ABC and empirical likelihood

1. Approximate Bayesian Computation (ABC) and empirical likelihood Christian P. Robert Structure and uncertainty, Bristol, Sept. 25, 2012 Universit´ Paris-Dauphine, IuF, & CREST e Joint work with Kerrie L. Mengersen and P. Pudlo

2. Outline Introduction ABC ABC as an inference machine ABCel

3. Intractable likelihood Case of a well-deﬁned statistical model where the likelihood function (θ|y) = f (y1 , . . . , yn |θ) is (really!) not available in closed form can (easily!) be neither completed nor demarginalised cannot be estimated by an unbiased estimator c Prohibits direct implementation of a generic MCMC algorithm like Metropolis–Hastings

4. Intractable likelihood Case of a well-deﬁned statistical model where the likelihood function (θ|y) = f (y1 , . . . , yn |θ) is (really!) not available in closed form can (easily!) be neither completed nor demarginalised cannot be estimated by an unbiased estimator c Prohibits direct implementation of a generic MCMC algorithm like Metropolis–Hastings

5. Diﬀerent perspectives on abc What is the (most) fundamental issue? a mere computational issue (that will eventually end up being solved by more powerful computers, &tc, even if too costly in the short term) an inferential issue (opening opportunities for new inference machine, with diﬀerent legitimity than classical B approach) a Bayesian conundrum (while inferencial methods available, how closely related to the B approach?)

8. Econom’ections Similar exploration of simulation-based and approximation techniques in Econometrics Simulated method of moments Method of simulated moments Simulated pseudo-maximum-likelihood Indirect inference [Gouri´roux & Monfort, 1996] e even though motivation is partly-deﬁned models rather than complex likelihoods

9. Econom’ections Similar exploration of simulation-based and approximation techniques in Econometrics Simulated method of moments Method of simulated moments Simulated pseudo-maximum-likelihood Indirect inference [Gouri´roux & Monfort, 1996] e even though motivation is partly-deﬁned models rather than complex likelihoods

10. Indirect inference ^ Minimise [in θ] a distance between estimators β based on a pseudo-model for genuine observations and for observations simulated under the true model and the parameter θ. [Gouri´roux, Monfort, & Renault, 1993; e Smith, 1993; Gallant & Tauchen, 1996]

11. Indirect inference (PML vs. PSE) Example of the pseudo-maximum-likelihood (PML) ^ β(y) = arg max log f (yt |β, y1:(t−1) ) β t leading to arg min ||β(yo ) − β(y1 (θ), . . . , yS (θ))||2 ^ ^ θ when ys (θ) ∼ f (y|θ) s = 1, . . . , S

12. Indirect inference (PML vs. PSE) Example of the pseudo-score-estimator (PSE) 2 ∂ log f ^ β(y) = arg min (yt |β, y1:(t−1) ) β t ∂β leading to arg min ||β(yo ) − β(y1 (θ), . . . , yS (θ))||2 ^ ^ θ when ys (θ) ∼ f (y|θ) s = 1, . . . , S

13. Consistent indirect inference “...in order to get a unique solution the dimension of the auxiliary parameter β must be larger than or equal to the dimension of the initial parameter θ. If the problem is just identified the different methods become easier...” Consistency depending on the criterion and on the asymptotic identifiability of θ [Gouri´roux & Monfort, 1996, p. 66] e

14. Consistent indirect inference “...in order to get a unique solution the dimension of the auxiliary parameter β must be larger than or equal to the dimension of the initial parameter θ. If the problem is just identified the different methods become easier...” Consistency depending on the criterion and on the asymptotic identifiability of θ [Gouri´roux & Monfort, 1996, p. 66] e

15. Choice of pseudo-model Arbitrariness of pseudo-model Pick model such that ^ 1. β(θ) not ﬂat (i.e. sensitive to changes in θ) ^ 2. β(θ) not dispersed (i.e. robust agains changes in ys (θ)) [Frigessi & Heggland, 2004]

16. Approximate Bayesian computation Introduction ABC Genesis of ABC ABC basics Advances and interpretations ABC as knn ABC as an inference machine ABCel

17. Genetic background of ABC skip genetics ABC is a recent computational technique that only requires being able to sample from the likelihood f (·|θ) This technique stemmed from population genetics models, about 15 years ago, and population geneticists still contribute signiﬁcantly to methodological developments of ABC. [Griﬃth & al., 1997; Tavar´ & al., 1999] e

18. Demo-genetic inference Each model is characterized by a set of parameters θ that cover historical (time divergence, admixture time ...), demographics (population sizes, admixture rates, migration rates, ...) and genetic (mutation rate, ...) factors The goal is to estimate these parameters from a dataset of polymorphism (DNA sample) y observed at the present time Problem: most of the time, we cannot calculate the likelihood of the polymorphism data f (y|θ)...

19. Demo-genetic inference Each model is characterized by a set of parameters θ that cover historical (time divergence, admixture time ...), demographics (population sizes, admixture rates, migration rates, ...) and genetic (mutation rate, ...) factors The goal is to estimate these parameters from a dataset of polymorphism (DNA sample) y observed at the present time Problem: most of the time, we cannot calculate the likelihood of the polymorphism data f (y|θ)...

20. Neutral model at a given microsatellite locus, in a closed panmictic population at equilibrium Mutations according to the Simple stepwise Mutation Model (SMM) • date of the mutations ∼ Poisson process with intensity θ/2 over the branches • MRCA = 100 • independent mutations: ±1 with pr. 1/2 Sample of 8 genes

21. Neutral model at a given microsatellite locus, in a closed panmictic population at equilibrium Kingman’s genealogy When time axis is normalized, T (k) ∼ Exp(k(k − 1)/2) Mutations according to the Simple stepwise Mutation Model (SMM) • date of the mutations ∼ Poisson process with intensity θ/2 over the branches • MRCA = 100 • independent mutations: ±1 with pr. 1/2

22. Neutral model at a given microsatellite locus, in a closed panmictic population at equilibrium Kingman’s genealogy When time axis is normalized, T (k) ∼ Exp(k(k − 1)/2) Mutations according to the Simple stepwise Mutation Model (SMM) • date of the mutations ∼ Poisson process with intensity θ/2 over the branches • MRCA = 100 • independent mutations: ±1 with pr. 1/2

