The Rate of Convergence for Approximate Bayesian Computation

By Stuart Barber, Jochen Voss, M. Webster. 2013

87 citations recorded by the source corpus.

Approximate Bayesian Computation (ABC) is a popular computational method for likelihood-free Bayesian inference. The term "likelihood-free" refers to problems where the likelihood is intractable to compute or estimate directly, but where it is possible to generate simulated data $X$ relatively easily given a candidate set of parameters $\theta$ simulated from a prior distribution. Parameters which generate simulated data within some tolerance $\delta$ of the observed data $x^*$ are regarded as plausible, and a collection of such $\theta$ is used to estimate the posterior distribution $\theta\,|\,X\!=\!x^*$. Suitable choice of $\delta$ is vital for ABC methods to return good approximations to $\theta$ in reasonable computational time. While ABC methods are widely used in practice, particularly in population genetics, study of the mathematical properties of ABC estimators is still in its infancy. We prove that ABC estimates converge to the exact solution under very weak assumptions and, under slightly stronger assumptions, quantify the rate of this convergence. Our results can be used to guide the choice of the tolerance parameter $\delta$.

Subfields: Artificial Intelligence, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistics and Probability

Citations in the corpus

K2-ABC: Approximate Bayesian Computation with Infinite Dimensional Summary Statistics via Kernel Embeddings (2015, background): “3Details on consistency of ABC schemes as well as their convergence properties are given in Barber et al. (2013).”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, methodology): “Barber et al. (2015) studies the rate of convergence for ABC algorithms through the mean square error when approximating a posterior moment.”
On the Asymptotic Efficiency of Approximate Bayesian Computation Estimators (2015, methodology): “The convergence rate of the bias is shown to be O(ε2) in various settings (e.g. Barber et al., 2015).”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, unknown): “Barber et al. (2015) studies the rate of convergence for ABC algorithms through the mean square error when approximating a posterior moment.”