This paper presents a continuous, information-theoretic extension of the Free Energy Principle through the concept of Markov blanket density, i.e., a scalar field that quantifies the degree of conditional independence between internal and external states at each point in space (ranging from 0 for full coupling to 1 for full separation). It demonstrates that active inference dynamics, including the minimization of variational and expected free energy, naturally emerge from spatial gradients in this density, making Markov blanket density a necessary foundation for the Free Energy Principle. These ideas are developed through a mathematically framework that links density gradients to precise and testable dynamics, offering a foundation for novel predictions and simulation paradigms.

翻译：本文通过马尔可夫毯密度这一概念，提出了自由能原理的一种连续信息论扩展。马尔可夫毯密度是一个标量场，用于量化空间各点处内部状态与外部状态之间的条件独立程度（取值范围从0表示完全耦合到1表示完全分离）。研究表明，主动推理动力学——包括变分自由能与期望自由能的最小化——自然地源于该密度的空间梯度，这使得马尔可夫毯密度成为自由能原理的必要基础。这些思想通过一个数学框架得以发展，该框架将密度梯度与精确且可检验的动力学联系起来，为新的预测与仿真范式提供了理论基础。