23. Neutral model at a given microsatellite locus, in a closed panmictic population at equilibrium Kingman’s genealogy When time axis is normalized, T (k) ∼ Exp(k(k − 1)/2) Mutations according to the Simple stepwise Mutation Model (SMM) • date of the mutations ∼ Poisson process with intensity θ/2 over the branches Observations: leafs of the tree • MRCA = 100 ^ θ=? • independent mutations: ±1 with pr. 1/2

24. Much more interesting models. . . several independent locus Independent gene genealogies and mutations diﬀerent populations linked by an evolutionary scenario made of divergences, admixtures, migrations between populations, etc. larger sample size usually between 50 and 100 genes MRCA τ2 τ1 A typical evolutionary scenario: POP 0 POP 1 POP 2

25. Intractable likelihood Missing (too missing!) data structure: f (y|θ) = f (y|G , θ)f (G |θ)dG G cannot be computed in a manageable way... The genealogies are considered as nuisance parameters This modelling clearly diﬀers from the phylogenetic perspective where the tree is the parameter of interest.

26. Intractable likelihood Missing (too missing!) data structure: f (y|θ) = f (y|G , θ)f (G |θ)dG G cannot be computed in a manageable way... The genealogies are considered as nuisance parameters This modelling clearly diﬀers from the phylogenetic perspective where the tree is the parameter of interest.

27. not-so-obvious ancestry... You went to school to learn, girl (. . . ) Why 2 plus 2 makes four Now, now, now, I’m gonna teach you (. . . ) All you gotta do is repeat after me! A, B, C! It’s easy as 1, 2, 3! Or simple as Do, Re, Mi! (. . . )

28. A?B?C? A stands for approximate [wrong likelihood / picture] B stands for Bayesian C stands for computation [producing a parameter sample]

29. A?B?C? A stands for approximate [wrong likelihood / picture] B stands for Bayesian C stands for computation [producing a parameter sample]

30. A?B?C? ESS=108.9 ESS=81.48 ESS=105.2 3.0 2.0 2.0 Density Density Density 1.5 1.0 1.0 0.0 0.0 0.0 A stands for approximate −0.4 0.0 0.2 ESS=133.3 θ 0.4 0.6 −0.8 −0.6 −0.4 ESS=87.75 θ −0.2 0.0 −0.2 0.0 0.2 ESS=72.89 θ 0.4 0.6 0.8 3.0 2.0 4 [wrong likelihood / 2.0 Density Density Density 1.0 1.0 2 0.0 0.0 0 picture] −0.8 −0.4 ESS=116.5 θ 0.0 0.2 0.4 −0.2 0.0 0.2 ESS=103.9 θ 0.4 0.6 0.8 −0.2 0.0 ESS=126.9 θ 0.2 0.4 3.0 2.0 3.0 2.0 Density Density Density 1.0 1.5 1.0 B stands for Bayesian 0.0 0.0 0.0 −0.4 0.0 0.2 0.4 0.6 −0.4 −0.2 0.0 0.2 0.4 −0.8 −0.4 0.0 0.4 ESS=113.3 θ ESS=92.99 θ ESS=121.4 θ 3.0 2.0 C stands for computation 2.0 Density Density Density 1.5 1.0 1.0 0.0 0.0 0.0 [producing a parameter −0.6 −0.2 ESS=133.6 θ 0.2 0.6 −0.2 0.0 0.2 ESS=116.4 θ 0.4 0.6 −0.5 ESS=131.6 θ 0.0 0.5 0.0 1.0 2.0 3.0 2.0 sample] Density Density Density 1.0 1.0 0.0 0.0 −0.6 −0.2 0.2 0.4 0.6 −0.4 −0.2 0.0 0.2 0.4 −0.5 0.0 0.5

31. How Bayesian is aBc? Could we turn the resolution into a Bayesian answer? ideally so (not meaningfull: requires ∞-ly powerful computer asymptotically so (when sample size goes to ∞: meaningfull?) approximation error unknown (w/o costly simulation) true Bayes for wrong model (formal and artiﬁcial) true Bayes for estimated likelihood (back to econometrics?)

32. Untractable likelihood Back to stage zero: what can we do when a likelihood function f (y|θ) is well-deﬁned but impossible / too costly to compute...? MCMC cannot be implemented! shall we give up Bayesian inference altogether?! or settle for an almost Bayesian inference/picture...?

33. Untractable likelihood Back to stage zero: what can we do when a likelihood function f (y|θ) is well-deﬁned but impossible / too costly to compute...? MCMC cannot be implemented! shall we give up Bayesian inference altogether?! or settle for an almost Bayesian inference/picture...?

34. ABC methodology Bayesian setting: target is π(θ)f (x|θ) When likelihood f (x|θ) not in closed form, likelihood-free rejection technique: Foundation For an observation y ∼ f (y|θ), under the prior π(θ), if one keeps jointly simulating θ ∼ π(θ) , z ∼ f (z|θ ) , until the auxiliary variable z is equal to the observed value, z = y, then the selected θ ∼ π(θ|y) [Rubin, 1984; Diggle & Gratton, 1984; Tavar´ et al., 1997] e

37. A as A...pproximative When y is a continuous random variable, strict equality z = y is replaced with a tolerance zone ρ(y, z) where ρ is a distance Output distributed from def π(θ) Pθ {ρ(y, z) < } ∝ π(θ|ρ(y, z) < ) [Pritchard et al., 1999]

38. A as A...pproximative When y is a continuous random variable, strict equality z = y is replaced with a tolerance zone ρ(y, z) where ρ is a distance Output distributed from def π(θ) Pθ {ρ(y, z) < } ∝ π(θ|ρ(y, z) < ) [Pritchard et al., 1999]

39. ABC algorithm In most implementations, further degree of A...pproximation: Algorithm 1 Likelihood-free rejection sampler for i = 1 to N do repeat generate θ from the prior distribution π(·) generate z from the likelihood f (·|θ ) until ρ{η(z), η(y)} set θi = θ end for where η(y) deﬁnes a (not necessarily suﬃcient) statistic

40. Output The likelihood-free algorithm samples from the marginal in z of: π(θ)f (z|θ)IA ,y (z) π (θ, z|y) = , A ,y ×Θ π(θ)f (z|θ)dzdθ where A ,y = {z ∈ D|ρ(η(z), η(y)) < }. The idea behind ABC is that the summary statistics coupled with a small tolerance should provide a good approximation of the posterior distribution: π (θ|y) = π (θ, z|y)dz ≈ π(θ|y) . ...does it?!

