A Sequential Markov Probabilistic Aggregation Algorithm for Causal Emergence in Markov Aggregation

Causal emergence (CE) is a phenomenon in which macrodynamics provide better effective information (EI) than microdynamics. The CE is widely used as the objective function in Markov aggregation. The existing works focus on deterministic aggregation, which may not offer a good solution since the search space of each step is finite. To solve this problem, we propose a sequential Markov probabilistic aggregation (SMPA) algorithm. We first express the aggregation problem as an optimization problem, then find that the EI is maximized when the transition probability matrix is a permutation matrix, and prove that the optimization problem is a nonconvex function of the probabilistic aggregation matrix. In the SMPA algorithm, the optimization problem is split into multiple univariate optimizations. Compared with the deterministic aggregation algorithm, SMPA can achieve better greedy solutions. The experimental results indicate that probabilistic aggregation generally performs better than deterministic aggregation.