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

Alexandra Lefebvre (1) , Grégory Nuel (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We consider a Bayesian network with a parameter θ. It is well known that the probability of an evidence conditional on θ (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 O(C × 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 O(C × p 2). These complexities are similar to the numerical method with the main advantage that it computes exact derivatives instead of approximations.

Mots clés: derivatives of the likelihood; Bayesian network; sum-product algorithm; generating functions

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Alexandra Lefebvre, Grégory Nuel. A sum-product algorithm with polynomials for computing exact derivatives of the likelihood in Bayesian networks. Proceedings of Machine Learning Research, 2018, 72, pp.201 - 212. ⟨hal-02350422⟩