From MCMC to ABC Methods                       From MCMC to ABC Methods                                    Christian P. Ro...
From MCMC to ABC MethodsOutlineComputational issues in Bayesian statisticsThe Metropolis-Hastings AlgorithmThe Gibbs Sampl...
From MCMC to ABC Methods  Computational issues in Bayesian statisticsA typology of Bayes computational problems       (i)....
From MCMC to ABC Methods  Computational issues in Bayesian statisticsA typology of Bayes computational problems       (i)....
From MCMC to ABC Methods  Computational issues in Bayesian statisticsA typology of Bayes computational problems       (i)....
From MCMC to ABC Methods  Computational issues in Bayesian statisticsA typology of Bayes computational problems       (i)....
From MCMC to ABC Methods  Computational issues in Bayesian statisticsA typology of Bayes computational problems       (i)....
From MCMC to ABC Methods  The Metropolis-Hastings AlgorithmThe Metropolis-Hastings Algorithm      Computational issues in ...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsGeneral purpose      Given a density π kn...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsMonte Carlo 101      Generate an iid samp...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsMonte Carlo 101      Generate an iid samp...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsImportance Sampling      For Q proposal d...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsImportance Sampling      For Q proposal d...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsProperties of importance      Then       ...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsProperties of importance      Then       ...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsSelf-Normalised Importance Sampling      ...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsSelf-Normalised Importance Sampling      ...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo basicsSelf-Normalised Importance Sampling      ...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo Methods based on Markov ChainsRunning Monte Car...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo Methods based on Markov ChainsRunning Monte Car...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo Methods based on Markov ChainsRunning Monte Car...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Monte Carlo Methods based on Markov ChainsRunning Monte Car...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    The Metropolis–Hastings algorithmThe Metropolis–Hastings al...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    The Metropolis–Hastings algorithmThe MH algorithm      Algo...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    The Metropolis–Hastings algorithmFeatures             Indep...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    The Metropolis–Hastings algorithmConvergence properties    ...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    The Metropolis–Hastings algorithmConvergence properties    ...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    The Metropolis–Hastings algorithmConvergence properties (2)...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    The Metropolis–Hastings algorithmConvergence properties (2)...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    The Metropolis–Hastings algorithmConvergence properties (2)...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsRandom walk Metro...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithms      Algorithm (...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsOptimizing the Ac...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsCase of the rando...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsCase of the rando...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsCase of the rando...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsCase of the rando...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithms      Example (No...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithms      Example (No...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithms      Example (No...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithms      Markov chai...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithms      Markov chai...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsMA(2)            ...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsBasic RWHM for MA...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsOutcome      Resu...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsOutcome      Resu...
From MCMC to ABC Methods  The Metropolis-Hastings Algorithm    Random-walk Metropolis-Hastings algorithmsOutcome      Resu...
From MCMC to ABC Methods  The Gibbs SamplerThe Gibbs Sampler     The Gibbs Sampler        General Principles        Slice ...
From MCMC to ABC Methods  The Gibbs Sampler    General PrinciplesGeneral Principles     A very specific simulation algorith...
From MCMC to ABC Methods  The Gibbs Sampler    General PrinciplesGeneral Principles     A very specific simulation algorith...
From MCMC to ABC Methods  The Gibbs Sampler    General PrinciplesGeneral Principles     A very specific simulation algorith...
From MCMC to ABC Methods  The Gibbs Sampler    General Principles     Algorithm (Gibbs sampler)                           ...
From MCMC to ABC Methods  The Gibbs Sampler    General PrinciplesProperties     The full conditionals densities f1 , . . ....
From MCMC to ABC Methods  The Gibbs Sampler    General PrinciplesProperties     The full conditionals densities f1 , . . ....
From MCMC to ABC Methods  The Gibbs Sampler    General PrinciplesLimitations of the Gibbs sampler     Formally, a special ...
From MCMC to ABC Methods  The Gibbs Sampler    General PrinciplesLimitations of the Gibbs sampler     Formally, a special ...
