ABC in London, May 5, 2011
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Those are the slides shown at the ABC in London meeting, Imperial College London, May 5, 2011

Those are the slides shown at the ABC in London meeting, Imperial College London, May 5, 2011

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ABC in London, May 5, 2011 ABC in London, May 5, 2011 Presentation Transcript

  • ABC Methods for Bayesian Model Choice ABC Methods for Bayesian Model Choice Christian P. Robert Universit´ Paris-Dauphine, IuF, & CREST e http://www.ceremade.dauphine.fr/~xian May 3, 2011
  • ABC Methods for Bayesian Model ChoiceABC for model choice Model choice Gibbs random fields Generic ABC model choice
  • ABC Methods for Bayesian Model Choice Model choiceBayesian model choice Several models M1 , M2 , . . . are considered simultaneously for a dataset y and the model index M is part of the inference. Use of a prior distribution. π(M = m), plus a prior distribution on the parameter conditional on the value m of the model index, πm (θ m ) Goal is to derive the posterior distribution of M , challenging computational target when models are complex.
  • ABC Methods for Bayesian Model Choice Model choiceGeneric ABC for model choice Algorithm 1 Likelihood-free model choice sampler (ABC-MC) for t = 1 to T do repeat Generate m from the prior π(M = m) Generate θ m from the prior πm (θ m ) Generate z from the model fm (z|θ m ) until ρ{η(z), η(y)} < Set m(t) = m and θ (t) = θ m end for [Toni, Welch, Strelkowa, Ipsen & Stumpf, 2009]
  • ABC Methods for Bayesian Model Choice Model choiceABC estimates Posterior probability π(M = m|y) approximated by the frequency of acceptances from model m T 1 Im(t) =m . T t=1 Early issues with implementation: should tolerances be the same for all models? should summary statistics vary across models? (incl. their dimension)? should the distance measure ρ vary as well?
  • ABC Methods for Bayesian Model Choice Model choiceABC estimates Posterior probability π(M = m|y) approximated by the frequency of acceptances from model m T 1 Im(t) =m . T t=1 Early issues with implementation: then needs to become part of the model this is incompatible with Bayesian model choice same difficulty Extension to a weighted polychotomous logistic regression estimate of π(M = m|y), with non-parametric kernel weights [Cornuet et al., DIYABC, 2009]
  • ABC Methods for Bayesian Model Choice Model choiceThe Great ABC controversy On-going controvery in phylogeographic genetics about the validity of using ABC for testing Against: Templeton, 2008, 2009, 2010a, 2010b, 2010c argues that nested hypotheses cannot have higher probabilities than nesting hypotheses (!)
  • ABC Methods for Bayesian Model Choice Model choiceThe Great ABC controversy On-going controvery in phylogeographic genetics about the validity of using ABC for testing Replies: Fagundes et al., 2008, Against: Templeton, 2008, Beaumont et al., 2010, Berger et 2009, 2010a, 2010b, 2010c al., 2010, Csill`ry et al., 2010 e argues that nested hypotheses point out that the criticisms are cannot have higher probabilities addressed at [Bayesian] than nesting hypotheses (!) model-based inference and have nothing to do with ABC...
