Marginal Likelihood Integrals for Mixtures of Independence Models
  Inference in Bayesian statistics involves the evaluation of marginal likelihood integrals. We present algebraic algorithms for computing such integrals exactly for discrete data of small sample size. Our methods apply to both uniform priors and Dirichlet priors. The underlying statistical models are mixtures of independent distributions, or, in geometric language, secant varieties of Segre-Veronese varieties. 
Shaowei Lin, Bernd Sturmfels and Zhiqiang Xu