The study of complex biological relationships is aided by large and high-dimensional data sets whose analysis often involves dimension reduction to highlight representative or informative directions of variation. In principle, information theory provides a general framework for quantifying complex statistical relationships for dimension reduction. Unfortunately, direct estimation of high-dimensional information theoretic quantities, such as entropy and mutual information (MI), is often unreliable given the relatively small sample sizes available for biological problems. Here, we develop and evaluate a hierarchy of approximations for high-dimensional information theoretic statistics from associated low-order terms, which can be more reliably estimated from limited samples. Due to a relationship between this metric and the minimum spanning tree over a graph representation of the system, we refer to these approximations as MIST (Maximum Information Spanning Trees).