Likelihood-free Bayesian inference for dynamic, stochastic simulators in the social sciences
Keywords: time series
machine learning
probabilistic machine learning
agent-based models
deep learning
Bayesian inference
density estimation
stochastic simulators
path signatures

Simulation models – such as agent-based models (abms) in the social sciences – are now used widely across scientific and commercial domains. However, such models often lack a tractable likelihood function, precluding standard likelihood-based statistical inference. In response to this challenge, the past two decades have seen the development of likelihood-free, simulation-based procedures for inferring simulator parameters within the computational statistics and machine learning communities. A prototypical and theoretically appealing approach is approximate Bayesian computation (abc), in which the pertinence of parameter values is determined on the basis of a meaningful notion of distance between the observed data and output generated by the simulator at those parameter values. However, abc typically requires many hundreds of thousands of calls to the simulator to construct accurate posterior densities, making it unsuitable for computationally expensive simulators such as macroeconomic abms. Furthermore, there are few abc approaches – and likelihoodfree inference (lfi) approaches more generally – that are compatible with generic time-series simulators: many approaches involve reducing the data to hand-crafted summary statistics – which can require substantial domain expertise and can lead to a deleterious loss of information – or making inappropriate assumptions, such as independent and identically distributed (iid) or regularly spaced observations.
In this thesis, we aim to develop the literature on likelihood-free inference algorithms for generic time-series simulators, with a focus on simulation models in economics and the social sciences. To this end, we present the following contributions:
Section 1: Introduction
In the first section of this thesis, we introduce the basic problem of (Bayesian) statistical inference, and the problem of extending this to generic simulation models for which standard likelihood-based inference procedures are not immediately available. We then provide a review of the literature in this area, both as it appears in the computational statistics community and the social sciences, before setting out the aims and objectives of the thesis.
Section 2: Approximate Bayesian Inference with Path Signatures
In the second section, we will introduce the use of the path signature as a natural, automatic feature set for approximate Bayesian inference algorithms involving generic timeseries simulators. Besides introducing path signatures, this section will consist of three main chapters. In the first of these, we will motivate and present experiments on the use of path signatures as automatic summary statistics in abc. In this way, we will demonstrate that signatures permit a natural notion of distance between sequential data and a powerful means to performing so-called semi-automatic abc, with competitive empirical performance in a set of benchmarking experiments. In the second and third, we will consider the problem of performing density ratio estimation (dre) – an alternative approach to lfi – using path signatures. In particular, we will investigate their use in neural dre methods in Chapter 4 as a natural means to learning low-dimensional, approximately sufficient summary statistics for sequential data; we will then demonstrate in Chapter 5 that their use in kernel logistic regression is able to yield more accurate parameter posteriors relative to competing methods in low–simulation-budget regimes.
Section 3: Black-box Neural Posterior Inference for Agent-based Models
In this section, we will concentrate more specifically on parameter inference for agentbased models, which is a class of simulation models that is growing in popularity in economics and the social sciences. We will argue for the use of neural dre and a further class of neural simulation-based inference methods – neural posterior estimation (npe) – in parameter inference tasks for complex and expensive simulation models such as abms. To do so, we will present experiments in which we demonstrate the greater simulation efficiency and accuracy of dre and npe, and thus their potential as generic inference procedures for expensive abms in economics and the social sciences. Finally, we will discuss how such approaches naturally extend to less typical but more complex inference tasks that may be encountered in abm settings, such as when sequences of graphs – rather than the more typically encountered sequences of Euclidean data – are observed. This can be the case when e.g. dynamic social networks or occupational mobility networks in labour markets are the target of the abm. Our contributions demonstrate that these methods are better tailored to the key challenges faced when calibrating arbitrary abms to data than the current most popular methods in the social scientific abm literature, and that these methods enable agent-based modellers to automatically perform parameter inference for abms that model complex and high-dimensional temporal datasets.