Winckler, Nicolas

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Bayesian analysis of the EC-decay rate oscillations - Part II ∗N. Winckler1 and the Two-Body Weak Decay collaboration11GSI, DarmstadtBayesian model selectionWe present the results of the model probabilities andBayes factor of the model and data defined in Ref. [1, 2].The table 1 shows the results obtained for uniform priorsand different data sets. In table 1, the uniform prior of ωis defined in [0, 3], which exclude the contribution of theglobal maximum likelihood at ω = 5.35.EC data β+ data EC data(245 MHz res.) (245 MHz res.) (cap. pick-up)Sample size N 3616 2912 2989Range (s) [6,60] [10,60] [6,60]P (M0|data) 66.3% 84.94% 0.03%P (M1|data) 33.7% 15.06% 99.97%B01 1.97 5.64 0.0003B10 0.5 0.17 3645.4Table 1: Model probabilities and Bayes factors results.Prior sensitivity analysisIn order to study the the prior sensitivities, other priordistributions have been used in the analysis . For exam-ple, increasing the range of the uniform prior of the angu-lar frequency to [0, 7] increases the probability of M1 toabout 43%, as larger frequency components in the likeli-hood function are taken into account.The use of gaussian priors with mean and width obtainedfrom the 2007 experimental results – i.e. a = 0.23(4),ω = 0.88(3), φ = −1.6(5) – have shown, for the modelM1, small probabilities of about 0.3%. This small prob-ability can be explained when considering the likelihoodfunction. The maximum contribution of the likelihoodfunction is found for a = 0.09(2) in the resonator data.The strength of the likelihood is reduced for amplitudesoutside the range a = 0.09± 0.02. Using a gaussian priorNa(0.23, 0.04) results in a posterior with small weights inthe a = 0.09± 0.02 region, reducing the M1 posterior, andhence the Bayes factor. As a consequence, we note thatprevious measurements are not supported by the presentanalysis of the resonantor data, and that this latter does notexhibit a sufficiently strong likelihood at a = 0.09 ± 0.02to overcome the Na(0.23, 0.04) subjective prior.DiscussionAccording to Jeffreys’ scale, the Bayes factor for the 245MHz resonator β+- and EC- data have no support for themodulated decay model. This result is independent on thechosen priors. However the EC-data set obtained from thecapacitive Schottky pick-up present decisive Bayesian evi-dence for the modulated decay, which is in complete con-tradiction with results obtained in resonator data.∗This work is part of the EMMI Rapid Reaction Task Force presenta-tion given in Jena, July 2014As mentioned in Ref. [2], the observable of the pick-up and resonator data are not the same. The pick up dataconsist of three components, i.e. the decay time, the delayrequired to electron-cool the ion, and the systematic errorin the determination of the observable. The decay time +delay observable has been analyzed as well in the resonatordata. The obtained posterior probability is about 33% forthe oscillation model, which a priori exlude the delay dis-tribution as being responsible for the difference in the pick-up and resonator data. The main difference remaining be-tween these two data sets are the distribution of the system-atic errors in the determination of the observables. Thesesystematic errors are assumed to be small enough to be ne-glected in the resonator data because of the large signal-to-noise ratio of the ion signatures in the time resolved spec-tra. This assumption may not be valid for the pick-up timeresolved spectra, which present poor signal-to-noise ratio.ConclusionWe have shown that Bayesian model selection meth-ods do not support the oscillation model in the 245 MHzresonator data but support, nevertheless, oscillation in thepick-up data. This conclusion is corroborated by the re-sults of frequentist hypothesis testing and likelihood anal-ysis [2], by Bayesian information criterion [3] and by anindependent Bayesian analysis using the binned likelihoodmethod and the nested sampling algorithms [4]. We notethat AIC analysis presents oscillation in the resonator dataas well. However, it has been shown [5] that the associateddecision making in the AIC framework has a 99% Type IError rate, discrediting its reliability in our analysis.Bayesian and frequentist analysis of the decay time +delay in the resonator data have been performed, and do notsupport the oscillation model. Since the main differencebetween the significant and non significant data set remainsin the systematic error distributions, this result point outto possible uncontrolled systematic effects in the pick-updata.References[1] P. Kienle et al., “High-resolution measurement of the time-modulated orbital electron capture and of the β+ decay ofhydrogen-like 142Pm60+ ions”, PLB 726 (2013) 638[2] N. Winckler, et al., “Unbinned likelihood Analysis of the EC-decay rate oscillations - Part I & II”, GSI Report (2014)[3] N. Winckler, et al., “Further insight into Bayesian and Akaikeinformation criteria ”, GSI Report (2013)[4] M. Trassinelli, “ nested fit”, private communication (2014)[5] N. Winckler, “Bayesian and Akaike information criteria ofthe EC-decay rate oscillations”, GSI Report (2014)APPA-MML-AP-25 GSI SCIENTIFIC REPORT 2014240 DOI:10.15120/GR-2015-1-APPA-MML-AP-25