43. Output The likelihood-free algorithm samples from the marginal in z of: π(θ)f (z|θ)IA ,y (z) π (θ, z|y) = , A ,y ×Θ π(θ)f (z|θ)dzdθ where A ,y = {z ∈ D|ρ(η(z), η(y)) < }. The idea behind ABC is that the summary statistics coupled with a small tolerance should provide a good approximation of the restricted posterior distribution: π (θ|y) = π (θ, z|y)dz ≈ π(θ|η(y)) . Not so good..! skip convergence details!

46. Convergence (do not attempt!) ...and the above does not apply to insuﬃcient statistics: If η(y) is not a suﬃcient statistics, the best one can hope for is π(θ|η(y)) , not π(θ|y) If η(y) is an ancillary statistic, the whole information contained in y is lost!, the “best” one can “hope” for is π(θ|η(y)) = π(θ) Bummer!!!

50. MA example Inference on the parameters of a MA(q) model q xt = t + ϑi t−i t−i i.i.d.w.n. i=1 bypass MA illustration Simple prior: uniform over the inverse [real and complex] roots in q Q(u) = 1 − ϑi u i i=1 under the identiﬁability conditions

51. MA example Inference on the parameters of a MA(q) model q xt = t + ϑi t−i t−i i.i.d.w.n. i=1 bypass MA illustration Simple prior: uniform prior over the identiﬁability zone in the parameter space, i.e. triangle for MA(2)

52. MA 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 distance between summary statistics like the q = 2 autocorrelations T τj = xt xt−j t=j+1

53. MA 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 distance between summary statistics like the q = 2 autocorrelations T τj = xt xt−j t=j+1

54. Comparison of distance impact Impact of tolerance on ABC sample against either distance ( = 100%, 10%, 1%, 0.1%) for an MA(2) model

55. Comparison 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 Impact of tolerance on ABC sample against either distance ( = 100%, 10%, 1%, 0.1%) for an MA(2) model

56. Comparison 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 Impact of tolerance on ABC sample against either distance ( = 100%, 10%, 1%, 0.1%) for an MA(2) model

57. Comments Role of distance paramount (because = 0) Scaling of components of η(y) is also determinant matters little if “small enough” representative of “curse of dimensionality” small is beautiful! the data as a whole may be paradoxically weakly informative for ABC

58. ABC (simul’) advances how approximative is ABC? ABC as knn Simulating from the prior is often poor in eﬃciency Either modify the proposal distribution on θ to increase the density of x’s within the vicinity of y ... [Marjoram et al, 2003; Bortot et al., 2007, Sisson et al., 2007] ...or by viewing the problem as a conditional density estimation and by developing techniques to allow for larger [Beaumont et al., 2002] .....or even by including in the inferential framework [ABCµ ] [Ratmann et al., 2009]

62. ABC-NP Better usage of [prior] simulations by adjustement: instead of throwing away θ such that ρ(η(z), η(y)) > , replace θ’s with locally regressed transforms θ∗ = θ − {η(z) − η(y)}T β ^ [Csill´ry et al., TEE, 2010] e ^ where β is obtained by [NP] weighted least square regression on (η(z) − η(y)) with weights Kδ {ρ(η(z), η(y))} [Beaumont et al., 2002, Genetics]

63. ABC-NP (regression) Also found in the subsequent literature, e.g. in Fearnhead-Prangle (2012) : weight directly simulation by Kδ {ρ(η(z(θ)), η(y))} or S 1 Kδ {ρ(η(zs (θ)), η(y))} S s=1 [consistent estimate of f (η|θ)] Curse of dimensionality: poor estimate when d = dim(η) is large...

64. ABC-NP (regression) Also found in the subsequent literature, e.g. in Fearnhead-Prangle (2012) : weight directly simulation by Kδ {ρ(η(z(θ)), η(y))} or S 1 Kδ {ρ(η(zs (θ)), η(y))} S s=1 [consistent estimate of f (η|θ)] Curse of dimensionality: poor estimate when d = dim(η) is large...

65. ABC-NP (density estimation) Use of the kernel weights Kδ {ρ(η(z(θ)), η(y))} leads to the NP estimate of the posterior expectation i θi Kδ {ρ(η(z(θi )), η(y))} i Kδ {ρ(η(z(θi )), η(y))} [Blum, JASA, 2010]

66. ABC-NP (density estimation) Use of the kernel weights Kδ {ρ(η(z(θ)), η(y))} leads to the NP estimate of the posterior conditional density i ˜ Kb (θi − θ)Kδ {ρ(η(z(θi )), η(y))} i Kδ {ρ(η(z(θi )), η(y))} [Blum, JASA, 2010]

67. ABC-NP (density estimations) Other versions incorporating regression adjustments i ˜ Kb (θ∗ − θ)Kδ {ρ(η(z(θi )), η(y))} i i Kδ {ρ(η(z(θi )), η(y))} In all cases, error E[^ (θ|y)] − g (θ|y) = cb 2 + cδ2 + OP (b 2 + δ2 ) + OP (1/nδd ) g c var(^ (θ|y)) = g (1 + oP (1)) nbδd

68. ABC-NP (density estimations) Other versions incorporating regression adjustments i ˜ Kb (θ∗ − θ)Kδ {ρ(η(z(θi )), η(y))} i i Kδ {ρ(η(z(θi )), η(y))} In all cases, error E[^ (θ|y)] − g (θ|y) = cb 2 + cδ2 + OP (b 2 + δ2 ) + OP (1/nδd ) g c var(^ (θ|y)) = g (1 + oP (1)) nbδd [Blum, JASA, 2010]

69. ABC-NP (density estimations) Other versions incorporating regression adjustments i ˜ Kb (θ∗ − θ)Kδ {ρ(η(z(θi )), η(y))} i i Kδ {ρ(η(z(θi )), η(y))} In all cases, error E[^ (θ|y)] − g (θ|y) = cb 2 + cδ2 + OP (b 2 + δ2 ) + OP (1/nδd ) g c var(^ (θ|y)) = g (1 + oP (1)) nbδd [standard NP calculations]