From MCMC to ABC Methods  The Gibbs Sampler    General PrinciplesLimitations of the Gibbs sampler     Formally, a special ...
From MCMC to ABC Methods  The Gibbs Sampler    General PrinciplesLimitations of the Gibbs sampler     Formally, a special ...
From MCMC to ABC Methods  The Gibbs Sampler        General PrinciplesA wee mixture problem        4        3        2   µ2...
From MCMC to ABC Methods  The Gibbs Sampler        General PrinciplesA wee mixture problem                                ...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingSlice sampler as generic Gibbs     If f (θ) can be written as...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingSlice sampler as generic Gibbs     If f (θ) can be written as...
From MCMC to ABC Methods  The Gibbs Sampler    Slice sampling     Algorithm (Slice sampler)     Simulate                (t...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingExample of results with a truncated N (−3, 1) distribution   ...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingExample of results with a truncated N (−3, 1) distribution   ...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingExample of results with a truncated N (−3, 1) distribution   ...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingExample of results with a truncated N (−3, 1) distribution   ...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingExample of results with a truncated N (−3, 1) distribution   ...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingExample of results with a truncated N (−3, 1) distribution   ...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingExample of results with a truncated N (−3, 1) distribution   ...
From MCMC to ABC Methods  The Gibbs Sampler    Slice samplingGood slices, tough slices     The slice sampler usually enjoy...
From MCMC to ABC Methods  The Gibbs Sampler    ConvergenceProperties of the Gibbs sampler     Theorem (Convergence)     Fo...
From MCMC to ABC Methods  The Gibbs Sampler    ConvergenceProperties of the Gibbs sampler (2)     Consequences     (i) If ...
From MCMC to ABC Methods  The Gibbs Sampler    ConvergenceHammersley-Clifford theorem     An illustration that conditionals...
From MCMC to ABC Methods  The Gibbs Sampler    ConvergenceGeneral HC decomposition     Under the positivity condition, the...
From MCMC to ABC Methods  Approximate Bayesian computationApproximate Bayesian computation     Computational issues in Bay...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsUntractable likelihoods     Cases when the likelih...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsIllustrations     Example    Stochastic volatility...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsIllustrations     Example     Potts model: if y ta...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsIllustrations     Example     Inference on CMB: in...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsIllustrations     Example   Phylogenetic tree: in ...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsThe ABC method     Bayesian setting: target is π(θ...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsThe ABC method     Bayesian setting: target is π(θ...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsThe ABC method     Bayesian setting: target is π(θ...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsWhy does it work?!     The proof is trivial:      ...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsA as approximative     When y is a continuous rand...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsA as approximative     When y is a continuous rand...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsABC algorithm     Algorithm 2 Likelihood-free reje...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsOutput     The likelihood-free algorithm samples f...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsOutput     The likelihood-free algorithm samples f...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsMA example     Back to the MA(q) model            ...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsMA example     Back to the MA(q) model            ...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsMA example (2)     ABC algorithm thus made of     ...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsMA example (2)     ABC algorithm thus made of     ...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsComparison of distance impact     Evaluation of th...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsComparison of distance impact     Evaluation of th...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsComparison of distance impact     Evaluation of th...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsABC advances     Simulating from the prior is ofte...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsABC advances     Simulating from the prior is ofte...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsABC advances     Simulating from the prior is ofte...
