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- 1. Uncertainties within some Bayesian concepts: Examples from classnotes Christian P. Robert Universit´ Paris-Dauphine, IuF, and CREST-INSEE e http://www.ceremade.dauphine.fr/~xian July 31, 2011Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 1 / 30
- 2. Outline Anyone not shocked by the Bayesian theory of inference has not understood it. — S. Senn, Bayesian Analysis, 20081 Testing2 Fully speciﬁed models?3 Model choiceChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 2 / 30
- 3. Add: Call for vignettesKerrie Mengersen and myself are collecting proposals towards a collectionof vignettes on the theme When is Bayesian analysis really successfull?celebrating notable achievements of Bayesian analysis. [deadline: September 30]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 3 / 30
- 4. Bayes factors The Jeﬀreys-subjective synthesis betrays a much more dangerous confusion than the Neyman-Pearson-Fisher synthesis as regards hypothesis tests — S. Senn, BA, 2008Deﬁnition (Bayes factors)When testing H0 : θ ∈ Θ0 vs. Ha : θ ∈ Θ0 use f (x|θ)π0 (θ)dθ π(Θ0 |x) π(Θ0 ) Θ0 B01 = = π(Θc |x) 0 π(Θc ) 0 f (x|θ)π1 (θ)dθ Θc 0 [Good, 1958 & Jeﬀreys, 1939]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 4 / 30
- 5. Self-contained concept Derived from 0 − 1 loss and Bayes rule: acceptance if B01 > {(1 − π(Θ0 ))/a1 }/{π(Θ0 )/a0 }Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 5 / 30
- 6. Self-contained concept Derived from 0 − 1 loss and Bayes rule: acceptance if B01 > {(1 − π(Θ0 ))/a1 }/{π(Θ0 )/a0 } but used outside decision-theoretic environment eliminates choice of π(Θ0 )Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 5 / 30
- 7. Self-contained concept Derived from 0 − 1 loss and Bayes rule: acceptance if B01 > {(1 − π(Θ0 ))/a1 }/{π(Θ0 )/a0 } but used outside decision-theoretic environment eliminates choice of π(Θ0 ) but still depends on the choice of (π0 , π1 )Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 5 / 30
- 8. Self-contained concept Derived from 0 − 1 loss and Bayes rule: acceptance if B01 > {(1 − π(Θ0 ))/a1 }/{π(Θ0 )/a0 } but used outside decision-theoretic environment eliminates choice of π(Θ0 ) but still depends on the choice of (π0 , π1 ) Jeﬀreys’ [arbitrary] scale of evidence: π if log10 (B10 ) between 0 and 0.5, evidence against H0 weak, π if log10 (B10 ) 0.5 and 1, evidence substantial, π if log10 (B10 ) 1 and 2, evidence strong and π if log10 (B10 ) above 2, evidence decisive convergent if used with proper statisticsChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 5 / 30
- 9. Diﬃculties with ABC-Bayes factors ‘This is also why focus on model discrimination typically (...) proceeds by (...) accepting that the Bayes Factor that one obtains is only derived from the summary statistics and may in no way correspond to that of the full model.’ — S. Sisson, Jan. 31, 2011, X.’OgChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 6 / 30
- 10. Diﬃculties with ABC-Bayes factors ‘This is also why focus on model discrimination typically (...) proceeds by (...) accepting that the Bayes Factor that one obtains is only derived from the summary statistics and may in no way correspond to that of the full model.’ — S. Sisson, Jan. 31, 2011, X.’OgIn the Poisson versus geometric case, if E[yi ] = θ0 > 0, η (θ0 + 1)2 −θ0 lim B12 (y) = e n→∞ θ0Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 6 / 30
- 11. Diﬃculties with ABC-Bayes factorsLaplace vs. Normal models:Comparing a sample x1 , . . . , xn from the Laplace (double-exponential) √L(µ, 1/ 2) distribution 1 √ f (x|µ) = √ exp{− 2|x − µ|} . 2or from the Normal N (µ, 1)Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 7 / 30
