Adaptive Approximate Bayesian Computation Tolerance Selection

Abstract

Approximate Bayesian Computation (ABC) methods are increasingly used for inference in situations in which the likelihood function is either computationally costly or intractable to evaluate. Extensions of the basic ABC rejection algorithm have improved the computational efficiency of the procedure and broadened its applicability. The ABC -Population Monte Carlo (ABC-PMC) approach has become a popular choice for approximate sampling from the posterior. ABC-PMC is a sequential sampler with an iteratively decreasing value of the tolerance, which specifies how close the simulated data need to be to the real data for acceptance. We propose a method for adaptively selecting a sequence of tolerances that improves the computational efficiency of the algorithm over other common techniques. In addition we define a stopping rule as a by-product of the adaptation procedure, which assists in automating termination of sampling. The proposed automatic ABC-PMC algorithm can be easily implemented and we present several examples demonstrating its benefits in terms of computational efficiency.

Approximate Bayesian computation for finite mixture models

“…On the other hand, selecting a quantile that is too low can also contribute to computational inefficiencies because more draws from the proposal are needed within each iteration to find datasets that achieve the small tolerance. The rule used to shrink the tolerance is important as imprudent selection can lead to the ABC-PMC algorithm getting stuck in local modes [49,50]. This latter potential issue is particularly crucial when working with mixture models, since the multimodality of the likelihood function is a well-known problem that needs to be addressed in order to produce reliable statistical inference results.…”

“…Using this adaptive initialization, the parameter space is better sampled than if we only considered N draws from the prior distribution. Second, we use lower quantiles for the first several iterations of the ABC-PMC algorithm per the suggestion in Silk et al [49] and Simola et al [50], which can help the algorithm avoid local modes.…”

Sequentially Guided MCMC Proposals for Synthetic Likelihoods and Correlated Synthetic Likelihoods

“…The sequence of approximations depends on the sequence of tolerances 1:T , where T is the final iteration of the procedure. The different approaches used in order to create the series of decreasing tolerances (Beaumont et al, 2009;Del Moral et al, 2012), together with the choice for T , can lead to inefficient sampling (Simola et al, 2020). For this reason, rather than using the ABC-SMC algorithm, we employed one of its extensions, the "adaptive ABC Population Monte Carlo" (hereafter aABC-PMC) found in Simola et al (2020).…”

“…The different approaches used in order to create the series of decreasing tolerances (Beaumont et al, 2009;Del Moral et al, 2012), together with the choice for T , can lead to inefficient sampling (Simola et al, 2020). For this reason, rather than using the ABC-SMC algorithm, we employed one of its extensions, the "adaptive ABC Population Monte Carlo" (hereafter aABC-PMC) found in Simola et al (2020). When using the aABC-PMC algorithm both the series of decreasing tolerances and T are automatically selected, by looking at the online behaviour of the approximations to the posterior distribution (aABC-PMC is also implemented in ELFI).…”

Compressed Particle Methods for Expensive Models With Application in Astronomy and Remote Sensing

“…The second approach is the so-called approximate Bayesian computation (ABC) [19][20][21]. In the standard ABC scheme, model evaluation is substituted by evaluating a distance between the observed data and some artificial data generated according to the model.…”