From MCMC to ABC Methods  Approximate Bayesian computation    ABC basicsABC advances     Simulating from the prior is ofte...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABC-NP    Better usage of [prior] simulations b...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABC-MCMC     Markov chain (θ(t) ) created via t...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABC-MCMC     Markov chain (θ(t) ) created via t...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABC-MCMC (2)     Algorithm 3 Likelihood-free MC...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupWhy does it work?     Acceptance probability th...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABCµ                       [Ratmann, Andrieu, W...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABCµ                       [Ratmann, Andrieu, W...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABCµ details     Multidimensional distances ρk ...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABCµ details     Multidimensional distances ρk ...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABCµ multiple errors                           ...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABCµ for model choice                          ...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupQuestions about ABCµ     For each model under c...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupQuestions about ABCµ     For each model under c...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupA PMC version     Generate a sample at iteratio...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupThe ABC-PMC algorithm     Given a decreasing se...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupSequential Monte Carlo     SMC is a simulation ...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupSequential Monte Carlo (2)     Algorithm 4 SMC ...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABC-SMC                                        ...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupABC-SMCM     Modification: Makes M repeated simu...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupProperties of ABC-SMC     The ABC-SMC method pr...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupProperties of ABC-SMC     The ABC-SMC method pr...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupSemi-automatic ABC     Fearnhead and Prangle (2...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupSummary statistics     Optimality of the poster...
From MCMC to ABC Methods  Approximate Bayesian computation    Alphabet soupSummary statistics     Optimality of the poster...
From MCMC to ABC Methods  Approximate Bayesian computation    Calibration of ABCWhich summary?     Fundamental difficulty of...
From MCMC to ABC Methods  Approximate Bayesian computation    Calibration of ABCWhich summary?     Fundamental difficulty of...
From MCMC to ABC Methods  Approximate Bayesian computation    Calibration of ABCWhich summary?     Fundamental difficulty of...
From MCMC to ABC Methods  ABC for model choiceABC for model choice     Computational issues in Bayesian statistics     The...
From MCMC to ABC Methods  ABC for model choice    Model choiceBayesian model choice     Several models M1 , M2 , . . . are...
From MCMC to ABC Methods  ABC for model choice    Model choiceGeneric ABC for model choice     Algorithm 5 Likelihood-free...
From MCMC to ABC Methods  ABC for model choice    Model choiceABC estimates     Posterior probability π(M = m|y) approxima...
From MCMC to ABC Methods  ABC for model choice    Model choiceABC estimates     Posterior probability π(M = m|y) approxima...
From MCMC to ABC Methods  ABC for model choice    Model choiceThe Great ABC controversy     On-going controvery in phyloge...
From MCMC to ABC Methods  ABC for model choice    Model choiceThe Great ABC controversy     On-going controvery in phyloge...
From MCMC to ABC Methods  ABC for model choice    Gibbs random fieldsGibbs random fields     Gibbs distribution     The rv y...
From MCMC to ABC Methods  ABC for model choice    Gibbs random fieldsGibbs random fields     Gibbs distribution     The rv y...
From MCMC to ABC Methods  ABC for model choice    Gibbs random fieldsPotts model     Potts model     Vc (y) is of the form ...
From MCMC to ABC Methods  ABC for model choice    Gibbs random fieldsPotts model     Potts model     Vc (y) is of the form ...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCBayesian Model Choice     Comparing a model with pot...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCBayesian Model Choice     Comparing a model with pot...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCNeighbourhood relations     Choice to be made betwee...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCModel index     Formalisation via a model index M th...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCModel index     Formalisation via a model index M th...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCSufficient statistics     By definition, if S(x) sufficie...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCSufficient statistics     By definition, if S(x) sufficie...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCSufficient statistics     By definition, if S(x) sufficie...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCABC model choice Algorithm     ABC-MC             Ge...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCABC approximation to the Bayes factor     Frequency ...
From MCMC to ABC Methods  ABC for model choice    Model choice via ABCABC approximation to the Bayes factor     Frequency ...
From MCMC to ABC Methods  ABC for model choice    IllustrationsToy example     iid Bernoulli model versus two-state first-o...