- 12. Diﬃculties with ABC-Bayes factorsEmpirical mean, median and variance have the same mean under bothmodels: useless!Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 7 / 30
- 13. Diﬃculties with ABC-Bayes factorsMedian absolute deviation: priceless!Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 7 / 30
- 14. Point null hypotheses I have no patience for statistical methods that assign positive probability to point hypotheses of the θ = 0 type that can never actually be true — A. Gelman, BA, 2008Particular case H0 : θ = θ0Take ρ0 = Prπ (θ = θ0 ) and π1 prior density under Ha .Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 8 / 30
- 15. Point null hypotheses I have no patience for statistical methods that assign positive probability to point hypotheses of the θ = 0 type that can never actually be true — A. Gelman, BA, 2008Particular case H0 : θ = θ0Take ρ0 = Prπ (θ = θ0 ) and π1 prior density under Ha .Posterior probability of H0 f (x|θ0 )ρ0 f (x|θ0 )ρ0 π(Θ0 |x) = = f (x|θ)π(θ) dθ f (x|θ0 )ρ0 + (1 − ρ0 )m1 (x)and marginal under Ha m1 (x) = f (x|θ)g1 (θ) dθ. Θ1Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 8 / 30
- 16. Point null hypotheses (cont’d)Example (Normal mean)Test of H0 : θ = 0 when x ∼ N (θ, 1): we take π1 as N (0, τ 2 ) then −1 1 − ρ0 σ2 τ 2 x2 π(θ = 0|x) = 1 + exp ρ0 σ2 + τ 2 2σ 2 (σ 2 + τ 2 )Inﬂuence of τ : τ /x 0 0.68 1.28 1.96 1 0.586 0.557 0.484 0.351 10 0.768 0.729 0.612 0.366Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 9 / 30
- 17. A fundamental diﬃcultyImproper priors are not allowed in this settingIf π1 (dθ1 ) = ∞ or π2 (dθ2 ) = ∞ Θ1 Θ2then either π1 or π2 cannot be coherently normalisedChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 10 / 30
- 18. A fundamental diﬃcultyImproper priors are not allowed in this settingIf π1 (dθ1 ) = ∞ or π2 (dθ2 ) = ∞ Θ1 Θ2then either π1 or π2 cannot be coherently normalised but thenormalisation matters in the Bayes factorChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 10 / 30
- 19. Jeﬀreys unaware of the problem??Example of testing for a zero normal mean: If σ is the standard error and λ the true value, λ is 0 on q. We want a suitable form for its prior on q . (...) Then we should take P (qdσ|H) ∝ dσ/σ λ P (q dσdλ|H) ∝ f dσ/σdλ/λ σ where f [is a true density] (ToP, V, §5.2).Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 11 / 30
- 20. Jeﬀreys unaware of the problem??Example of testing for a zero normal mean: If σ is the standard error and λ the true value, λ is 0 on q. We want a suitable form for its prior on q . (...) Then we should take P (qdσ|H) ∝ dσ/σ λ P (q dσdλ|H) ∝ f dσ/σdλ/λ σ where f [is a true density] (ToP, V, §5.2).Unavoidable fallacy of the “same” σ?!Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 11 / 30
- 21. Puzzling alternativesWhen taking two normal samples x11 , . . . , x1n1 and x21 , . . . , x2n2 withmeans λ1 and λ2 and same variance σ, testing for H0 : λ1 = λ2 getsoutwordly: ...we are really considering four hypotheses, not two as in the test for agreement of a location parameter with zero; for neither may be disturbed, or either, or both may.ToP then uses parameters (λ, σ) in all versions of the alternativehypotheses, with π0 (λ, σ) ∝ 1/σ π1 (λ, σ, λ1 ) ∝ 1/π{σ 2 + (λ1 − λ)2 } π2 (λ, σ, λ2 ) ∝ 1/π{σ 2 + (λ2 − λ)2 } π12 (λ, σ, λ1 , λ2 ) ∝ σ/π 2 {σ 2 + (λ1 − λ)2 }{σ 2 + (λ2 − λ)2 }Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 12 / 30
- 22. Puzzling alternativesToP misses the points that 1 λ does not have the same meaning under q, under q1 (= λ2 ) and under q2 (= λ1 ) 2 λ has no precise meaning under q12 [hyperparameter?] On q12 , since λ does not appear explicitely in the likelihood we can integrate it (V, §5.41). 3 even σ has a varying meaning over hypotheses 4 integrating over measures 2 dσdλ1 dλ2 P (q12 dσdλ1 dλ2 |H) ∝ π 4σ 2 + (λ1 − λ2 )2 simply deﬁnes a new improper prior...Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 13 / 30
