A sum-product algorithm with polynomials for computing exact derivatives of the likelihood in Bayesian networks

We consider a Bayesian network with a parameter $\theta$. It is well known that the probability of an \emph{evidence} conditional on $\theta$ (the likelihood) can be computed through a sum-product of potentials. In this work we propose a polynomial version of the sum-product algorithm based on generating functions for computing both the likelihood function and all its exact derivatives. For a unidimensional parameter we obtain the derivatives up to order $d$ with a complexity $\mathcal{O} (C \times d^2)$ where $C$ is the complexity for computing the likelihood alone. For a parameter of $p$ dimensions we obtain the likelihood, the gradient and the Hessian with a complexity $\mathcal{O} (C \times p^2)$. These complexities are similar to the numerical method with the main advantage that it computes exact derivatives instead of approximations.