From MCMC to ABC Methods  ABC for model choice    IllustrationsToy example (2)     (left) Comparison of the true BF m0 /m1...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceBack to sufficiency     If η1 (x) sufficient statist...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceBack to sufficiency     If η1 (x) sufficient statist...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceLimiting behaviour of B12 (T → ∞)     ABC approx...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceLimiting behaviour of B12 (T → ∞)     ABC approx...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceLimiting behaviour of B12 ( → 0)     When       ...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceLimiting behaviour of B12 ( → 0)     When       ...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceLimiting behaviour of B12 (under sufficiency)     ...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceLimiting behaviour of B12 (under sufficiency)     ...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choicePoisson/geometric example     Sample            ...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choicePoisson/geometric discrepancy                   ...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceFormal recovery     Creating an encompassing exp...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceFormal recovery     Creating an encompassing exp...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceFormal recovery     Creating an encompassing exp...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceMeaning of the ABC-Bayes factor     In the Poiss...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceMA(q) divergence     Evolution [against ] of ABC...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceMA(q) divergence     Evolution [against ] of ABC...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceA population genetics evaluation     Population ...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceA population genetics evaluation     Population ...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceStability of importance sampling
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceComparison with ABC     Use of 24 summary statis...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceComparison with ABC     Use of 15 summary statis...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceComparison with ABC     Use of 24 summary statis...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceThe only safe cases     Besides specific models l...
From MCMC to ABC Methods  ABC for model choice    Generic ABC model choiceThe only safe cases     Besides specific models l...
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Shanghai tutorial

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Shanghai tutorial

  1. 1. From MCMC to ABC Methods From MCMC to ABC Methods Christian P. Robert Universit´ Paris-Dauphine, IuF, & CREST e http://www.ceremade.dauphine.fr/~xian O’Bayes 11, Shanghai, June 10, 2011
  2. 2. From MCMC to ABC MethodsOutlineComputational issues in Bayesian statisticsThe Metropolis-Hastings AlgorithmThe Gibbs SamplerApproximate Bayesian computationABC for model choice
  3. 3. From MCMC to ABC Methods Computational issues in Bayesian statisticsA typology of Bayes computational problems (i). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models;
  4. 4. From MCMC to ABC Methods Computational issues in Bayesian statisticsA typology of Bayes computational problems (i). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models; (ii). use of a complex sampling model with an intractable likelihood, as for instance in some latent variable or graphical models or in inverse problems;
  5. 5. From MCMC to ABC Methods Computational issues in Bayesian statisticsA typology of Bayes computational problems (i). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models; (ii). use of a complex sampling model with an intractable likelihood, as for instance in some latent variable or graphical models or in inverse problems; (iii). use of a huge dataset;
  6. 6. From MCMC to ABC Methods Computational issues in Bayesian statisticsA typology of Bayes computational problems (i). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models; (ii). use of a complex sampling model with an intractable likelihood, as for instance in some latent variable or graphical models or in inverse problems; (iii). use of a huge dataset; (iv). use of a complex prior distribution (which may be the posterior distribution associated with an earlier sample);
  7. 7. From MCMC to ABC Methods Computational issues in Bayesian statisticsA typology of Bayes computational problems (i). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models; (ii). use of a complex sampling model with an intractable likelihood, as for instance in some latent variable or graphical models or in inverse problems; (iii). use of a huge dataset; (iv). use of a complex prior distribution (which may be the posterior distribution associated with an earlier sample); (v). use of a particular inferential procedure as for instance, Bayes factors π P (θ ∈ Θ0 | x) π(θ ∈ Θ0 ) B01 (x) = . P (θ ∈ Θ1 | x) π(θ ∈ Θ1 )
  8. 8. From MCMC to ABC Methods The Metropolis-Hastings AlgorithmThe Metropolis-Hastings Algorithm Computational issues in Bayesian statistics The Metropolis-Hastings Algorithm Monte Carlo basics Monte Carlo Methods based on Markov Chains The Metropolis–Hastings algorithm Random-walk Metropolis-Hastings algorithms The Gibbs Sampler Approximate Bayesian computation ABC for model choice
  9. 9. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsGeneral purpose Given a density π known up to a normalizing constant, and an integrable function h, compute h(x)˜ (x)µ(dx) π Π(h) = h(x)π(x)µ(dx) = π (x)µ(dx) ˜ when h(x)˜ (x)µ(dx) is intractable. π
  10. 10. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsMonte Carlo 101 Generate an iid sample x1 , . . . , xN from π and estimate Π(h) by N ˆN ΠM C (h) = N −1 h(xi ). i=1 ˆN as LLN: ΠM C (h) −→ Π(h) If Π(h2 ) = h2 (x)π(x)µ(dx) < ∞, √ L CLT: ˆN N ΠM C (h) − Π(h) N 0, Π [h − Π(h)]2 .