70. ABC-NCH Incorporating non-linearities and heterocedasticities: σ(η(y)) ^ θ∗ = m(η(y)) + [θ − m(η(z))] ^ ^ σ(η(z)) ^ where m(η) estimated by non-linear regression (e.g., neural network) ^ σ(η) estimated by non-linear regression on residuals ^ log{θi − m(ηi )}2 = log σ2 (ηi ) + ξi ^ [Blum & Fran¸ois, 2009] c

71. ABC-NCH Incorporating non-linearities and heterocedasticities: σ(η(y)) ^ θ∗ = m(η(y)) + [θ − m(η(z))] ^ ^ σ(η(z)) ^ where m(η) estimated by non-linear regression (e.g., neural network) ^ σ(η) estimated by non-linear regression on residuals ^ log{θi − m(ηi )}2 = log σ2 (ηi ) + ξi ^ [Blum & Fran¸ois, 2009] c

72. ABC as knn Practice of ABC: determine tolerance as a quantile on observed distances, say 10% or 1% quantile, = N = qα (d1 , . . . , dN ) Interpretation of ε as non- parametric bandwidth only approximation of the actual practice [Blum & Fran¸ois, 2010] c ABC is a k-nearest neighbour (knn) method with kN = N N [Loftsgaarden & Quesenberry, 1965] [Biau et al., 2012, arxiv:1207.6461]

75. ABC consistency Provided kN / log log N −→ ∞ and kN /N −→ 0 as N → ∞, for almost all s0 (with respect to the distribution of S), with probability 1, kN 1 ϕ(θj ) −→ E[ϕ(θj )|S = s0 ] kN j=1 [Devroye, 1982] Biau et al. (2012) also recall pointwise and integrated mean square error consistency results on the corresponding kernel estimate of the conditional posterior distribution, under constraints p kN → ∞, kN /N → 0, hN → 0 and hN kN → ∞,

76. ABC consistency Provided kN / log log N −→ ∞ and kN /N −→ 0 as N → ∞, for almost all s0 (with respect to the distribution of S), with probability 1, kN 1 ϕ(θj ) −→ E[ϕ(θj )|S = s0 ] kN j=1 [Devroye, 1982] Biau et al. (2012) also recall pointwise and integrated mean square error consistency results on the corresponding kernel estimate of the conditional posterior distribution, under constraints p kN → ∞, kN /N → 0, hN → 0 and hN kN → ∞,

77. Rates of convergence Further assumptions (on target and kernel) allow for precise (integrated mean square) convergence rates (as a power of the sample size N), derived from classical k-nearest neighbour regression, like 4 when m = 1, 2, 3, kN ≈ N (p+4)/(p+8) and rate N − p+8 4 when m = 4, kN ≈ N (p+4)/(p+8) and rate N − p+8 log N 4 when m > 4, kN ≈ N (p+4)/(m+p+4) and rate N − m+p+4 [Biau et al., 2012, arxiv:1207.6461] Only applies to suﬃcient summary statistics

78. Rates of convergence Further assumptions (on target and kernel) allow for precise (integrated mean square) convergence rates (as a power of the sample size N), derived from classical k-nearest neighbour regression, like 4 when m = 1, 2, 3, kN ≈ N (p+4)/(p+8) and rate N − p+8 4 when m = 4, kN ≈ N (p+4)/(p+8) and rate N − p+8 log N 4 when m > 4, kN ≈ N (p+4)/(m+p+4) and rate N − m+p+4 [Biau et al., 2012, arxiv:1207.6461] Only applies to suﬃcient summary statistics

79. ABC inference machine Introduction ABC ABC as an inference machine Error inc. Exact BC and approximate targets summary statistic ABCel

80. How much Bayesian aBc is..? maybe a convergent method of inference (meaningful? suﬃcient? foreign?) approximation error unknown (w/o simulation) pragmatic Bayes (there is no other solution!) many calibration issues (tolerance, distance, statistics)

81. How much Bayesian aBc is..? maybe a convergent method of inference (meaningful? suﬃcient? foreign?) approximation error unknown (w/o simulation) pragmatic Bayes (there is no other solution!) many calibration issues (tolerance, distance, statistics) ...should Bayesians care?!

82. How much Bayesian aBc is..? maybe a convergent method of inference (meaningful? suﬃcient? foreign?) approximation error unknown (w/o simulation) pragmatic Bayes (there is no other solution!) many calibration issues (tolerance, distance, statistics) yes they should!!!

83. How much Bayesian aBc is..? maybe a convergent method of inference (meaningful? suﬃcient? foreign?) approximation error unknown (w/o simulation) pragmatic Bayes (there is no other solution!) many calibration issues (tolerance, distance, statistics) to ABCel

84. ABCµ Idea Infer about the error as well as about the parameter: Use of a joint density f (θ, |y) ∝ ξ( |y, θ) × πθ (θ) × π ( ) where y is the data, and ξ( |y, θ) is the prior predictive density of ρ(η(z), η(y)) given θ and y when z ∼ f (z|θ) Warning! Replacement of ξ( |y, θ) with a non-parametric kernel approximation. [Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS]

87. ABCµ details Multidimensional distances ρk (k = 1, . . . , K ) and errors k = ρk (ηk (z), ηk (y)), with 1 k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = ^ K [{ k −ρk (ηk (zb ), ηk (y))}/hk ] Bhk b then used in replacing ξ( |y, θ) with mink ξk ( |y, θ) ^ ABCµ involves acceptance probability π(θ , ) q(θ , θ)q( , ) mink ξk ( |y, θ ) ^ π(θ, ) q(θ, θ )q( , ) mink ξk ( |y, θ) ^

88. ABCµ details Multidimensional distances ρk (k = 1, . . . , K ) and errors k = ρk (ηk (z), ηk (y)), with 1 k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = ^ K [{ k −ρk (ηk (zb ), ηk (y))}/hk ] Bhk b then used in replacing ξ( |y, θ) with mink ξk ( |y, θ) ^ ABCµ involves acceptance probability π(θ , ) q(θ , θ)q( , ) mink ξk ( |y, θ ) ^ π(θ, ) q(θ, θ )q( , ) mink ξk ( |y, θ) ^

