A New Probabilistic Tree Expression for Probabilistic Model Building Genetic Programming

Abstract

This paper proposes a new expression of probabilistic tree for probabilistic model building GPs (PMBGP). Tree-structured PMBGPs estimate the probability of appearance of symbols at each node of the tree from past search information, and decide the symbol based on the probability at each node in generating a solution. The probabilistic tree is a key component of PMBGPs to keep appearance frequencies of symbols at each node by probabilistic tables, and the probabilistic prototype tree (PPT) expressed as a perfect tree has been often employed in order to include any breadth of trees. The depth of PPT is an important parameter that involves trade-off between the search accuracy and the computational cost. The appropriate depth depends on problems and is difficult to be estimated in advance. Our proposed probabilistic tree is constructed by a union operator between probabilistic trees, and its depth and breadth are extendable, so that the depth of the probabilistic tree is not required as a parameter and it needs not necessarily be a perfect tree. Through numerical experiments, we show the effectiveness of the proposed probabilistic tree by incorporating it to a local search-based crossover in symbolic regression problems.