  11. 11. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsMonte Carlo 101 Generate an iid sample x1 , . . . , xN from π and estimate Π(h) by N ˆN ΠM C (h) = N −1 h(xi ). i=1 ˆN as LLN: ΠM C (h) −→ Π(h) If Π(h2 ) = h2 (x)π(x)µ(dx) < ∞, √ L CLT: ˆN N ΠM C (h) − Π(h) N 0, Π [h − Π(h)]2 . Caveat announcing MCMC Often impossible or inefficient to simulate directly from Π
  12. 12. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsImportance Sampling For Q proposal distribution such that Q(dx) = q(x)µ(dx), alternative representation Π(h) = h(x){π/q}(x)q(x)µ(dx).
  13. 13. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsImportance Sampling For Q proposal distribution such that Q(dx) = q(x)µ(dx), alternative representation Π(h) = h(x){π/q}(x)q(x)µ(dx). Principle of importance Generate an iid sample x1 , . . . , xN ∼ Q and estimate Π(h) by N ˆ IS ΠQ,N (h) = N −1 h(xi ){π/q}(xi ). i=1
  14. 14. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsProperties of importance Then ˆ as LLN: ΠIS (h) −→ Π(h) Q,N and if Q((hπ/q)2 ) < ∞, √ L CLT: ˆ Q,N N (ΠIS (h) − Π(h)) N 0, Q{(hπ/q − Π(h))2 } .
  15. 15. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsProperties of importance Then ˆ as LLN: ΠIS (h) −→ Π(h) Q,N and if Q((hπ/q)2 ) < ∞, √ L CLT: ˆ Q,N N (ΠIS (h) − Π(h)) N 0, Q{(hπ/q − Π(h))2 } . Caveat ˆ Q,N If normalizing constant of π unknown, impossible to use ΠIS Generic problem in Bayesian Statistics: π(θ|x) ∝ f (x|θ)π(θ).
  16. 16. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsSelf-Normalised Importance Sampling Self normalized version N −1 N ˆ Q,N ΠSN IS (h) = {π/q}(xi ) h(xi ){π/q}(xi ). i=1 i=1
  17. 17. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsSelf-Normalised Importance Sampling Self normalized version N −1 N ˆ Q,N ΠSN IS (h) = {π/q}(xi ) h(xi ){π/q}(xi ). i=1 i=1 ˆ as LLN : ΠSN IS (h) −→ Π(h) Q,N and if Π((1 + h2 )(π/q)) < ∞, √ L CLT : ˆ Q,N N (ΠSN IS (h) − Π(h)) N 0, π {(π/q)(h − Π(h)}2 ) .