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsGibbs random fields Gibbs distribution The rv y = (y1 , . . . , yn ) is a Gibbs random field associated with the graph G if 1 f (y) = exp − Vc (yc ) , Z c∈C where Z is the normalising constant, C is the set of cliques of G and Vc is any function also called potential U (y) = c∈C Vc (yc ) is the energy function
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsGibbs random fields Gibbs distribution The rv y = (y1 , . . . , yn ) is a Gibbs random field associated with the graph G if 1 f (y) = exp − Vc (yc ) , Z c∈C where Z is the normalising constant, C is the set of cliques of G and Vc is any function also called potential U (y) = c∈C Vc (yc ) is the energy function c Z is usually unavailable in closed form
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsPotts model Potts model Vc (y) is of the form Vc (y) = θS(y) = θ δyl =yi l∼i where l∼i denotes a neighbourhood structure
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsPotts model Potts model Vc (y) is of the form Vc (y) = θS(y) = θ δyl =yi l∼i where l∼i denotes a neighbourhood structure In most realistic settings, summation Zθ = exp{θ T S(x)} x∈X involves too many terms to be manageable and numerical approximations cannot always be trusted
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsBayesian Model Choice Comparing a model with energy S0 taking values in Rp0 versus a model with energy S1 taking values in Rp1 can be done through the Bayes factor corresponding to the priors π0 and π1 on each parameter space exp{θ T S0 (x)}/Zθ 0 ,0 π0 (dθ 0 ) 0 Bm0 /m1 (x) = exp{θ T S1 (x)}/Zθ 1 ,1 π1 (dθ 1 ) 1
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsNeighbourhood relations Choice to be made between M neighbourhood relations m i∼i (0 ≤ m ≤ M − 1) with Sm (x) = I{xi =xi } m i∼i driven by the posterior probabilities of the models.
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsModel index Computational target: P(M = m|x) ∝ fm (x|θm )πm (θm ) dθm π(M = m) , Θm
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsModel index Computational target: P(M = m|x) ∝ fm (x|θm )πm (θm ) dθm π(M = m) , Θm If S(x) sufficient statistic for the joint parameters (M, θ0 , . . . , θM −1 ), P(M = m|x) = P(M = m|S(x)) .
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsSufficient statistics in Gibbs random fields
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsSufficient statistics in Gibbs random fields Each model m has its own sufficient statistic Sm (·) and S(·) = (S0 (·), . . . , SM −1 (·)) is also (model-)sufficient.
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsSufficient statistics in Gibbs random fields Each model m has its own sufficient statistic Sm (·) and S(·) = (S0 (·), . . . , SM −1 (·)) is also (model-)sufficient. For Gibbs random fields, 1 2 x|M = m ∼ fm (x|θm ) = fm (x|S(x))fm (S(x)|θm ) 1 = f 2 (S(x)|θm ) n(S(x)) m where n(S(x)) = {˜ ∈ X : S(˜ ) = S(x)} x x c S(x) is therefore also sufficient for the joint parameters
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsABC model choice Algorithm ABC-MC Generate m∗ from the prior π(M = m). ∗ Generate θm∗ from the prior πm∗ (·). Generate x∗ from the model fm∗ (·|θm∗ ). ∗ Compute the distance ρ(S(x0 ), S(x∗ )). Accept (θm∗ , m∗ ) if ρ(S(x0 ), S(x∗ )) < . ∗ Note When = 0 the algorithm is exact
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsToy example iid Bernoulli model versus two-state first-order Markov chain, i.e. n f0 (x|θ0 ) = exp θ0 I{xi =1} {1 + exp(θ0 )}n , i=1 versus n 1 f1 (x|θ1 ) = exp θ1 I{xi =xi−1 } {1 + exp(θ1 )}n−1 , 2 i=2 with priors θ0 ∼ U(−5, 5) and θ1 ∼ U(0, 6) (inspired by “phase transition” boundaries).
  • ABC Methods for Bayesian Model Choice Gibbs random fieldsToy example (2) ^ BF01 −5 0 5 −40 −20 BF01 0 10 ^ BF01 −10 −5 0 5 10 −40 −20 BF01 0 10 (left) Comparison of the true BF m0 /m1 (x0 ) with BF m0 /m1 (x0 ) (in logs) over 2, 000 simulations and 4.106 proposals from the prior. (right) Same when using tolerance corresponding to the 1% quantile on the distances.