- 23. Addiction to modelsOne potential diﬃculty with Bayesian analysis is its ultimate dependenceon model(s) speciﬁcation π(θ) ∝ π(θ)f (x|θ)Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 14 / 30
- 24. Addiction to modelsOne potential diﬃculty with Bayesian analysis is its ultimate dependenceon model(s) speciﬁcation π(θ) ∝ π(θ)f (x|θ)While Bayesian analysis allows for model variability, prunning,improvement, comparison, embedding, &tc., there always is a basicreliance [or at least conditioning] on the ”truth” of an overall model.Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 14 / 30
- 25. Addiction to modelsOne potential diﬃculty with Bayesian analysis is its ultimate dependenceon model(s) speciﬁcation π(θ) ∝ π(θ)f (x|θ)While Bayesian analysis allows for model variability, prunning,improvement, comparison, embedding, &tc., there always is a basicreliance [or at least conditioning] on the ”truth” of an overall model. Maysound paradoxical because of the many tools oﬀered by Bayesian analysis,however method is blind once ”out of the model”, in the sense that itcannot assess the validity of a model without imbedding this model insideanother model.Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 14 / 30
- 26. ABCµ multiple errors [ c Ratmann et al., PNAS, 2009]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 15 / 30
- 27. ABCµ multiple errors [ c Ratmann et al., PNAS, 2009]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 15 / 30
- 28. No proper goodness-of-ﬁt test ‘There is not the slightest use in rejecting any hypothesis unless we can do it in favor of some deﬁnite alternative that better ﬁts the facts.” — E.T. Jaynes, Probability TheoryWhile the setting H 0 : M = M0 versus H a : M = M0is rather artiﬁcial, there is no satisfactory way of answering the questionChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 16 / 30
- 29. An approximate goodness-of-ﬁt testTesting H 0 : M = Mθ versus H a : M = Mθrephrased as H0 : min d(Fθ , U(0,1) ) = 0 versus Ha : min d(Fθ , U(0,1) ) > 0 θ θ [Verdinelli and Wasserman, 98; Rousseau and Robert, 01]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 17 / 30
- 30. An approximate goodness-of-ﬁt testTesting H 0 : M = Mθ versus H a : M = Mθrephrased as H0 : Fθ (x) ∼ U(0, 1) versus k ωi Ha : Fθ (x) ∼ p0 U(0, 1) + (1 − p0 ) Be(αi i , αi (1 − i )) i=1 ωwith (αi , i ) ∼ [1 − exp{−(αi − 2)2 − ( i − .5)2 }] 2 2 × exp[−1/(αi i (1 − i )) − 0.2αi /2] [Verdinelli and Wasserman, 98; Rousseau and Robert, 01]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 17 / 30
- 31. RobustnessModels only partly deﬁned through moments Eθ [hi (x)] = Hi (θ) i = 1, . . .i.e., no complete construction of the underlying modelExample (White noise in AR)The relation xt = ρxt−1 + σ toften makes no assumption on t besides its ﬁrst two moments...How can we run Bayesian analysis in such settings? Should we? [Lazar, 2005; Cornuet et al., 2011, in prep.]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 18 / 30
- 32. [back to] Bayesian model choiceHaving a high relative probability does not mean that a hypothesis is true or supportedby the data — A. Templeton, Mol. Ecol., 2009The formal Bayesian approach put probabilities all over the entiremodel/parameter spaceChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 19 / 30
- 33. [back to] Bayesian model choiceHaving a high relative probability does not mean that a hypothesis is true or supportedby the data — A. Templeton, Mol. Ecol., 2009The formal Bayesian approach put probabilities all over the entiremodel/parameter spaceThis means: allocating probabilities pi to all models Mi deﬁning priors πi (θi ) for each parameter space Θi pick largest p(Mi |x) to determine “best” modelChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 19 / 30
