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ABC workshop: 17w5025

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ABC workshop: 17w5025

  1. 1. ABCD : ABC in High Dimensions Christian P. Robert Universit´e Paris-Dauphine, University of Warwick, & IUF workshop 17w5025, BIRS, Banff, Alberta, Canada
  2. 2. Approximate Bayesian computation 1 Approximate Bayesian computation ABC basics Remarks on high D Non-parametric roots Automated summary selection Series B discussion
  3. 3. The ABC method Bayesian setting: target is π(θ)f (x|θ)
  4. 4. The ABC method Bayesian setting: target is π(θ)f (x|θ) When likelihood f (x|θ) not in closed form, likelihood-free rejection technique:
  5. 5. The ABC method Bayesian setting: target is π(θ)f (x|θ) When likelihood f (x|θ) not in closed form, likelihood-free rejection technique: ABC algorithm For an observation y ∼ f (y|θ), under the prior π(θ), keep jointly simulating θ ∼ π(θ) , z ∼ f (z|θ ) , until the auxiliary variable z is equal to the observed value, z = y. [Tavar´e et al., 1997]
  6. 6. A as A...pproximative When y is a continuous random variable, equality z = y is replaced with a tolerance condition, (y, z) ≤ where is a distance
  7. 7. A as A...pproximative When y is a continuous random variable, equality z = y is replaced with a tolerance condition, (y, z) ≤ where is a distance Output distributed from π(θ) Pθ{ (y, z) < } ∝ π(θ| (y, z) < ) [Pritchard et al., 1999]
  8. 8. ABC algorithm Algorithm 1 Likelihood-free rejection sampler 2 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
  9. 9. Output The likelihood-free algorithm samples from the marginal in z of: π (θ, z|y) = π(θ)f (z|θ)IA ,y (z) A ,y×Θ π(θ)f (z|θ)dzdθ , where A ,y = {z ∈ D|ρ(η(z), η(y)) < }.
  10. 10. Output The likelihood-free algorithm samples from the marginal in z of: π (θ, z|y) = π(θ)f (z|θ)IA ,y (z) 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|η(y)) .
  11. 11. MA example Consider the MA(q) model xt = t + q i=1 ϑi t−i Simple prior(s): uniform over the inverse [real and complex] roots in Q(u) = 1 − q i=1 ϑi ui under the identifiability conditions
  12. 12. MA example Consider the MA(q) model xt = t + q i=1 ϑi t−i Simple prior(s): uniform prior over the identifiability zone, e.g. triangle for MA(2)
  13. 13. 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
  14. 14. 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 ρ((xt)1≤t≤T , (xt)1≤t≤T ) = T t=1 (xt − xt)2 or distance between summary statistics like the q autocorrelations τj = T t=j+1 xtxt−j
  15. 15. Comparison of distance impact Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
  16. 16. Comparison of distance impact 0.0 0.2 0.4 0.6 0.8 01234 θ1 −2.0 −1.0 0.0 0.5 1.0 1.5 0.00.51.01.5 θ2 Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
  17. 17. Comparison of distance impact 0.0 0.2 0.4 0.6 0.8 01234 θ1 −2.0 −1.0 0.0 0.5 1.0 1.5 0.00.51.01.5 θ2 Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
  18. 18. Approximate Bayesian computation 1 Approximate Bayesian computation ABC basics Remarks on high D Non-parametric roots Automated summary selection Series B discussion
  19. 19. ABC[and the]D[curse] “These theoretical results suggests that when performing ABC in high dimensions, the most important problem is how to address the poor performance of the ABC estimator. This is often bourne out in practice.” R. Everitt • how to most effectively make use of the sample from the joint distribution on (θ, t) that is generated by ABC algorithms • algorithms that estimate the distribution or statistics of θ given η(y) • methods that estimate the conditional η(y)|θ that is often less important in this context (but is still relevant in some applications): that of the • design of inference algorithms that are computationally scalable to high dimensional cases