  18. 18. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo basicsSelf-Normalised Importance Sampling Self normalized version N −1 N ˆ Q,N ΠSN IS (h) = {π/q}(xi ) h(xi ){π/q}(xi ). i=1 i=1 ˆ as LLN : ΠSN IS (h) −→ Π(h) Q,N and if Π((1 + h2 )(π/q)) < ∞, √ L CLT : ˆ Q,N N (ΠSN IS (h) − Π(h)) N 0, π {(π/q)(h − Π(h)}2 ) . c The quality of the SNIS approximation depends on the choice of Q
  19. 19. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov ChainsRunning Monte Carlo via Markov Chains (MCMC) It is not necessary to use a sample from the distribution f to approximate the integral I= h(x)f (x)dx ,
  20. 20. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov ChainsRunning Monte Carlo via Markov Chains (MCMC) It is not necessary to use a sample from the distribution f to approximate the integral I= h(x)f (x)dx , We can obtain X1 , . . . , Xn ∼ f (approx) without directly simulating from f , using an ergodic Markov chain with stationary distribution f
  21. 21. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov ChainsRunning Monte Carlo via Markov Chains (2) Idea For an arbitrary starting value x(0) , an ergodic chain (X (t) ) is generated using a transition kernel with stationary distribution f
  22. 22. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov ChainsRunning Monte Carlo via Markov Chains (2) Idea For an arbitrary starting value x(0) , an ergodic chain (X (t) ) is generated using a transition kernel with stationary distribution f Insures the convergence in distribution of (X (t) ) to a random variable from f . For a “large enough” T0 , X (T0 ) can be considered as distributed from f Produce a dependent sample X (T0 ) , X (T0 +1) , . . ., which is generated from f , sufficient for most approximation purposes.
  23. 23. From MCMC to ABC Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmThe Metropolis–Hastings algorithm Basics The algorithm uses the target density f and a conditional density q(y|x) called the instrumental (or proposal) distribution
  24. 24. From MCMC to ABC Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmThe MH algorithm Algorithm (Metropolis–Hastings) Given x(t) , 1. Generate Yt ∼ q(y|x(t) ). 2. Take Yt with prob. ρ(x(t) , Yt ), X (t+1) = x(t) with prob. 1 − ρ(x(t) , Yt ), where f (y) q(x|y) ρ(x, y) = min ,1 . f (x) q(y|x)
  25. 25. From MCMC to ABC Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmFeatures Independent of normalizing constants for both f and q(·|x) (ie, those constants independent of x) Never move to values with f (y) = 0 The chain (x(t) )t may take the same value several times in a row, even though f is a density wrt Lebesgue measure The sequence (yt )t is usually not a Markov chain
  26. 26. From MCMC to ABC Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmConvergence properties Under irreducibility, 1. The M-H Markov chain is reversible, with invariant/stationary density f since it satisfies the detailed balance condition f (y) K(y, x) = f (x) K(x, y)
  27. 27. From MCMC to ABC Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmConvergence properties Under irreducibility, 1. The M-H Markov chain is reversible, with invariant/stationary density f since it satisfies the detailed balance condition f (y) K(y, x) = f (x) K(x, y) 2. As f is a probability measure, the chain is positive recurrent
  28. 28. From MCMC to ABC Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmConvergence properties (2) 4. If q(y|x) > 0 for every (x, y), (2) the chain is irreducible
  29. 29. From MCMC to ABC Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmConvergence properties (2) 4. If q(y|x) > 0 for every (x, y), (2) the chain is irreducible 5. For M-H, f -irreducibility implies Harris recurrence
  30. 30. From MCMC to ABC Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithmConvergence properties (2) 4. If q(y|x) > 0 for every (x, y), (2) the chain is irreducible 5. For M-H, f -irreducibility implies Harris recurrence 6. Thus, for M-H satisfying (1) and (2) (i) For h, with Ef |h(X)| < ∞, T 1 lim h(X (t) ) = h(x)df (x) a.e. f. T →∞ T t=1 (ii) and lim K n (x, ·)µ(dx) − f =0 n→∞ TV for every initial distribution µ, where K n (x, ·) denotes the kernel for n transitions.
  31. 31. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsRandom walk Metropolis–Hastings Use of a local perturbation as proposal Yt = X (t) + εt , where εt ∼ g, independent of X (t) . The instrumental density is of the form g(y − x) and the Markov chain is a random walk if we take g to be symmetric g(x) = g(−x)
  32. 32. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Algorithm (Random walk Metropolis) Given x(t) 1. Generate Yt ∼ g(y − x(t) ) 2. Take  Y f (Yt ) (t+1) t with prob. min 1, , X = f (x(t) )  x(t) otherwise.
