Approximate Bayesian computation: Difference between revisions

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Revision as of 11:31, 5 April 2021

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To assess how severely the dimensionality of a data set affects the analysis within the context of ABC, analytical formulas have been derived for the error of the ABC estimators as functions of the dimension of the summary statistics.<ref name="Blum10" /><ref name="Fearnhead12" /> In addition, Blum and François have investigated how the dimension of the summary statistics is related to the mean squared error for different correction adjustments to the error of ABC estimators. It was also argued that dimension reduction techniques are useful to avoid the curse-of-dimensionality, due to a potentially lower-dimensional underlying structure of summary statistics.<ref name="Blum10" /> Motivated by minimizing the quadratic loss of ABC estimators, Fearnhead and Prangle have proposed a scheme to project (possibly high-dimensional) data into estimates of the parameter posterior means; these means, now having the same dimension as the parameters, are then used as summary statistics for ABC.<ref name="Fearnhead12" />

ABC can be used to infer problems in high-dimensional parameter spaces, although one should account for the possibility of overfitting (e.g., see the model selection methods in <ref name="Ratmann" /> and <ref name="Francois" />). However, the probability of accepting the simulated values for the parameters under a given tolerance with the ABC rejection algorithm typically decreases exponentially with increasing dimensionality of the parameter space (due to the global acceptance criterion).<ref name="Csillery" /> Although no computational method (based on ABC or not) seems to be able to break the curse-of-dimensionality, methods have recently been developed to handle high-dimensional parameter spaces under certain assumptions (e.g., based on polynomial approximation on sparse grids,<ref name="Gerstner" /> which could potentially heavily reduce the simulation times for ABC). However, the applicability of such methods is problem dependent, and the difficulty of exploring parameter spaces should in general not be underestimated. For example, the introduction of deterministic global parameter estimation led to reports that the global optima obtained in several previous studies of low-dimensional problems were incorrect.<ref name="Singer" /> For certain problems, it might therefore be difficult to know whether the model is incorrect or, [[#Small number of models|as discussed above]], whether the explored region of the parameter space is inappropriate.<ref name="Templeton2009a" /> More pragmatic approaches are to cut the scope of the problem through model reduction,<ref name="Csillery" /> discretisation of variables and the use of canonical models such as noisy models. Noisy models exploit information on the conditional independence between variables.<ref>{{cite journal|last1= Cardenas |first1=IC|title= On the use of Bayesian networks as a meta-modeling approach to analyse uncertainties in slope stability analysis|journal =Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards|date=2019|volume=13|issue=1|pages=53–65|doi=10.1080/17499518.2018.1498524|s2cid=216590427}}</ref>

==Software==

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<ref name="Marin">Marin J-M, Pillai NS, Robert CP, Rousseau J (2011) Relevant statistics for Bayesian model choice. ArXiv:11104700v1 [mathST] 21 Oct 2011: 1-24.</ref>

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<ref name="Toni2010">Toni T, Stumpf MPH (2010). Simulation-based model selection for dynamical systems in systems and population biology, ''Bioinformatics' 26 (1):104–10.</ref>

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