- 34. Several types of problemsConcentrate on selection perspective: how to integrate loss function/decision/consequences representation of parsimony/sparcity (Occam’s rule) how to ﬁght overﬁtting for nested modelsChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 20 / 30
- 35. Several types of problemsIncoherent methods, such as ABC, Bayes factor, or any simulation approach that treatsall hypotheses as mutually exclusive, should never be used with logically overlappinghypotheses. — A. Templeton, PNAS, 2010Choice of prior structures adequate weights pi : > if M1 = M2 ∪ M3 , p(M1 ) = p(M2 ) + p(M3 ) ? priors distributions πi (·) deﬁned for every i ∈ I πi (·) proper (Jeﬀreys) πi (·) coherent (?) for nested models prior modelling inﬂationChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 20 / 30
- 36. Compatibility principleDiﬃculty of ﬁnding simultaneously priors on a collection of models Mi(i ∈ I)Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 21 / 30
- 37. Compatibility principleDiﬃculty of ﬁnding simultaneously priors on a collection of models Mi(i ∈ I)Easier to start from a single prior on a “big” model and to derive theothers from a coherence principle [Dawid & Lauritzen, 2000]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 21 / 30
- 38. Projection approach ⊥For M2 submodel of M1 , π2 can be derived as the distribution of θ2 (θ1 ) ⊥ (θ ) is a projection of θ on M , e.g.when θ1 ∼ π1 (θ1 ) and θ2 1 1 2 d(f (· |θ1 ), f (· |θ1 ⊥ )) = inf d(f (· |θ1 ) , f (· |θ2 )) . θ2 ∈Θ2where d is a divergence measure [McCulloch & Rossi, 1992]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 22 / 30
- 39. Projection approach ⊥For M2 submodel of M1 , π2 can be derived as the distribution of θ2 (θ1 ) ⊥ (θ ) is a projection of θ on M , e.g.when θ1 ∼ π1 (θ1 ) and θ2 1 1 2 d(f (· |θ1 ), f (· |θ1 ⊥ )) = inf d(f (· |θ1 ) , f (· |θ2 )) . θ2 ∈Θ2where d is a divergence measure [McCulloch & Rossi, 1992]Or we can look instead at the posterior distribution of d(f (· |θ1 ), f (· |θ1 ⊥ )) [Goutis & Robert, 1998; Dupuis & Robert, 2001]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 22 / 30
- 40. Kullback proximityAlternative projection to the aboveDeﬁnition (Compatible prior)Given a prior π1 on a model M1 and a submodel M2 , a prior π2 on M2 iscompatible with π1Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 23 / 30
- 41. Kullback proximityAlternative projection to the aboveDeﬁnition (Compatible prior)Given a prior π1 on a model M1 and a submodel M2 , a prior π2 on M2 iscompatible with π1 when it achieves the minimum Kullback divergencebetween the corresponding marginals: m1 (x; π1 ) = Θ1 f1 (x|θ)π1 (θ)dθand m2 (x); π2 = Θ2 f2 (x|θ)π2 (θ)dθ,Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 23 / 30
- 42. Kullback proximityAlternative projection to the aboveDeﬁnition (Compatible prior)Given a prior π1 on a model M1 and a submodel M2 , a prior π2 on M2 iscompatible with π1 when it achieves the minimum Kullback divergencebetween the corresponding marginals: m1 (x; π1 ) = Θ1 f1 (x|θ)π1 (θ)dθand m2 (x); π2 = Θ2 f2 (x|θ)π2 (θ)dθ, m1 (x; π1 ) π2 = arg min log m1 (x; π1 ) dx π2 m2 (x; π2 )Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 23 / 30
- 43. Diﬃculties Further complicating dimensionality of test statistics is the fact that the models are often not nested, and one model may contain parameters that do not have analogues in the other models and vice versa. — A. Templeton, Mol. Ecol., 2009 Does not give a working principle when M2 is not a submodel M1 [Perez & Berger, 2000; Cano, Salmer´n & Robert, 2006] o Depends on the choice of π1 Prohibits the use of improper priors Worse: useless in unconstrained settings...Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 24 / 30
