate predictions possible, which can be achieved by choosing actions that reduce uncertainty or surprise. The mathematical formulation of this process is still an active area of research.

Active inference, a corollary process theory of FEP, is a framework for understanding perception and action that is based on the principles of enactivism. It is a computational approach to cognition that emphasizes the active, embodied nature of perception and action. According to the theory of active inference, perception and action are not two separate processes, but rather two sides of the same coin. Perception is not just a matter of passively receiving sensory input from the environment, but rather of actively seeking out sensory information that is relevant to the organism's goals and plans. Action, in turn, is not just a matter of executing pre-determined motor programs, but rather of actively seeking out sensory information that is relevant to the organism's goals and plans.