89. ABCµ multiple errors [ c Ratmann et al., PNAS, 2009]

90. ABCµ for model choice [ c Ratmann et al., PNAS, 2009]

91. Wilkinson’s exact BC (not exactly!) ABC approximation error (i.e. non-zero tolerance) replaced with exact simulation from a controlled approximation to the target, convolution of true posterior with kernel function π(θ)f (z|θ)K (y − z) π (θ, z|y) = , π(θ)f (z|θ)K (y − z)dzdθ with K kernel parameterised by bandwidth . [Wilkinson, 2008] Theorem The ABC algorithm based on the assumption of a randomised observation y = y + ξ, ξ ∼ K , and an acceptance probability of ˜ K (y − z)/M gives draws from the posterior distribution π(θ|y).

92. Wilkinson’s exact BC (not exactly!) ABC approximation error (i.e. non-zero tolerance) replaced with exact simulation from a controlled approximation to the target, convolution of true posterior with kernel function π(θ)f (z|θ)K (y − z) π (θ, z|y) = , π(θ)f (z|θ)K (y − z)dzdθ with K kernel parameterised by bandwidth . [Wilkinson, 2008] Theorem The ABC algorithm based on the assumption of a randomised observation y = y + ξ, ξ ∼ K , and an acceptance probability of ˜ K (y − z)/M gives draws from the posterior distribution π(θ|y).

93. How exact a BC? “Using to represent measurement error is straightforward, whereas using to model the model discrepancy is harder to conceptualize and not as commonly used” [Richard Wilkinson, 2008]

94. How exact a BC? Pros Pseudo-data from true model and observed data from noisy model Interesting perspective in that outcome is completely controlled Link with ABCµ and assuming y is observed with a measurement error with density K Relates to the theory of model approximation [Kennedy & O’Hagan, 2001] Cons Requires K to be bounded by M True approximation error never assessed Requires a modiﬁcation of the standard ABC algorithm

95. ABC for HMMs Speciﬁc case of a hidden Markov model Xt+1 ∼ Qθ (Xt , ·) Yt+1 ∼ gθ (·|xt ) where only y0 is observed. 1:n [Dean, Singh, Jasra, & Peters, 2011] Use of speciﬁc constraints, adapted to the Markov structure: y1 ∈ B(y1 , ) × · · · × yn ∈ B(yn , ) 0 0

96. ABC for HMMs Speciﬁc case of a hidden Markov model Xt+1 ∼ Qθ (Xt , ·) Yt+1 ∼ gθ (·|xt ) where only y0 is observed. 1:n [Dean, Singh, Jasra, & Peters, 2011] Use of speciﬁc constraints, adapted to the Markov structure: y1 ∈ B(y1 , ) × · · · × yn ∈ B(yn , ) 0 0

97. ABC-MLE for HMMs ABC-MLE deﬁned by θn = arg max Pθ Y1 ∈ B(y1 , ), . . . , Yn ∈ B(yn , ) ^ 0 0 θ Exact MLE for the likelihood same basis as Wilkinson! 0 pθ (y1 , . . . , yn ) corresponding to the perturbed process (xt , yt + zt )1 t n zt ∼ U(B(0, 1) [Dean, Singh, Jasra, & Peters, 2011]

98. ABC-MLE for HMMs ABC-MLE deﬁned by θn = arg max Pθ Y1 ∈ B(y1 , ), . . . , Yn ∈ B(yn , ) ^ 0 0 θ Exact MLE for the likelihood same basis as Wilkinson! 0 pθ (y1 , . . . , yn ) corresponding to the perturbed process (xt , yt + zt )1 t n zt ∼ U(B(0, 1) [Dean, Singh, Jasra, & Peters, 2011]

99. ABC-MLE is biased ABC-MLE is asymptotically (in n) biased with target l (θ) = Eθ∗ [log pθ (Y1 |Y−∞:0 )] but ABC-MLE converges to true value in the sense l n (θn ) → l (θ) for all sequences (θn ) converging to θ and n

100. ABC-MLE is biased ABC-MLE is asymptotically (in n) biased with target l (θ) = Eθ∗ [log pθ (Y1 |Y−∞:0 )] but ABC-MLE converges to true value in the sense l n (θn ) → l (θ) for all sequences (θn ) converging to θ and n

101. Noisy ABC-MLE Idea: Modify instead the data from the start 0 (y1 + ζ1 , . . . , yn + ζn ) [ see Fearnhead-Prangle ] noisy ABC-MLE estimate arg max Pθ Y1 ∈ B(y1 + ζ1 , ), . . . , Yn ∈ B(yn + ζn , ) 0 0 θ [Dean, Singh, Jasra, & Peters, 2011]

102. Consistent noisy ABC-MLE Degrading the data improves the estimation performances: Noisy ABC-MLE is asymptotically (in n) consistent under further assumptions, the noisy ABC-MLE is asymptotically normal increase in variance of order −2 likely degradation in precision or computing time due to the lack of summary statistic [curse of dimensionality]

103. SMC for ABC likelihood Algorithm 2 SMC ABC for HMMs Given θ for k = 1, . . . , n do 1 1 N N generate proposals (xk , yk ), . . . , (xk , yk ) from the model weigh each proposal with ωk l =I l B(yk + ζk , ) (yk ) 0 l renormalise the weights and sample the xk ’s accordingly end for approximate the likelihood by n N ωlk N k=1 l=1 [Jasra, Singh, Martin, & McCoy, 2010]

104. Which summary? Fundamental diﬃculty of the choice of the summary statistic when there is no non-trivial suﬃcient statistics Starting from a large collection of summary statistics is available, Joyce and Marjoram (2008) consider the sequential inclusion into the ABC target, with a stopping rule based on a likelihood ratio test Not taking into account the sequential nature of the tests Depends on parameterisation Order of inclusion matters likelihood ratio test?!