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceBack to sufficiency ‘Sufficient statistics for individual models are unlikely to be very informative for the model probability. This is already well known and understood by the ABC-user community.’ [Scott Sisson, Jan. 31, 2011, ’Og]
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceBack to sufficiency ‘Sufficient statistics for individual models are unlikely to be very informative for the model probability. This is already well known and understood by the ABC-user community.’ [Scott Sisson, Jan. 31, 2011, ’Og] If η1 (x) sufficient statistic for model m = 1 and parameter θ1 and η2 (x) sufficient statistic for model m = 2 and parameter θ2 , (η1 (x), η2 (x)) is not always sufficient for (m, θm )
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceBack to sufficiency ‘Sufficient statistics for individual models are unlikely to be very informative for the model probability. This is already well known and understood by the ABC-user community.’ [Scott Sisson, Jan. 31, 2011, ’Og] If η1 (x) sufficient statistic for model m = 1 and parameter θ1 and η2 (x) sufficient statistic for model m = 2 and parameter θ2 , (η1 (x), η2 (x)) is not always sufficient for (m, θm ) c Potential loss of information at the testing level
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceLimiting behaviour of B12 (T → ∞) ABC approximation T t=1 Imt =1 Iρ{η(zt ),η(y)}≤ B12 (y) = T , t=1 Imt =2 Iρ{η(zt ),η(y)}≤ where the (mt , z t )’s are simulated from the (joint) prior
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceLimiting behaviour of B12 (T → ∞) ABC approximation T t=1 Imt =1 Iρ{η(zt ),η(y)}≤ B12 (y) = T , t=1 Imt =2 Iρ{η(zt ),η(y)}≤ where the (mt , z t )’s are simulated from the (joint) prior As T go to infinity, limit Iρ{η(z),η(y)}≤ π1 (θ 1 )f1 (z|θ 1 ) dz dθ 1 B12 (y) = Iρ{η(z),η(y)}≤ π2 (θ 2 )f2 (z|θ 2 ) dz dθ 2 η Iρ{η,η(y)}≤ π1 (θ 1 )f1 (η|θ 1 ) dη dθ 1 = η , Iρ{η,η(y)}≤ π2 (θ 2 )f2 (η|θ 2 ) dη dθ 2 η η where f1 (η|θ 1 ) and f2 (η|θ 2 ) distributions of η(z)
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceLimiting behaviour of B12 ( → 0) When goes to zero, η η π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 B12 (y) = η , π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceLimiting behaviour of B12 ( → 0) When goes to zero, η η π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 B12 (y) = η , π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 Bayes factor based on the sole observation of η(y)
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceLimiting behaviour of B12 (under sufficiency) If η(y) sufficient statistic for both models, fi (y|θ i ) = gi (y)fiη (η(y)|θ i ) Thus η Θ1 π(θ 1 )g1 (y)f1 (η(y)|θ 1 ) dθ 1 B12 (y) = η Θ2 π(θ 2 )g2 (y)f2 (η(y)|θ 2 ) dθ 2 η g1 (y) π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 = η g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g1 (y) η = B (y) . g2 (y) 12 [Didelot, Everitt, Johansen & Lawson, 2011]
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceLimiting behaviour of B12 (under sufficiency) If η(y) sufficient statistic for both models, fi (y|θ i ) = gi (y)fiη (η(y)|θ i ) Thus η Θ1 π(θ 1 )g1 (y)f1 (η(y)|θ 1 ) dθ 1 B12 (y) = η Θ2 π(θ 2 )g2 (y)f2 (η(y)|θ 2 ) dθ 2 η g1 (y) π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 = η g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g1 (y) η = B (y) . g2 (y) 12 [Didelot, Everitt, Johansen & Lawson, 2011] No discrepancy only when cross-model sufficiency
  • ABC Methods for Bayesian Model Choice Generic ABC model choicePoisson/geometric example Sample x = (x1 , . . . , xn ) from either a Poisson P(λ) or from a geometric G(p) Sum n S= yi = η(x) i=1 sufficient statistic for either model but not simultaneously Discrepancy ratio g1 (x) S!n−S / i yi ! = g2 (x) 1 n+S−1 S
  • ABC Methods for Bayesian Model Choice Generic ABC model choicePoisson/geometric discrepancy η Range of B12 (x) versus B12 (x) B12 (x): The values produced have nothing in common.