  20. 20. Parameter given summary • local regression correction `a la Beaumont et al. (2002) • non-parametric regression `a la Blum et al. (2013) • ridge regression `a la Blum et al. (2013) • semi-automatic ABC `a la Fernhead and Prangle (2012) • neural networks `a la Blum and Fran¸cois (2010) • random forest `a la Marin et al. (2016) • not particularly dimension resistant • restricted to estimation
  21. 21. Summary given parameter • non-parametric regression `a la Blum et al. (2013) • synthetic likelihood `a la Wood (2010) and Dutta et al. (2016) • Gaussian processes `a la Wilkinson (2014), Meeds and Welling (2013), J¨arvenp¨a¨a et al. (2016)
  22. 22. Reproducing kernel Hilbert spaces Several papers suggesting moving from kernel estimation to RKHS regression: • Fukumizu, Le & Gretton (2013) • Nakagome, Fukumizu, & Mano (2013) • Park, Jitkrittum, & Sejdinovic (2016) • Mitrovic, Sejdinovic, & Teh (2016)
  23. 23. Reproducing kernel Hilbert spaces Several papers suggesting moving from kernel estimation to RKHS regression: with arguments • removing dependence on summary statistics by using “the data themselves” • correlatively no loss of information • providing a form of distance between empirical measures • learning directly the regression • relatively easy and fast implementation (with linear time MMD)
  24. 24. K-ABC = ABC + RKHS ”A notable property of KBR is that the prior and likelihood are represented in terms of samples.” [Park et al., 2016] Concept of a kernel function h(·, ·) that serves as a discrepancy between two samples or two sets of summary statistics, h(y, yobs)
  25. 25. K-ABC = ABC + RKHS ”A notable property of KBR is that the prior and likelihood are represented in terms of samples.” [Park et al., 2016] Concept of a kernel function h(·, ·) that serves as a discrepancy between two samples or two sets of summary statistics, h(y, yobs) Estimates based on learning sample (θt, yt)T t=1 expressed as T t=1 ωth(θt) with G = (h(η(yi ), η(yj )))i,j and ω = [G + n nIn]−1 (h(η(y1), η(yobs )) · · · h(η(yT ), η(yobs )))T [Fukumizu et al., 2013]
  26. 26. K2-ABC = ABC + RKHS + RKHS Second level RKHS when h(yi , yj )) = exp − −1 MMD2 (yi , yj ) • no need of summary statistics • require maximum mean discrepancy or other distance between distributions • extension to non-iid data problematic [Park et al., 2016]
  27. 27. DR-ABC = ABC + RKHS + RKHS Kernel-based distribution regression (DR): instead of kernel-weighted regression for posterior expectations, construction of ABC summaries Claim optimality of a summary statistics against given loss function but unclear to me Ridge-like summary Θ[G + n nIn]−1 h close to above derivation of weights [Mitrovic et al., 2016]
  28. 28. Approximate Bayesian computation 1 Approximate Bayesian computation ABC basics Remarks on high D Non-parametric roots Automated summary selection Series B discussion
  29. 29. 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 (use with BIC) θ∗ = θ − {η(z) − η(y)}T ˆβ [Csill´ery et al., TEE, 2010] where ˆβ 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]
  30. 30. 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 1 S S s=1 Kδ {ρ(η(zs (θ)), η(y))} [consistent estimate of f (η|θ)]
  31. 31. 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 1 S S s=1 Kδ {ρ(η(zs (θ)), η(y))} [consistent estimate of f (η|θ)] Curse of dimensionality: poor estimate when d = dim(η) is large...