107. Which summary for model choice? Depending on the choice of η(·), the Bayes factor based on this insuﬃcient statistic, η π1 (θ1 )f1η (η(y)|θ1 ) dθ1 B12 (y) = , π2 (θ2 )f2η (η(y)|θ2 ) dθ2 is consistent or not. [X, Cornuet, Marin, & Pillai, 2012] Consistency only depends on the range of Ei [η(y)] under both models. [Marin, Pillai, X, & Rousseau, 2012]

108. Which summary for model choice? Depending on the choice of η(·), the Bayes factor based on this insuﬃcient statistic, η π1 (θ1 )f1η (η(y)|θ1 ) dθ1 B12 (y) = , π2 (θ2 )f2η (η(y)|θ2 ) dθ2 is consistent or not. [X, Cornuet, Marin, & Pillai, 2012] Consistency only depends on the range of Ei [η(y)] under both models. [Marin, Pillai, X, & Rousseau, 2012]

109. Semi-automatic ABC Fearnhead and Prangle (2010) study ABC and the selection of the summary statistic in close proximity to Wilkinson’s proposal ABC considered as inferential method and calibrated as such randomised (or ‘noisy’) version of the summary statistics ˜ η(y) = η(y) + τ derivation of a well-calibrated version of ABC, i.e. an algorithm that gives proper predictions for the distribution associated with this randomised summary statistic

110. Summary [of F&P/statistics) optimality of the posterior expectation E[θ|y] of the parameter of interest as summary statistics η(y)! use of the standard quadratic loss function (θ − θ0 )T A(θ − θ0 ) . recent extension to model choice, optimality of Bayes factor B12 (y) [F&P, ISBA 2012 talk]

111. Summary [of F&P/statistics) optimality of the posterior expectation E[θ|y] of the parameter of interest as summary statistics η(y)! use of the standard quadratic loss function (θ − θ0 )T A(θ − θ0 ) . recent extension to model choice, optimality of Bayes factor B12 (y) [F&P, ISBA 2012 talk]

112. Conclusion Choice of summary statistics is paramount for ABC validation/performance At best, ABC approximates π(. | η(y)) Model selection feasible with ABC [with caution!] For estimation, consistency if {θ; µ(θ) = µ0 } = θ0 For testing consistency if {µ1 (θ1 ), θ1 ∈ Θ1 } ∩ {µ2 (θ2 ), θ2 ∈ Θ2 } = ∅ [Marin et al., 2011]

117. Empirical likelihood (EL) Introduction ABC ABC as an inference machine ABCel ABC and EL Composite likelihood Illustrations

118. Empirical likelihood (EL) Dataset x made of n independent replicates x = (x1 , . . . , xn ) of some X ∼ F Generalized moment condition model EF h(X , φ) = 0, where h is a known function, and φ an unknown parameter Corresponding empirical likelihood n Lel (φ|x) = max pi p i=1 for all p such that 0 pi 1, pi = 1, i pi h(xi , φ) = 0. [Owen, 1988, Bio’ka; Owen, 2001]

119. Empirical likelihood (EL) Dataset x made of n independent replicates x = (x1 , . . . , xn ) of some X ∼ F Generalized moment condition model EF h(X , φ) = 0, where h is a known function, and φ an unknown parameter Corresponding empirical likelihood n Lel (φ|x) = max pi p i=1 for all p such that 0 pi 1, pi = 1, i pi h(xi , φ) = 0. [Owen, 1988, Bio’ka; Owen, 2001]

120. Convergence of EL [3.4] Theorem 3.4 Let X , Y1 , . . . , Yn be independent rv’s with common distribution F0 . For θ ∈ Θ, and the function h(X , θ) ∈ Rs , let θ0 ∈ Θ be such that Var(h(Yi , θ0 )) is ﬁnite and has rank q > 0. If θ0 satisﬁes E(h(X , θ0 )) = 0, then Lel (θ0 |Y1 , . . . , Yn ) −2 log → χ2 (q) n−n in distribution when n → ∞. [Owen, 2001]

121. Convergence of EL [3.4] “...The interesting thing about Theorem 3.4 is what is not there. It ^ includes no conditions to make θ a good estimate of θ0 , nor even conditions to ensure a unique value for θ0 , nor even that any solution θ0 exists. Theorem 3.4 applies in the just determined, over-determined, and under-determined cases. When we can prove that our estimating ^ equations uniquely deﬁne θ0 , and provide a consistent estimator θ of it, then conﬁdence regions and tests follow almost automatically through Theorem 3.4.”. [Owen, 2001]

122. Raw ABCel sampler We act as if EL was an exact likelihood for i = 1 → N do generate φi from the prior distribution π(·) set the weight ωi = Lel (φi |xobs ) end for return (φi , ωi ), i = 1, . . . , N The output is sample of parameters of size N with associated weights [Cornuet et al., 2012]

123. Raw ABCel sampler We act as if EL was an exact likelihood for i = 1 → N do generate φi from the prior distribution π(·) set the weight ωi = Lel (φi |xobs ) end for return (φi , ωi ), i = 1, . . . , N Performance of the output evaluated through eﬀective sample size  2 N  N  ESS = 1 ωi ωj   i=1 j=1 [Cornuet et al., 2012]

124. Raw ABCel sampler We act as if EL was an exact likelihood for i = 1 → N do generate φi from the prior distribution π(·) set the weight ωi = Lel (φi |xobs ) end for return (φi , ωi ), i = 1, . . . , N Other classical sampling algorithms might be adapted to use EL. We resorted to the adaptive multiple importance sampling (AMIS) of Cornuet to speed up computations [Cornuet et al., 2012]

125. Moment condition in population genetics? EL does not require a fully deﬁned and often complex (hence debatable) parametric model Main diﬃculty Derive a constraint EF h(X , φ) = 0, on the parameters of interest φ when X is made of the genotypes of the sample of individuals at a given locus E.g., in phylogeography, φ is composed of dates of divergence between populations, ratio of population sizes, mutation rates, etc. None of them are moments of the distribution of the allelic states of the sample

126. Moment condition in population genetics? EL does not require a fully deﬁned and often complex (hence debatable) parametric model Main diﬃculty Derive a constraint EF h(X , φ) = 0, on the parameters of interest φ when X is made of the genotypes of the sample of individuals at a given locus c h pairwise composite scores whose zero is the pairwise maximum likelihood estimator