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceFormal recovery Creating an encompassing exponential family T T f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)} leads to a sufficient statistic (η1 (x), η2 (x), t1 (x), t2 (x)) [Didelot, Everitt, Johansen & Lawson, 2011]
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceFormal recovery Creating an encompassing exponential family T T f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)} leads to a sufficient statistic (η1 (x), η2 (x), t1 (x), t2 (x)) [Didelot, Everitt, Johansen & Lawson, 2011] In the Poisson/geometric case, if i xi ! is added to S, no discrepancy
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceFormal recovery Creating an encompassing exponential family T T f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)} leads to a sufficient statistic (η1 (x), η2 (x), t1 (x), t2 (x)) [Didelot, Everitt, Johansen & Lawson, 2011] Only applies in genuine sufficiency settings... c Inability to evaluate loss brought by summary statistics
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceThe Pitman–Koopman lemma Lemma A necessary and sufficient condition for the existence of a sufficient statistic of a fixed dimension whatever the sample size is that the sampling distribution belongs to an exponential family. [Pitman, 1933; Koopman, 1933]
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceThe Pitman–Koopman lemma Lemma A necessary and sufficient condition for the existence of a sufficient statistic of a fixed dimension whatever the sample size is that the sampling distribution belongs to an exponential family. [Pitman, 1933; Koopman, 1933] Provision of fixed support (consider U(0, θ) counterexample)
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceMeaning of the ABC-Bayes factor ‘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.’ [Scott Sisson, Jan. 31, 2011, ’Og]
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceMeaning of the ABC-Bayes factor ‘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.’ [Scott Sisson, Jan. 31, 2011, ’Og] In the Poisson/geometric case, if E[yi ] = θ0 > 0, η (θ0 + 1)2 −θ0 lim B12 (y) = e n→∞ θ0
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceMA example Consider the MA(q) model q xt = t+ ϑi t−i i=1 Simple prior: uniform prior over the identifiability zone, e.g. triangle for MA(2)
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceMA example (2) ABC algorithm thus made of 1. picking a new value (ϑ1 , ϑ2 ) in the triangle 2. generating an iid sequence ( t )−q<t≤T 3. producing a simulated series (xt )1≤t≤T
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceMA example (2) ABC algorithm thus made of 1. picking a new value (ϑ1 , ϑ2 ) in the triangle 2. generating an iid sequence ( t )−q<t≤T 3. producing a simulated series (xt )1≤t≤T Distance: basic distance between the series T ρ((xt )1≤t≤T , (xt )1≤t≤T ) = (xt − xt )2 t=1 or between summary statistics like the first q autocorrelations T τj = xt xt−j t=j+1
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceComparison of distance impact Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceComparison of distance impact 4 1.5 3 1.0 2 0.5 1 0.0 0 0.0 0.2 0.4 0.6 0.8 −2.0 −1.0 0.0 0.5 1.0 1.5 θ1 θ2 Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceComparison of distance impact 4 1.5 3 1.0 2 0.5 1 0.0 0 0.0 0.2 0.4 0.6 0.8 −2.0 −1.0 0.0 0.5 1.0 1.5 θ1 θ2 Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceMA(q) divergence 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 1 2 1 2 1 2 1 2 Evolution [against ] of ABC Bayes factor, in terms of frequencies of visits to models MA(1) (left) and MA(2) (right) when equal to 10, 1, .1, .01% quantiles on insufficient autocovariance distances. Sample of 50 points from a MA(2) with θ1 = 0.6, θ2 = 0.2. True Bayes factor equal to 17.71.
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceMA(q) divergence 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 1 2 1 2 1 2 1 2 Evolution [against ] of ABC Bayes factor, in terms of frequencies of visits to models MA(1) (left) and MA(2) (right) when equal to 10, 1, .1, .01% quantiles on insufficient autocovariance distances. Sample of 50 points from a MA(1) model with θ1 = 0.6. True Bayes factor B21 equal to .004.