  32. 32. 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]
  33. 33. 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]
  34. 34. ABC-NP (density estimations) Other versions incorporating regression adjustments i ˜Kb(θ∗ i − θ)Kδ {ρ(η(z(θi )), η(y))} i Kδ {ρ(η(z(θi )), η(y))}
  35. 35. ABC-NP (density estimations) Other versions incorporating regression adjustments i ˜Kb(θ∗ i − θ)Kδ {ρ(η(z(θi )), η(y))} i Kδ {ρ(η(z(θi )), η(y))} In all cases, error E[ˆg(θ|y)] − g(θ|y) = cb2 + cδ2 + OP(b2 + δ2 ) + OP(1/nδd ) var(ˆg(θ|y)) = c nbδd (1 + oP(1)) [Blum, JASA, 2010]
  36. 36. ABC-NP (density estimations) Other versions incorporating regression adjustments i ˜Kb(θ∗ i − θ)Kδ {ρ(η(z(θi )), η(y))} i Kδ {ρ(η(z(θi )), η(y))} In all cases, error E[ˆg(θ|y)] − g(θ|y) = cb2 + cδ2 + OP(b2 + δ2 ) + OP(1/nδd ) var(ˆg(θ|y)) = c nbδd (1 + oP(1)) [standard NP calculations]
  37. 37. ABC-NCH Incorporating non-linearities and heterocedasticities: θ∗ = ˆm(η(y)) + [θ − ˆm(η(z))] ˆσ(η(y)) ˆσ(η(z))
  38. 38. ABC-NCH Incorporating non-linearities and heterocedasticities: θ∗ = ˆm(η(y)) + [θ − ˆm(η(z))] ˆσ(η(y)) ˆσ(η(z)) where • ˆm(η) estimated by non-linear regression (e.g., neural network) • ˆσ(η) estimated by non-linear regression on residuals log{θi − ˆm(ηi )}2 = log σ2 (ηi ) + ξi [Blum & Fran¸cois, 2009]
  39. 39. ABC-NCH (2) Why neural network?
  40. 40. ABC-NCH (2) Why neural network? • fights curse of dimensionality • selects relevant summary statistics • provides automated dimension reduction • offers a model choice capability • improves upon multinomial logistic [Blum & Fran¸cois, 2009]
  41. 41. ABC as knn [Biau et al., 2013, Annales de l’IHP] Practice of ABC: determine tolerance as a quantile on observed distances, say 10% or 1% quantile, = N = qα(d1, . . . , dN)
  42. 42. ABC as knn [Biau et al., 2013, Annales de l’IHP] Practice of ABC: determine tolerance as a quantile on observed distances, say 10% or 1% quantile, = N = qα(d1, . . . , dN) • Interpretation of ε as nonparametric bandwidth only approximation of the actual practice [Blum & Fran¸cois, 2010]
  43. 43. ABC as knn [Biau et al., 2013, Annales de l’IHP] Practice of ABC: determine tolerance as a quantile on observed distances, say 10% or 1% quantile, = N = qα(d1, . . . , dN) • Interpretation of ε as nonparametric bandwidth only approximation of the actual practice [Blum & Fran¸cois, 2010] • ABC is a k-nearest neighbour (knn) method with kN = N N [Loftsgaarden & Quesenberry, 1965]
  44. 44. 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, 1 kN kN j=1 ϕ(θj ) −→ E[ϕ(θj )|S = s0] [Devroye, 1982]
  45. 45. 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, 1 kN kN j=1 ϕ(θj ) −→ E[ϕ(θj )|S = s0] [Devroye, 1982] Biau et al. (2013) also recall pointwise and integrated mean square error consistency results on the corresponding kernel estimate of the conditional posterior distribution, under constraints kN → ∞, kN /N → 0, hN → 0 and hp N kN → ∞,
  46. 46. 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 • when m = 1, 2, 3, kN ≈ N(p+4)/(p+8) and rate N − 4 p+8 • when m = 4, kN ≈ N(p+4)/(p+8) and rate N − 4 p+8 log N • when m > 4, kN ≈ N(p+4)/(m+p+4) and rate N − 4 m+p+4 [Biau et al., 2013]
  47. 47. Noisy ABC Idea: Modify the data from the start ˜y = y0 + ζ1 with the same scale as ABC [ see Fearnhead-Prangle ] run ABC on ˜y