127. Pairwise composite likelihood The intra-locus pairwise likelihood j 2 (xk |φ) 2 (xk , xk |φ) i = i<j 1 n with xk , . . . , xk : allelic states of the gene sample at the k-th locus The pairwise score function j φ log 2 (xk |φ) φ log 2 (xk , xk |φ) i = i<j Composite likelihoods are often much narrower than the original likelihood of the model Safe with EL because we only use position of its mode

128. Pairwise likelihood: a simple case 1 Assumptions 2 (δ|θ) =√ ρ (θ)|δ| 1 + 2θ sample ⊂ closed, panmictic with θ population at equilibrium ρ(θ) = √ 1 + θ + 1 + 2θ marker: microsatellite mutation rate: θ/2 Pairwise score function ∂θ log 2 (δ|θ) = i j 1 |δ| if xk et xk are two genes of the − + √ sample, 1 + 2θ θ 1 + 2θ j 2 (xk , xk |θ) i depends only on i j δ = xk − xk

131. Pairwise likelihood: 2 diverging populations MRCA Then 2 (δ|θ, τ) = +∞ e−τθ √ ρ(θ)|k| Iδ−k (τθ). τ 1 + 2θ k=−∞ where In (z) nth-order modiﬁed POP a POP b Bessel function of the ﬁrst Assumptions kind τ: divergence date of pop. a and b θ/2: mutation rate i j Let xk and xk be two genes coming resp. from pop. a and b i j Set δ = xk − xk .

133. Example: normal posterior ABCel with two constraints ESS=108.9 ESS=81.48 ESS=105.2 3.0 2.0 2.0 Density Density Density 1.5 1.0 1.0 0.0 0.0 0.0 −0.4 0.0 0.2 0.4 0.6 −0.8 −0.6 −0.4 −0.2 0.0 −0.2 0.0 0.2 0.4 0.6 0.8 ESS=133.3 θ ESS=87.75 θ ESS=72.89 θ 3.0 2.0 4 2.0 Density Density Density 1.0 1.0 2 0.0 0.0 0 −0.8 −0.4 0.0 0.2 0.4 −0.2 0.0 0.2 0.4 0.6 0.8 −0.2 0.0 0.2 0.4 ESS=116.5 θ ESS=103.9 θ ESS=126.9 θ 3.0 2.0 3.0 2.0 Density Density Density 1.0 1.5 1.0 0.0 0.0 −0.4 0.0 0.2 0.4 0.6 −0.4 −0.2 0.0 0.2 0.4 0.0 −0.8 −0.4 0.0 0.4 ESS=113.3 θ ESS=92.99 θ ESS=121.4 θ 3.0 2.0 2.0 Density Density Density 1.5 1.0 1.0 0.0 0.0 0.0 −0.6 −0.2 0.2 0.6 −0.2 0.0 0.2 0.4 0.6 −0.5 0.0 0.5 ESS=133.6 θ ESS=116.4 θ ESS=131.6 θ 0.0 1.0 2.0 3.0 2.0 Density Density Density 1.0 1.0 0.0 0.0 −0.6 −0.2 0.2 0.4 0.6 −0.4 −0.2 0.0 0.2 0.4 −0.5 0.0 0.5 Sample sizes are of 21 (column 3), 41 (column 1) and 61 (column 2) observations

134. Example: normal posterior ABCel with three constraints ESS=233.8 ESS=186.6 ESS=370.1 3.0 3.0 2.0 2.0 Density Density Density 1.5 1.0 1.0 0.0 0.0 0.0 −0.4 −0.2 0.0 0.2 0.4 −0.8 −0.6 −0.4 −0.2 0.0 0.2 −0.4 0.0 0.2 0.4 0.6 0.8 ESS=248.2 θ ESS=189.9 θ ESS=274.1 θ 0.0 1.0 2.0 3.0 4 2.0 Density Density Density 3 2 1.0 1 0.0 0 0.0 0.2 0.4 0.6 0.8 −0.4 −0.2 0.0 0.2 0.4 −0.3 −0.2 −0.1 0.0 0.1 ESS=228.0 θ ESS=166.4 θ ESS=348.2 θ 3.0 4 0.0 1.0 2.0 3.0 3 2.0 Density Density Density 2 1.0 1 0.0 0 −0.6 −0.2 0.2 0.4 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −0.8 −0.6 −0.4 −0.2 ESS=202.5 θ ESS=163.3 θ ESS=228.7 θ 4 8 2.0 3 Density Density Density 6 2 4 1.0 1 2 0.0 0 0 −0.4 −0.2 0.0 0.2 0.4 −0.3 −0.1 0.1 0.2 0.3 −0.55 −0.45 −0.35 ESS=223.7 θ ESS=133.7 θ ESS=352.1 θ 3.0 2.0 2.0 Density Density Density 1.5 1.0 1.0 0.0 0.0 0.0 −0.6 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 0.4 −0.8 −0.6 −0.4 −0.2 0.0 0.2 Sample sizes are of 21 (column 3), 41 (column 1) and 61 (column 2) observations

135. Example: Superposition of gamma processes Example of superposition of N renewal processes with waiting times τij (i = 1, . . . , M), j = 1, . . .) ∼ G(α, β), when N is unknown. Renewal processes ζi1 = τi1 , ζi2 = ζi1 + τi2 , . . . with observations made of ﬁrst n values of the ζij ’s, z1 = min{ζij }, z2 = min{ζij ; ζij > z1 }, . . . , ending with zn = min{ζij ; ζij > zn−1 } . [Cox & Kartsonaki, B’ka, 2012]

136. Example: Superposition of gamma processes (ABC) Interesting testing ground for ABCel since data (zt ) neither iid nor Markov Recovery of an iid structure by 1. simulating a pseudo-dataset, Comparison of ABC and (z , . . . , z ), as in regular ABCel posteriors 1 n ABC, 0.08 1.4 1.5 1.2 0.06 1.0 1.0 0.8 Density Density Density 2. deriving sequence of 0.04 0.6 0.5 0.4 0.02 0.2 indicators (ν1 , . . . , νn ), as 0.00 0.0 0.0 0 1 2 3 4 0 1 2 3 4 0 5 10 15 20 α β N z1 = ζν1 1 , z2 = ζν2 j2 , . . . 1.5 0.06 1.0 0.05 0.8 1.0 0.04 0.6 Density Density Density 0.03 0.4 0.5 0.02 3. exploiting that those 0.2 0.01 0.00 0.0 0.0 indicators are distributed 0 1 2 α 3 4 0 1 2 β 3 4 0 5 10 N 15 20 from the prior distribution Top: ABCel on the νt ’s leading to an iid Bottom: regular ABC sample of G(α, β) variables

