A Family of Distributions of Random Subsets for Controlling Positive and Negative Dependence
This work addresses a fundamental problem in probabilistic modeling for researchers and practitioners dealing with dependence structures, though it appears incremental as it builds on existing models like determinantal point processes.
The authors tackled the challenge of modeling both positive and negative dependence in random subsets by introducing the discrete kernel point process (DKPP), a new family of distributions that includes determinantal point processes and parts of Boltzmann machines, and demonstrated its controllability and effectiveness through numerical experiments.
Positive and negative dependence are fundamental concepts that characterize the attractive and repulsive behavior of random subsets. Although some probabilistic models are known to exhibit positive or negative dependence, it is challenging to seamlessly bridge them with a practicable probabilistic model. In this study, we introduce a new family of distributions, named the discrete kernel point process (DKPP), which includes determinantal point processes and parts of Boltzmann machines. We also develop some computational methods for probabilistic operations and inference with DKPPs, such as calculating marginal and conditional probabilities and learning the parameters. Our numerical experiments demonstrate the controllability of positive and negative dependence and the effectiveness of the computational methods for DKPPs.