We study the generalization behavior of Markov Logic Networks (MLNs) across relational structures of different sizes. Multiple works have noticed that MLNs learned on a given domain generalize poorly across domains of different sizes. This behavior emerges from a lack of internal consistency within an MLN when used across different domain sizes. In this paper, we quantify this inconsistency and bound it in terms of the variance of the MLN parameters. The parameter variance also bounds the KL divergence between an MLN's marginal distributions taken from different domain sizes. We use these bounds to show that maximizing the data log-likelihood while simultaneously minimizing the parameter variance corresponds to two natural notions of generalization across domain sizes. Our theoretical results apply to Exponential Random Graphs and other Markov network based relational models. Finally, we observe that solutions known to decrease the variance of the MLN parameters, like regularization and Domain-Size Aware MLNs, increase the internal consistency of the MLNs. We empirically verify our results on four different datasets, with different methods to control parameter variance, showing that controlling parameter variance leads to better generalization.

The Gaussian process is a powerful and flexible technique for interpolating spatiotemporal data, especially with its ability to capture complex trends and uncertainty from the input signal. This chapter describes Gaussian processes as an interpolation technique for geospatial trajectories. A Gaussian process models measurements of a trajectory as coming from a multidimensional Gaussian, and it produces for each timestamp a Gaussian distribution as a prediction. We discuss elements that need to be considered when applying Gaussian process to trajectories, common choices for those elements, and provide a concrete example of implementing a Gaussian process.