Constrained Adjusted Maximum a Posteriori Estimation of Bayesian Network Parameters

Abstract

Maximum a posteriori estimation (MAP) with Dirichlet prior has been shown to be effective in improving the parameter learning of Bayesian networks when the available data are insufficient. Given no extra domain knowledge, uniform prior is often considered for regularization. However, when the underlying parameter distribution is non-uniform or skewed, uniform prior does not work well, and a more informative prior is required. In reality, unless the domain experts are extremely unfamiliar with the network, they would be able to provide some reliable knowledge on the studied network. With that knowledge, we can automatically refine informative priors and select reasonable equivalent sample size (ESS). In this paper, considering the parameter constraints that are transformed from the domain knowledge, we propose a Constrained adjusted Maximum a Posteriori (CaMAP) estimation method, which is featured by two novel techniques. First, to draw an informative prior distribution (or prior shape), we present a novel sampling method that can construct the prior distribution from the constraints. Then, to find the optimal ESS (or prior strength), we derive constraints on the ESS from the parameter constraints and select the optimal ESS by cross-validation. Numerical experiments show that the proposed method is superior to other learning algorithms.

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“…Gao [21] proposes a constrained Bayesian estimation (CBE) algorithm that enhances learning accuracy by introducing expert criteria. Di [22] proposes a constrained adjusted MAP (CaMAP) algorithm by choosing a reasonable equivalent sample size. The qualitative maximum a posteriori estimation (QMAP) algorithm proposed by Chang [23] performs Monte Carlo (MC) sampling on the feasible domain of the parameters determined based on constraints.…”

“…The prior distribution of [21] is set to be the BDeu priors rather than the transferred priors, which are more meaningful. When no parameter constraints are available, the CaMAP [22] is inferior to the MAP. The FC-QMAP algorithm [24] only outperforms QMAP when dealing with small datasets.…”