Approximate Bayesian computation: Difference between revisions

Revision as of 00:47, 15 November 2023

Revision as of 07:19, 28 March 2024

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===Approximation of the posterior===

A non-negligible <math>\epsilon</math> comes with the price that one samples from <math>p(\theta|\rho(\hat{D},D)\le\epsilon)</math> instead of the true posterior <math>p(\theta|D)</math>. With a sufficiently small tolerance, and a sensible distance measure, the resulting distribution <math>p(\theta|\rho(\hat{D},D)\le\epsilon)</math> should often approximate the actual target distribution <math>p(\theta|D)</math> reasonably well. On the other hand, a tolerance that is large enough that every point in the parameter space becomes accepted will yield a replica of the prior distribution. There are empirical studies of the difference between <math>p(\theta|\rho(\hat{D},D)\le\epsilon)</math> and <math>p(\theta|D)</math> as a function of <math>\epsilon</math>,<ref name="Sisson" /> <ref>{{Cite journal |last=Peters |first=Gareth |date=2009 |title=Advances in Approximate Bayesian Computation and Trans-Dimensional Sampling Methodology |url=https://www.ssrn.com/abstract=3785580 |journal=SSRN Electronic Journal |language=en |doi=10.2139/ssrn.3785580 |issn=1556-5068}}</ref> and theoretical results for an upper <math>\epsilon</math>-dependent bound for the error in parameter estimates.<ref name="Dean" /> The accuracy of the posterior (defined as the expected quadratic loss) delivered by ABC as a function of <math>\epsilon</math> has also been investigated.<ref name="Fearnhead" /> However, the convergence of the distributions when <math>\epsilon</math> approaches zero, and how it depends on the distance measure used, is an important topic that has yet to be investigated in greater detail. In particular, it remains difficult to disentangle errors introduced by this approximation from errors due to model mis-specification.<ref name="Beaumont2010" />

As an attempt to correct some of the error due to a non-zero <math>\epsilon</math>, the usage of local linear weighted regression with ABC to reduce the variance of the posterior estimates has been suggested.<ref name="Beaumont2002" /> The method assigns weights to the parameters according to how well simulated summaries adhere to the observed ones and performs linear regression between the summaries and the weighted parameters in the vicinity of observed summaries. The obtained regression coefficients are used to correct sampled parameters in the direction of observed summaries. An improvement was suggested in the form of nonlinear regression using a feed-forward neural network model.<ref name="Blum2010" /> However, it has been shown that the posterior distributions obtained with these approaches are not always consistent with the prior distribution, which did lead to a reformulation of the regression adjustment that respects the prior distribution.<ref name="Leuenberger2009" />

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===Choice and sufficiency of summary statistics===

Summary statistics may be used to increase the acceptance rate of ABC for high-dimensional data. Low-dimensional sufficient statistics are optimal for this purpose, as they capture all relevant information present in the data in the simplest possible form.<ref name="Csillery" /><ref>{{Cite journal |last=Peters |first=Gareth William |last2=Wuthrich |first2=Mario V. |last3=Shevchenko |first3=Pavel V. |date=2009 |title=Chain Ladder Method: Bayesian Bootstrap Versus Classical Bootstrap |url=https://dx.doi.org/10.2139/ssrn.2980411 |journal=SSRN Electronic Journal |doi=10.2139/ssrn.2980411 |issn=1556-5068}}</ref><ref>{{Citation |last=Peters |first=G. W. |title=Likelihood-free Bayesian inference for alpha-stable models |date=2009-12-23 |url=http://arxiv.org/abs/0912.4729 |access-date=2024-03-28 |doi=10.48550/arXiv.0912.4729 |last2=Sisson |first2=S. A. |last3=Fan |first3=Y.}}</ref> However, low-dimensional sufficient statistics are typically unattainable for statistical models where ABC-based inference is most relevant, and consequently, some [[heuristic]] is usually necessary to identify useful low-dimensional summary statistics. The use of a set of poorly chosen summary statistics will often lead to inflated [[credible interval]]s due to the implied loss of information,<ref name="Csillery" /> which can also bias the discrimination between models. A review of methods for choosing summary statistics is available,<ref name="Blum12" /> which may provide valuable guidance in practice.

One approach to capture most of the information present in data would be to use many statistics, but the accuracy and stability of ABC appears to decrease rapidly with an increasing numbers of summary statistics.<ref name="Beaumont2010" /><ref name="Csillery" /> Instead, a better strategy is to focus on the relevant statistics only—relevancy depending on the whole inference problem, on the model used, and on the data at hand.<ref name="Nunes" />