An efficient algorithm for accurate computation of the Dirichlet-multinomial log-likelihood function

The Dirichlet-multinomial (DMN) distribution is a fundamental model for multicategory count data with overdispersion. This distribution has many uses in bioinformatics including applications to metagenomics data, transctriptomics and alternative splicing. The DMN distribution reduces to the multinomial distribution when the overdispersion parameter Psi is 0. Unfortunately, numerical computation of the DMN log-likelihood function by conventional methods results in instability in the neighborhood of Psi = 0. An alternative formulation circumvents this instability, but it leads to long runtimes that make it impractical for large count data common in bioinformatics. We have developed a newmethod for computation of the DMN log-likelihood to solve the instability problem without incurring long runtimes. The new approach is composed of a novel formula and an algorithm to extend its applicability. Our numerical experiments show that this new method both improves the accuracy of log-likelihood evaluation and the runtime by several orders of magnitude, especially in high-count data situations that are common in deep sequencing data. Using real metagenomic data, our method achieves manyfold runtime improvement. Our method increases the feasibility of using the DMN distribution to model many high-throughput problems in bioinformatics. We have included in our work an R package giving access to this method and a vingette applying this approach to metagenomic data.