[Paper Review] An easy-to-use empirical likelihood ABC method

Sanjay Chaudhuri, Subhroshekhar Ghosh+2 more|arXiv (Cornell University)|Oct 3, 2018

Markov Chains and Monte Carlo Methods32 references4 citations

TL;DR

This paper proposes a novel, easy-to-use empirical likelihood-based ABC method that requires only a summary statistic, its observed value, and the ability to simulate the statistic under the model for any parameter value. The method achieves posterior consistency without relying on analytically tractable estimating equations or kernel approximations, offering a robust, data-driven alternative to existing ABC approaches with improved computational feasibility and interpretability.

ABSTRACT

Many scientifically well-motivated statistical models in natural, engineering and environmental sciences are specified through a generative process, but in some cases it may not be possible to write down a likelihood for these models analytically. Approximate Bayesian computation (ABC) methods, which allow Bayesian inference in these situations, are typically computationally intensive. Recently, computationally attractive empirical likelihood based ABC methods have been suggested in the literature. These methods heavily rely on the availability of a set of suitable analytically tractable estimating equations. We propose an easy-to-use empirical likelihood ABC method, where the only inputs required are a choice of summary statistic, it's observed value, and the ability to simulate summary statistics for any parameter value under the model. It is shown that the posterior obtained using the proposed method is consistent, and its performance is explored using various examples.

Motivation & Objective

• To develop a practical, user-friendly ABC method for models with intractable likelihoods, especially in natural and environmental sciences.
• To eliminate the need for analytically tractable estimating equations, which limits the applicability of existing empirical likelihood ABC methods.
• To ensure posterior consistency under mild regularity conditions, enabling reliable Bayesian inference in complex generative models.
• To provide a computationally efficient alternative to standard ABC and synthetic likelihood methods by avoiding multivariate normality assumptions.

Proposed method

• The method constructs an empirical likelihood function based on simulated summary statistics and the observed statistic, without requiring explicit estimating equations.
• It uses a constrained optimization framework to define a likelihood-like function over the parameter space, ensuring feasibility through convex hull constraints.
• The approach leverages data cloning principles, where increasing the number of simulated replicates concentrates the empirical likelihood on the true parameter value.
• The posterior is defined via a normalized empirical likelihood, avoiding kernel density approximations used in standard ABC.
• The method relies on the asymptotic behavior of empirical likelihood under repeated simulation, ensuring consistency as the number of simulations increases.
• It is fully data-dependent and interpretable, with no parametric assumptions on the distribution of summary statistics.

Experimental results

Research questions

• RQ1 Can a computationally efficient and easy-to-implement ABC method be developed that avoids the need for analytically tractable estimating equations?
• RQ2 Does an empirical likelihood-based ABC approach achieve posterior consistency when only summary statistics and simulation capability are available?
• RQ3 How does the proposed method compare in performance to existing ABC and synthetic likelihood methods in terms of accuracy and robustness?
• RQ4 Can the method maintain validity under non-normal dependence structures in summary statistics, without relying on normality assumptions?
• RQ5 Is the method robust to model misspecification and high-dimensional summary statistics?

Key findings

• The proposed empirical likelihood ABC method achieves posterior consistency, converging weakly to a point mass at the true parameter value as the number of simulations increases.
• The method does not require any analytically tractable estimating equations, making it applicable to a broader class of models with intractable likelihoods.
• Posterior concentration occurs even when the distribution of summary statistics is non-normal, demonstrating robustness to violations of normality assumptions.
• The method avoids kernel density approximation, reducing sensitivity to bandwidth selection and improving interpretability.
• Theoretical results show that the posterior assigns zero mass to any parameter set outside a shrinking neighborhood of the true parameter, confirming consistency.
• Numerical examples demonstrate the method’s practical feasibility and competitive performance across various models, including those with complex dependence structures.