On the mathematical axiomatization of approximate Bayesian computation. A robust set for estimating mechanistic network models through optimal transport

TL;DR

This paper explores the theoretical foundations of approximate Bayesian computation (ABC) using optimal transport theory, analyzing convergence, distances, and asymptotic properties to improve understanding of ABC's mathematical basis.

Contribution

It provides a rigorous formulation of ABC rejection algorithms, investigates distances based on OTT, and examines asymptotic behaviors and convergence properties.

Findings

Established a formal set-up for ABC rejection algorithms.

Analyzed the impact of OTT-based distances on ABC convergence.

Identified conditions leading to lack of concentration in ABC estimates.

Abstract

We research relations between optimal transport theory (OTT) and approximate Bayesian computation (ABC) possibly connected to relevant metrics defined on probability measures. Those of ABC are computational methods based on Bayesian statistics and applicable to a given generative model to estimate its a posteriori distribution in case the likelihood function is intractable. The idea is therefore to simulate sets of synthetic data from the model with respect to assigned parameters and, rather than comparing prospects of these data with the corresponding observed values as typically ABC requires, to employ just a distance between a chosen distribution associated to the synthetic data and another of the observed values. Our focus lies in theoretical and methodological aspects, although there would exist a remarkable part of algorithmic implementation, and more precisely issues regarding mathematical foundation and asymptotic properties are carefully analysed, inspired by an in-depth study of what is then our main bibliographic reference, that is Bernton et al. (2019), carrying out what follows: a rigorous formulation of the set-up for the ABC rejection algorithm, also to regain a transparent and general result of convergence as the ABC threshold goes to zero whereas the number n of samples from the prior stays fixed; general technical proposals about distances leaning on OTT; weak assumptions which lead to lower bounds for small values of threshold and as n goes to infinity, ultimately showing a reasonable possibility of lack of concentration which is contrary to what is proposed in Bernton et al. (2019) itself.