Approximate Bayesian Inference for Network Processes

Abstract

In network science and epidemiology, simulations are valuable for understanding real-world systems, informing predictions, and evaluating intervention strategies. However, for simulations to be useful, inference or calibration must be conducted on the internal parameters of the model. In many situations, this inference can be difficult, as many simulated models, even with relatively simple mechanistic rules, do not have computationally tractable likelihoods. For inferences on such models, one popular set of methods is Approximate Bayesian computation (ABC), an approach that has been utilized for models in population genetics (Tavaré et al., 1997; Beaumont et al., 2002), physics (Akeret et al., 2015), and ecology (Toni et al., 2009). In order to compare simulation outputs and observed data, ABC methods require the specification of a set of summary statistics. In this dissertation, we seek to combine ABC methods with literature on Mixture Density Networks (MDN), which are neural networks that aim to learn a parametrized approximation to the posterior distribution of the parameters, conditioned on observed data (Bishop, 1994). In Chapter 1, we will investigate the use of Mixture Density Network-Augmented ABC (MDN-ABC) (Hoffmann and Onnela, 2022) for inferences on epidemics where the event times (times of infection and recovery) are not observed. By learning informative summary statistics through an MDN, we show how valid Bayesian inferences can be obtained while circumventing the summary statistic selection step that most ABC methods rely on. Furthermore, we discuss the interpretability of the summary statistics obtained from MDNs. In Chapter 2, we will continue to explore the use of MDN-ABC for epidemics on networks, but in cases where the contact network itself is also unobserved. By adopting a framework for modeling noise and missingness on networks proposed by Young et al. (2020), we find that it is possible to account for contact network uncertainty in a statistically valid manner through an additional network sampling step. In this chapter, we apply this Network-Augmented MDN-ABC (NA-MDN-ABC) to conduct inferences on Tattoo Skin Disease (TSD) spreading among dolphins in Shark Bay, Australia (Powell et al., 2019) and estimate the per-contact transmissibility and infectious period of the disease. In Chapter 3, we will discuss Bayesian inferences for mixture-of-mechanisms models for networks. Bayesian inferences on the relative importance of network formation mechanisms remains a difficult problem, as mechanistic network models do not generally yield tractable likelihoods. Existing methods focus on utilizing network summary statistics (Ratmann et al., 2007; Raynal and Onnela, 2022), but it is not guaranteed that such statistics are informative or optimal. In this chapter, we will discuss the use of an MDN that utilizes a Graph Neural Network (GNN) to extract network information and conduct Bayesian inferences. Approximate Bayesian inference tools provide a flexible framework with which researchers can study complex systems through simulations. By leveraging neural networks to extract relevant information from datasets, the methods we present allow for the automated learning of summary statistics, which avoids the use of ad hoc summary statistics and circumventing the summary statistic selection step. Simulations, in principle, can be designed to emulate reality as closely as possible, given the available computational budget. However, parameter inference and estimation for such simulations often require automated, statistically principled methods for likelihood-free inferences. By developing Bayesian methods capable of accommodating a wide range of data inputs, we seek to bridge this existing gap between realistic simulations and valid statistical inferences.