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceFurther comments ‘There should be the possibility that for the same model, but different (non-minimal) [summary] statistics (so ∗ different η’s: η1 and η1 ) the ratio of evidences may no longer be equal to one.’ [Michael Stumpf, Jan. 28, 2011, ’Og] Using different summary statistics [on different models] may indicate the loss of information brought by each set but agreement does not lead to trustworthy approximations.
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceA population genetics evaluation Population genetics example with 3 populations 2 scenari 15 individuals 5 loci single mutation parameter
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceA population genetics evaluation Population genetics example with 3 populations 2 scenari 15 individuals 5 loci single mutation parameter 24 summary statistics 2 million ABC proposal importance [tree] sampling alternative
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceA population genetics evaluation Population genetics example with 3 populations 2 scenari 15 individuals 5 loci single mutation parameter 24 summary statistics 2 million ABC proposal importance [tree] sampling alternative
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceStability of importance sampling 1.0 1.0 1.0 1.0 1.0 q q 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 q 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceComparison with ABC Use of 24 summary statistics and DIY-ABC logistic correction 1.0 qq qq qq qq qqq qq q q qq q q qq q q q q q q q q q q qq q q qq q q q qq qq qq q q q q qq q q q q q q q q q q q qq q q q qq q q 0.8 q q q q q q q q q q q q qq q q q q q q q qq q q q q q qqq q q qq q ABC direct and logistic q q qq q q q q q q q 0.6 q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q qq 0.4 qq q q q q q q q q q q q q q q q q q 0.2 q qq q q q q q q q q q q 0.0 qq 0.0 0.2 0.4 0.6 0.8 1.0 importance sampling
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceComparison with ABC Use of 15 summary statistics and DIY-ABC logistic correction 6 4 q q q q q q q qq 2 q q q q qq ABC direct q q q qq q qq q q q q qq qq q qq q q q q q qq q q q q q qq q q qq q qq q q qq q q q qqq q q q q q q q q 0 qq qq q q q q qq qq q q qq q q q q q q q q −2 −4 −4 −2 0 2 4 6 importance sampling
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceComparison with ABC Use of 15 summary statistics and DIY-ABC logistic correction qq 6 q q q q q q q q q q q 4 q qq q qq qq q q q q q q q q q q qq q q q q q q q ABC direct and logistic q q q q q q q qq q q q q q q q qq q 2 q q q q q qq q qq q q q qq qq q qq qq q q q qq q q qq qq qq q q qqqq qq q q q qqqqq qq q q q qq q q q q qq q qq q q qqq q q qqqqq q q qq q q qq q q q qq 0 qq qqq q q qq q qqq q q q q q q q q q q q q q q q q q q q q qq q q −2 q q q q q −4 q q q −4 −2 0 2 4 6 importance sampling
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceComparison with ABC Use of 15 summary statistics and DIY-ABC logistic correction q 2 q q 1 q q q q q q q q q q log−ratio q q q q q qq q q q qqq q q q q q q q qq q q q q q qq qq q qq q q qqqqqq q q qqqq qq qq q qq qq q qq q qq qq q q q qq q qq q q qqq q 0 q q q q qq q q q qq q q q q q q qq q qq q qq qq q qq qqq qq qq q q q q qq q qqqq q q q q q q q q qq qqq q qq qq q q qq q q q q qq q q qq q q q q q q q q qq qqqq q q q q q q q q q −1 q q q q −2 q 0 20 40 60 80 100 index
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceThe only safe cases Besides specific models like Gibbs random fields, using distances over the data itself escapes the discrepancy... [Toni & Stumpf, 2010;Sousa et al., 2009]
  • ABC Methods for Bayesian Model Choice Generic ABC model choiceThe only safe cases Besides specific models like Gibbs random fields, using distances over the data itself escapes the discrepancy... [Toni & Stumpf, 2010;Sousa et al., 2009] ...and so does the use of more informal model fitting measures [Ratmann, Andrieu, Richardson and Wiujf, 2009]