A Family of Distributions of Random Subsets for Controlling Positive and Negative Dependence

TL;DR

This paper introduces the discrete kernel point process (DKPP), a new family of distributions that models positive and negative dependence in random subsets, with computational methods and demonstrated controllability.

Contribution

The paper proposes DKPP, unifying models like determinantal point processes and Boltzmann machines, and develops computational techniques for inference and learning.

Findings

Demonstrates controllability of dependence types in DKPP

Develops methods for marginal and conditional probability calculations

Shows effectiveness of computational algorithms through numerical experiments

Abstract

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.