Bayesian analysis of the EC-decay rate oscillations - Part II

Bayesian analysis of the EC-decay rate oscillations - Part I ∗N. Winckler1 and the Two-Body Weak Decay collaboration11GSI, DarmstadtIntroductionIn this contribution we perform a Bayesian analysis withthe models and data presented in Ref. [1]. We first re-mind the definitions of the Bayesian quantities computedhere, and present the results of parameter estimation anal-ysis. The results of model selection and the conclusion arepresented in the part II of this GSI scientific report.ComputationIn the following M0 denotes the pure exponential decay,and M1 the modulated exponential decay. This analysishas been performed using the BAT [2] and CUBA [3] C++packages. All probabilities have been computed from un-binned likelihood, reducing the information loss.Parameter probabilitiesThe posterior probability density functions of the param-eter(s) of interest θ is given by :P (θ|data) = P (data|θ)P (θ)P (data) , (1)where P (θ) is the prior probability of θ, P (data) the nor-malization term, and P (data|θ) the marginal likelilihood.The Metropolis algorithm has been used to sample themarginal likelihood distribution. The high resolution of theobtained posterior distributions, as the one shown in figure1, have been achieved using 10 Markov chains, each with10 millions iterations. The convergence has been reachedafter about 140 thousands iterations.Model probabilitiesThe posterior probability of a model Mi given the datais given byP (Mi|data) = P (data|Mi)piiP (data) , (2)where pii is the prior probability of the model Mi, P (data)the normilzation term, and P (data|Mi) the Bayesian Evi-dence :P (data|Mi) =∫−→θiP (data|−→θi ,Mi)P (−→θi |Mi)d−→θi . (3)Given the modelsM0 and M1, the posterior odds is definedas :P (M0|data)P (M1|data) = B01pi0pi1(4)∗This work is part of the EMMI Rapid Reaction Task Force presenta-tion given in Jena, July 2014For equal model prior probabilities, the posterior odds re-duces to the Bayes factor B01. Both, model probabitlitiesand Bayes factors have been obtained using the VEGASMonte Carlo algorithm for the integration of the Bayesianevidence. The obtained relative precisions of the integralsare smaller than 10−3. The results of the computed modelprobabilities and Bayes factor are presented in the Part IIof this report.Figure 1: Two-dimensional posterior probability-densityfor the oscillation frequency ω and the oscillation ampli-tude a in the 245 MHz resonator EC-data set. The globalmode of the posterior P (a, ω|data,M1) is found for ω =5.35 and a = 0.09.Bayesian parameter estimationFor all parameters, and for all possible combination ofparameter pairs, the 1- and 2-dimensional posterior prob-abilities have been computed using the models M0 andM1 and the data presented in Ref. [1]. Different pri-ors have been used. The posteriors computed with uni-form prior distributions have shown, as expected, excellentagreements with the unbinned likelihood analysis (see e.g.figure 1 and Ref. [4]).References[1] P. Kienle et al., “High-resolution measurement of the time-modulated orbital electron capture and of the β+ decay ofhydrogen-like 142Pm60+ ions”, PLB 726 (2013) 638[2] A. Caldwell, D. Kollar, K. Kroeninger, “BAT - The BayesianAnalysis Toolkit”, Comput. Phys. Commun. 180 (2009)[3] T. Hanh, “Cuba—a library for multidimensional numericalintegration” Comput. Phys. Commun. 168 (2005)[4] N. Winckler, “Unbinned likelihood Analysis of the EC-decayrate oscillations - Part I and II”, GSI Report 2014 - 2015GSI SCIENTIFIC REPORT 2014 APPA-MML-AP-20DOI:10.15120/GR-2015-1-APPA-MML-AP-20 235