Approximate Bayesian computation (ABC) gives exact results under the assumption of model error

Area of Science:

• Statistics
• Computational Statistics
• Bayesian Inference

Background:

• Approximate Bayesian computation (ABC) methods approximate posterior distributions without explicit likelihood functions.
• ABC relies on simulating data from a model to compare with observed data.
• Existing ABC applications often involve approximations due to model or measurement errors.

Purpose of the Study:

• To demonstrate that ABC algorithms yield exact results under specific assumptions.
• To provide a theoretical framework for understanding approximations in prior ABC applications.
• To guide the selection of metrics and tolerances in future ABC studies.

Main Methods:

• Theoretical analysis of Approximate Bayesian computation (ABC) algorithms.
• Investigation of the impact of uniform additive model error terms.
• Generalization of ABC by introducing a variable acceptance probability based on data distance.
• Application of generalized ABC to approximate Markov chain Monte Carlo (MCMC) algorithms.

Main Results:

• ABC algorithms produce exact posterior approximations when sufficient summary statistics and a uniform additive model error are assumed.
• A generalized ABC framework allows for non-uniform error distributions and improved inference.
• The proposed generalization is applicable to approximate MCMC methods.

Conclusions:

• ABC algorithms can be viewed as calibration techniques for implicit stochastic models.
• This work clarifies the nature of approximations in ABC and suggests optimal choices for metrics and tolerances.
• The generalization of ABC offers a more flexible and powerful approach to likelihood-free inference.