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An algorithm has been proposed for identifying a representative subset of summary statistics, by iteratively assessing whether an additional statistic introduces a meaningful modification of the posterior.<ref name="Joyce" /> One of the challenges here is that a large ABC approximation error may heavily influence the conclusions about the usefulness of a statistic at any stage of the procedure. Another method<ref name="Nunes" /> decomposes into two main steps. First, a reference approximation of the posterior is constructed by minimizing the [[Entropy (statistical thermodynamics)|entropy]]. Sets of candidate summaries are then evaluated by comparing the ABC-approximated posteriors with the reference posterior.
With both of these strategies, a subset of statistics is selected from a large set of candidate statistics. Instead, the [[partial least squares regression]] approach uses information from all the candidate statistics, each being weighted appropriately.<ref name="Wegmann" /> Recently, a method for constructing summaries in a semi-automatic manner has attained a considerable interest.<ref name="Fearnhead" /> This method is based on the observation that the optimal choice of summary statistics, when minimizing the quadratic loss of the parameter point estimates, can be obtained through the posterior mean of the parameters, which is approximated by performing a linear regression based on the simulated data. Summary statistics for model selection have been obtained using [[multinomial logistic regression]] on simulated data, treating competing models as the label to predict.<ref name="Prangle" />
Methods for the identification of summary statistics that could also simultaneously assess the influence on the approximation of the posterior would be of substantial value.<ref name="Marjoram" /> This is because the choice of summary statistics and the choice of tolerance constitute two sources of error in the resulting posterior distribution. These errors may corrupt the ranking of models and may also lead to incorrect model predictions
===Bayes factor with ABC and summary statistics===
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<ref name="Klinger2017">Klinger, E.; Rickert, D.; Hasenauer, J. (2017). pyABC: distributed, likelihood-free inference.</ref>
<ref name="Salvatier2016">{{cite journal | doi=10.7717/peerj-cs.55 | doi-access=free | title=Probabilistic programming in Python using PyMC3 | date=2016 | last1=Salvatier | first1=John | last2=Wiecki | first2=Thomas V. | last3=Fonnesbeck | first3=Christopher | journal=PeerJ Computer Science | volume=2 | pages=e55 | arxiv=1507.08050 }}</ref>
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<ref name="Prangle">{{cite journal | doi=10.1515/sagmb-2013-0012 | title=Semi-automatic selection of summary statistics for ABC model choice| date=2014 | last1=Prangle | first1=Dennis | last2=Fearnhead | first2=Paul | last3=Cox | first3=Murray P. | last4=Biggs | first4=Patrick J. | last5=French | first5=Nigel P. | journal=Stat Appl Genet Mol Biol | pages=67–82 | arxiv=1302.5624 }}</ref>
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