Recalibration: A post-processing method for approximate Bayesian computation

Abstract

A new recalibration post-processing method is presented to improve the quality of the posterior approximation when using Approximate Bayesian Computation (ABC) algorithms. Recalibration may be used in conjunction with existing post-processing methods, such as regression-adjustments. In addition, this work extends and strengthens the links between ABC and indirect inference algorithms, allowing more extensive use of misspecified auxiliary models in the ABC context. The method is illustrated using simulated examples to demonstrate the effects of recalibration under various conditions, and through an application to an analysis of stereological extremes both with and without the use of auxiliary models. Code to implement recalibration postprocessing is available in the R package, abctools.

Model Misspecification in Approximate Bayesian Computation: Consequences and Diagnostics

“…Furthermore, we demonstrate that post-processing the ABC output by local regression may lead to poor inference in misspecified models, and we propose an alternative approach that is less sensitive to the correctness of the model specification. At this stage, it is unclear whether other post-processing ABC approaches, such as the marginal adjustment approach (Nott et al, 2014) or the recalibration approach (Rodrigues et al, 2018), will perform similarly to local regression post-processing under model misspecification, and more research on this front is necessary to obtain conclusive results.…”

“…Algorithm 1 in Table 1 details the common accept-reject implementation of ABC, which can be augmented with additional steps to increase sampling efficiency; see, for example the Markov chain Monte Carlo-ABC approach of Marjoram et al (2003), or the sequential Monte Carlo-ABC approach of Sisson et al (2007). Post-processing of the simulated pairs {θ i , η.z i /} has also been proposed as a means of obtaining more accurate posterior approximations (see, for example, the local linear regression adjustment approach of Beaumont et al (2002), the marginal adjustment approach of Nott et al (2014) or the recalibration approach of Rodrigues et al (2018)).…”

Calibration Procedures for Approximate Bayesian Credible Sets

“…When we approximate b y (α) with Pr(φ ≤q Y (α)|Y ∈ ∆ y ) we assume b y (α) does not depend on y for y ∈ ∆ y . If φ ∼ π(·) and y ∼ p(·|φ) so that φ|y ∼ π(·|y ) then Rodrigues et al (2018) have φ (adj) ∼ π(·|y) (approximately). It is straightforward to check that b y is invertible at each y.…”

“…It is straightforward to check that b y is invertible at each y. Rodrigues et al (2018) use the empirical estimatê G −1 y •Ĝ y (φ) to implement the map. We do not wish to make an adjustment of the kind Rodrigues et al (2018) make, as we do not need to map samples θ at y to samples φ (adj) at the data y.…”

“…Algorithm 4, given here for lower-tail intervals, can be extended to handle equal-tailed intervals and HPD regions (Rodrigues et al (2018) set this out in a general way). It is sufficient that credible sets at smaller nominal α are nested within credible sets at larger α.…”