  48. 48. Noisy ABC Idea: Modify the data from the start ˜y = y0 + ζ1 with the same scale as ABC [ see Fearnhead-Prangle ] run ABC on ˜y Then ABC produces an exact simulation from π(θ|˜y) = π(θ|˜y) [Dean et al., 2011; Fearnhead and Prangle, 2012]
  49. 49. Consistent noisy ABC • 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]
  50. 50. Approximate Bayesian computation 1 Approximate Bayesian computation ABC basics Remarks on high D Non-parametric roots Automated summary selection Series B discussion
  51. 51. Semi-automatic ABC Fearnhead and Prangle (2010) study ABC and the selection of the summary statistic in close proximity to Wilkinson’s proposal ABC then considered from a purely inferential viewpoint and calibrated for estimation purposes Use of a 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
  52. 52. Semi-automatic ABC Fearnhead and Prangle (2010) study ABC and the selection of the summary statistic in close proximity to Wilkinson’s proposal ABC then considered from a purely inferential viewpoint and calibrated for estimation purposes Use of a 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 [calibration constraint: ABC approximation with same posterior mean as the true randomised posterior]
  53. 53. Summary statistics • Optimality of the posterior expectation E[θ|y] of the parameter of interest as summary statistics η(y)!
  54. 54. Summary 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) . bare summary
  55. 55. Details on Fearnhead and Prangle (F&P) ABC Use of a summary statistic S(·), an importance proposal g(·), a kernel K(·) ≤ 1 and a bandwidth h > 0 such that (θ, ysim) ∼ g(θ)f (ysim|θ) is accepted with probability (hence the bound) K[{S(ysim) − sobs}/h] and the corresponding importance weight defined by π(θ) g(θ) [Fearnhead & Prangle, 2012]
  56. 56. Errors, errors, and errors Three levels of approximation • π(θ|yobs) by π(θ|sobs) loss of information [ignored] • π(θ|sobs) by πABC(θ|sobs) = π(s)K[{s − sobs}/h]π(θ|s) ds π(s)K[{s − sobs}/h] ds noisy observations • πABC(θ|sobs) by importance Monte Carlo based on N simulations, represented by var(a(θ)|sobs)/Nacc [expected number of acceptances] [M. Twain/B. Disraeli]
  57. 57. Average acceptance asymptotics For the average acceptance probability/approximate likelihood p(θ|sobs) = f (ysim|θ) K[{S(ysim) − sobs}/h] dysim , overall acceptance probability p(sobs) = p(θ|sobs) π(θ) dθ = π(sobs)hd + o(hd ) [F&P, Lemma 1]
  58. 58. Calibration of h “This result gives insight into how S(·) and h affect the Monte Carlo error. To minimize Monte Carlo error, we need hd to be not too small. Thus ideally we want S(·) to be a low dimensional summary of the data that is sufficiently informative about θ that π(θ|sobs) is close, in some sense, to π(θ|yobs)” (F&P, p.5) • turns h into an absolute value while it should be context-dependent and user-calibrated • only addresses one term in the approximation error and acceptance probability (“curse of dimensionality”) • h large prevents πABC(θ|sobs) to be close to π(θ|sobs) • d small prevents π(θ|sobs) to be close to π(θ|yobs) (“curse of [dis]information”)
  59. 59. Converging ABC Theorem (F&P) For noisy ABC, the expected noisy-ABC log-likelihood, E {log[p(θ|sobs)]} = log[p(θ|S(yobs) + )]π(yobs|θ0)K( )dyobsd , has its maximum at θ = θ0.
  60. 60. Converging ABC Corollary For noisy ABC, the ABC posterior converges onto a point mass on the true parameter value as m → ∞. For standard ABC, not always the case (unless h goes to zero).
