Combining Subjective Probabilities and Data in Training Markov Logic Networks

Markov logic is a rich language that allows one to specify a knowledge base (called a Markov logic network or MLN) as a set of weighted first-order logic formulas, and to define a probability distribution over truth assignments to ground atoms using this knowledge base. Unless the knowledge base satisfies certain special conditions, the weight of a formula cannot be related to the probability of the formula without taking into account the weights of the other formulas. In general, this is not an issue, since the weights are learned from training data. However, in many domains (e.g. healthcare, dependable systems, etc.), only little or no training data may be available. Fortunately, in many such cases, one has access to a domain expert whose knowledge is available in the form of subjective probabilities. Within the framework of Bayesian statistics, we present a formalism for using a domain expert’s knowledge for weight learning. Our approach defines priors that are different from and more general than previously used Gaussian priors over weights. Previous work has demonstrated that conditional probabilities could serve as input to log-linear models using probabilistic constraint logic. However, this previous approach could only handle the case when the subjective probabilities provided by the domain expert were consistent. We show how one can learn weights in an MLN by combining subjective probabilities and training data, without requiring that the domain expert provides consistent knowledge. Additionally, we also provide a formalism for capturing conditional subjective probabilities, which are often easier to obtain and more reliable than non-conditional probabilities. We demonstrate the effectiveness of our approach by extensive experiments in a domain that models failure dependencies in a cyber-physical system. Moreover, we demonstrate the advantages of using our proposed prior over that of using non-zero mean Gaussian priors in a commonly cited social network MLN testbed.