This paper presents a novel feature selection method based on the conditional mutual information (CMI). The proposed High Order Conditional Mutual Information Maximization (HOCMIM) incorporates high order dependencies into the feature selection procedure and has a straightforward interpretation due to its bottom-up derivation. The HOCMIM is derived from the CMI's chain expansion and expressed as a maximization optimization problem. The maximization problem is solved using a greedy search procedure, which speeds up the entire feature selection process. The experiments are run on a set of benchmark datasets (20 in total). The HOCMIM is compared with eighteen state-of-the-art feature selection algorithms, from the results of two supervised learning classifiers (Support Vector Machine and K-Nearest Neighbor). The HOCMIM achieves the best results in terms of accuracy and shows to be faster than high order feature selection counterparts.