- 44. A side remark: Zellner’s gUse of Zellner’s g-prior in linear regression, i.e. a normal prior for βconditional on σ 2 , ˜ β|σ 2 ∼ N (β, gσ 2 (X T X)−1 )and a Jeﬀreys prior for σ 2 , π(σ 2 ) ∝ σ −2Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 25 / 30
- 45. Variable selectionFor the hierarchical parameter γ, we use p π(γ) = τiγi (1 − τi )1−γi , i=1where τi corresponds to the prior probability that variable i is present inthe model (and a priori independence between the presence/absence ofvariables)Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 26 / 30
- 46. Variable selectionFor the hierarchical parameter γ, we use p π(γ) = τiγi (1 − τi )1−γi , i=1where τi corresponds to the prior probability that variable i is present inthe model (and a priori independence between the presence/absence ofvariables)Typically (?), when no prior information is available, τ1 = . . . = τp = 1/2,ie a uniform prior π(γ) = 2−pChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 26 / 30
- 47. Inﬂuence of g Taking ˜ β = 0p+1 and c large does not workChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 27 / 30
- 48. Inﬂuence of g Taking ˜ β = 0p+1 and c large does not workConsider the 10-predictor full model 3 3 2 2 2 y|β, σ ∼ N β0 + βi x i + βi+3 xi + β7 x1 x2 + β8 x1 x3 + β9 x2 x3 + β10 x1 x2 x3 , σ In i=1 i=1 where the xi s are iid U (0, 10) [Casella & Moreno, 2004]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 27 / 30
- 49. Inﬂuence of g Taking ˜ β = 0p+1 and c large does not workConsider the 10-predictor full model 3 3 2 2 2 y|β, σ ∼ N β0 + βi x i + βi+3 xi + β7 x1 x2 + β8 x1 x3 + β9 x2 x3 + β10 x1 x2 x3 , σ In i=1 i=1 where the xi s are iid U (0, 10) [Casella & Moreno, 2004]True model: two predictors x1 and x2 , i.e. γ ∗ = 110. . .0,(β0 , β1 , β2 ) = (5, 1, 3), and σ 2 = 4.Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 27 / 30
- 50. Inﬂuence of g 2 t1 (γ) g = 10 g = 100 g = 103 g = 104 g = 106 0,1,2 0.04062 0.35368 0.65858 0.85895 0.98222 0,1,2,7 0.01326 0.06142 0.08395 0.04434 0.00524 0,1,2,4 0.01299 0.05310 0.05805 0.02868 0.00336 0,2,4 0.02927 0.03962 0.00409 0.00246 0.00254 0,1,2,8 0.01240 0.03833 0.01100 0.00126 0.00126Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 28 / 30
- 51. Case for a noninformative hierarchical solution ˜Use the same compatible informative g-prior distribution with β = 0p+1and a hierarchical diﬀuse prior distribution on g, e.g. π(g) ∝ g −1 IN∗ (c) [Liang et al., 2007; Marin & Robert, 2007; Celeux et al., ca. 2011]Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 29 / 30
- 52. Occam’s razorPluralitas non est ponenda sine neccesitate Variation is random until the contrary is shown; and new parameters in laws, when they are suggested, must be tested one at a time, unless there is speciﬁc reason to the contrary. H. Jeﬀreys, ToP, 1939No well-accepted implementation behind the principle...Christian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 30 / 30
- 53. Occam’s razorPluralitas non est ponenda sine neccesitate Variation is random until the contrary is shown; and new parameters in laws, when they are suggested, must be tested one at a time, unless there is speciﬁc reason to the contrary. H. Jeﬀreys, ToP, 1939No well-accepted implementation behind the principle...besides the fact that the Bayes factor naturally penalises larger modelsChristian P. Robert (Paris-Dauphine) Uncertainties within Bayesian concepts July 31, 2011 30 / 30

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