137. Example: Superposition of gamma processes (ABC) Interesting testing ground for ABCel since data (zt ) neither iid nor Markov Recovery of an iid structure by 1. simulating a pseudo-dataset, Comparison of ABC and (z , . . . , z ), as in regular ABCel posteriors 1 n ABC, 0.08 1.4 1.5 1.2 0.06 1.0 1.0 0.8 Density Density Density 2. deriving sequence of 0.04 0.6 0.5 0.4 0.02 0.2 indicators (ν1 , . . . , νn ), as 0.00 0.0 0.0 0 1 2 3 4 0 1 2 3 4 0 5 10 15 20 α β N z1 = ζν1 1 , z2 = ζν2 j2 , . . . 1.5 0.06 1.0 0.05 0.8 1.0 0.04 0.6 Density Density Density 0.03 0.4 0.5 0.02 3. exploiting that those 0.2 0.01 0.00 0.0 0.0 indicators are distributed 0 1 2 α 3 4 0 1 2 β 3 4 0 5 10 N 15 20 from the prior distribution Top: ABCel on the νt ’s leading to an iid Bottom: regular ABC sample of G(α, β) variables

138. Pop’gen’: A ﬁrst experiment Evolutionary scenario: Comparison of the original MRCA ABC with ABCel ESS=7034 15 τ 10 Density 5 POP 0 POP 1 0 0.00 0.05 0.10 0.15 0.20 0.25 Dataset: log(theta) 50 genes per populations, 7 6 100 microsat. loci 5 Density 4 3 Assumptions: 2 1 0 Ne identical over all −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 log(tau1) populations φ = (log10 θ, log10 τ) histogram = ABCel curve = original ABC uniform prior over vertical line = “true” (−1., 1.5) × (−1., 1.) parameter

139. Pop’gen’: A ﬁrst experiment Evolutionary scenario: Comparison of the original MRCA ABC with ABCel ESS=7034 15 τ 10 Density 5 POP 0 POP 1 0 0.00 0.05 0.10 0.15 0.20 0.25 Dataset: log(theta) 50 genes per populations, 7 6 100 microsat. loci 5 Density 4 3 Assumptions: 2 1 0 Ne identical over all −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 log(tau1) populations φ = (log10 θ, log10 τ) histogram = ABCel curve = original ABC uniform prior over vertical line = “true” (−1., 1.5) × (−1., 1.) parameter

140. ABC vs. ABCel on 100 replicates of the 1st experiment Accuracy: log10 θ log10 τ ABC ABCel ABC ABCel (1) 0.097 0.094 0.315 0.117 (2) 0.071 0.059 0.272 0.077 (3) 0.68 0.81 1.0 0.80 (1) Root Mean Square Error of the posterior mean (2) Median Absolute Deviation of the posterior median (3) Coverage of the credibility interval of probability 0.8 Computation time: on a recent 6-core computer (C++/OpenMP) ABC ≈ 4 hours ABCel ≈ 2 minutes

141. Pop’gen’: Second experiment Evolutionary scenario: Comparison of the original ABC MRCA with ABCel τ2 histogram = ABCel curve = original ABC τ1 vertical line = “true” parameter POP 0 POP 1 POP 2 Dataset: 50 genes per populations, 100 microsat. loci Assumptions: Ne identical over all populations φ= (log10 θ, log10 τ1 , log10 τ2 ) non-informative uniform

142. Pop’gen’: Second experiment Evolutionary scenario: Comparison of the original ABC MRCA with ABCel τ2 τ1 POP 0 POP 1 POP 2 Dataset: 50 genes per populations, 100 microsat. loci Assumptions: Ne identical over all histogram = ABCel populations curve = original ABC φ= vertical line = “true” parameter (log10 θ, log10 τ1 , log10 τ2 ) non-informative uniform

143. Pop’gen’: Second experiment Evolutionary scenario: Comparison of the original ABC MRCA with ABCel τ2 τ1 POP 0 POP 1 POP 2 Dataset: 50 genes per populations, 100 microsat. loci Assumptions: Ne identical over all populations φ= histogram = ABCel (log10 θ, log10 τ1 , log10 τ2 ) curve = original ABC non-informative uniform vertical line = “true” parameter

144. Pop’gen’: Second experiment Evolutionary scenario: Comparison of the original ABC MRCA with ABCel τ2 τ1 POP 0 POP 1 POP 2 Dataset: 50 genes per populations, 100 microsat. loci Assumptions: Ne identical over all populations φ= histogram = ABCel (log10 θ, log10 τ1 , log10 τ2 ) curve = original ABC non-informative uniform vertical line = “true” parameter

145. ABC vs. ABCel on 100 replicates of the 2nd experiment Accuracy: log10 θ log10 τ1 log10 τ2 ABC ABCel ABC ABCel ABC ABCel (1) 0.0059 0.0794 0.472 0.483 29.3 4.76 (2) 0.048 0.053 0.32 0.28 4.13 3.36 (3) 0.79 0.76 0.88 0.76 0.89 0.79 (1) Root Mean Square Error of the posterior mean (2) Median Absolute Deviation of the posterior median (3) Coverage of the credibility interval of probability 0.8 Computation time: on a recent 6-core computer (C++/OpenMP) ABC ≈ 6 hours ABCel ≈ 8 minutes

146. Why? On large datasets, ABCel gives more accurate results than ABC ABC simpliﬁes the dataset through summary statistics Due to the large dimension of x, the original ABC algorithm estimates π θ η(xobs ) , where η(xobs ) is some (non-linear) projection of the observed dataset on a space with smaller dimension → Some information is lost ABCel simpliﬁes the model through a generalized moment condition model. → Here, the moment condition model is based on pairwise composition likelihood

147. To be continued...

ABC and empirical likelihood

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ABC and empirical likelihood