  61. 61. Loss motivated statistic Under quadratic loss function, Theorem (F&P) (i) The minimal posterior error E[L(θ, ˆθ)|yobs] occurs when ˆθ = E(θ|yobs) (!) (ii) When h → 0, EABC(θ|sobs) converges to E(θ|yobs) (iii) If S(yobs) = E[θ|yobs] then for ˆθ = EABC[θ|sobs] E[L(θ, ˆθ)|yobs] = trace(AΣ) + h2 xT AxK(x)dx + o(h2 ).
  62. 62. Optimal summary statistic “We take a different approach, and weaken the requirement for πABC to be a good approximation to π(θ|yobs). We argue for πABC to be a good approximation solely in terms of the accuracy of certain estimates of the parameters.” (F&P, p.5) From this result, F&P • derive their choice of summary statistic, S(y) = E(θ|y) [almost sufficient] • suggest h = O(N−1/(2+d) ) and h = O(N−1/(4+d) ) as optimal bandwidths for noisy and standard ABC.
  63. 63. Optimal summary statistic “We take a different approach, and weaken the requirement for πABC to be a good approximation to π(θ|yobs). We argue for πABC to be a good approximation solely in terms of the accuracy of certain estimates of the parameters.” (F&P, p.5) From this result, F&P • derive their choice of summary statistic, S(y) = E(θ|y) [wow! EABC[θ|S(yobs)] = E[θ|yobs]] • suggest h = O(N−1/(2+d) ) and h = O(N−1/(4+d) ) as optimal bandwidths for noisy and standard ABC.
  64. 64. Caveat Since E(θ|yobs) is most usually unavailable, F&P suggest (i) use a pilot run of ABC to determine a region of non-negligible posterior mass; (ii) simulate sets of parameter values and data; (iii) use the simulated sets of parameter values and data to estimate the summary statistic; and (iv) run ABC with this choice of summary statistic.
  65. 65. Approximate Bayesian computation 1 Approximate Bayesian computation ABC basics Remarks on high D Non-parametric roots Automated summary selection Series B discussion
  66. 66. More discussions about semi-automatic ABC [ End of section derived from comments on Read Paper, Series B, 2012] “The apparently arbitrary nature of the choice of summary statistics has always been perceived as the Achilles heel of ABC.” M. Beaumont
  67. 67. More discussions about semi-automatic ABC [ End of section derived from comments on Read Paper, Series B, 2012] “The apparently arbitrary nature of the choice of summary statistics has always been perceived as the Achilles heel of ABC.” M. Beaumont • “Curse of dimensionality” linked with the increase of the dimension of the summary statistic • Connection with principal component analysis [Itan et al., 2010] • Connection with partial least squares [Wegman et al., 2009] • Beaumont et al. (2002) postprocessed output is used as input by F&P to run a second ABC
  68. 68. Wood’s alternative Instead of a non-parametric kernel approximation to the likelihood 1 R r K {η(yr ) − η(yobs )} Wood (2010) suggests a normal approximation η(y(θ)) ∼ Nd (µθ, Σθ) whose parameters can be approximated based on the R simulations (for each value of θ).
  69. 69. Wood’s alternative Instead of a non-parametric kernel approximation to the likelihood 1 R r K {η(yr ) − η(yobs )} Wood (2010) suggests a normal approximation η(y(θ)) ∼ Nd (µθ, Σθ) whose parameters can be approximated based on the R simulations (for each value of θ). • Parametric versus non-parametric rate [Uh?!] • Automatic weighting of components of η(·) through Σθ • Dependence on normality assumption (pseudo-likelihood?) [Cornebise, Girolami & Kosmidis, 2012]
  70. 70. ABC and BIC Idea of applying BIC during the local regression : • Run regular ABC • Select summary statistics during local regression • Recycle the prior simulation sample (reference table) with those summary statistics • Rerun the corresponding local regression (low cost) [Pudlo & Sedki, 2012]

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