  33. 33. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsOptimizing the Acceptance Rate Problem of choice of the transition kernel from a practical point of view Most common alternatives: 1. an instrumental density g which approximates f , such that f /g is bounded for uniform ergodicity to apply; 2. a random walk In both cases, the choice of g is critical,
  34. 34. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsCase of the random walk Different approach to acceptance rates A high acceptance rate does not indicate that the algorithm is moving correctly since it indicates that the random walk is moving too slowly on the surface of f .
  35. 35. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsCase of the random walk Different approach to acceptance rates A high acceptance rate does not indicate that the algorithm is moving correctly since it indicates that the random walk is moving too slowly on the surface of f . If x(t) and yt are close, i.e. f (x(t) ) f (yt ) y is accepted with probability f (yt ) min ,1 1. f (x(t) ) For multimodal densities with well separated modes, the negative effect of limited moves on the surface of f clearly shows.
  36. 36. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsCase of the random walk (2) If the average acceptance rate is low, the successive values of f (yt ) tend to be small compared with f (x(t) ), which means that the random walk moves quickly on the surface of f since it often reaches the “borders” of the support of f
  37. 37. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsCase of the random walk (2) If the average acceptance rate is low, the successive values of f (yt ) tend to be small compared with f (x(t) ), which means that the random walk moves quickly on the surface of f since it often reaches the “borders” of the support of f In small dimensions, aim at an average acceptance rate of 50%. In large dimensions, at an average acceptance rate of 25%. [Gelman,Gilks and Roberts, 1995]
  38. 38. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Example (Noisy AR(1)) Hidden Markov chain from a regular AR(1) model, xt+1 = ϕxt + t+1 t ∼ N (0, τ 2 ) and observables yt |xt ∼ N (x2 , σ 2 ) t
  39. 39. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Example (Noisy AR(1)) Hidden Markov chain from a regular AR(1) model, xt+1 = ϕxt + t+1 t ∼ N (0, τ 2 ) and observables yt |xt ∼ N (x2 , σ 2 ) t The distribution of xt given xt−1 , xt+1 and yt is −1 τ2 exp (xt − ϕxt−1 )2 + (xt+1 − ϕxt )2 + (yt − x2 )2 t . 2τ 2 σ2
  40. 40. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Example (Noisy AR(1) continued) For a Gaussian random walk with scale ω small enough, the random walk never jumps to the other mode. But if the scale ω is sufficiently large, the Markov chain explores both modes and give a satisfactory approximation of the target distribution.
  41. 41. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Markov chain based on a random walk with scale ω = .1.
  42. 42. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Markov chain based on a random walk with scale ω = .5.
  43. 43. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsMA(2) xt = t − θ1 t−1 − θ2 t−2 Since the constraints on (ϑ1 , ϑ2 ) are well-defined, use of a flat prior over the triangle as prior. Simple representation of the likelihood library(mnormt) ma2like=function(theta){ n=length(y) sigma = toeplitz(c(1 +theta[1]^2+theta[2]^2, theta[1]+theta[1]*theta[2],theta[2],rep(0,n-3))) dmnorm(y,rep(0,n),sigma,log=TRUE) }
  44. 44. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsBasic RWHM for MA(2) Algorithm 1 RW-HM-MA(2) sampler set ω and ϑ(1) for i = 2 to T do ˜ (i−1) (i−1) generate ϑj ∼ U (ϑj − ω, ϑj + ω) set p = 0 and ϑ (i) = ϑ(i−1) ˜ if ϑ within the triangle then ˜ p = exp(ma2like(ϑ) − ma2like(ϑ(i−1) )) end if if U < p then ˜ ϑ(i) = ϑ end if end for
  45. 45. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsOutcome Result with a simulated sample of 100 points and ϑ1 = 0.6, ϑ2 = 0.2 and scale ω = 0.2
  46. 46. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsOutcome Result with a simulated sample of 100 points and ϑ1 = 0.6, ϑ2 = 0.2 and scale ω = 0.5
  47. 47. From MCMC to ABC Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithmsOutcome Result with a simulated sample of 100 points and ϑ1 = 0.6, ϑ2 = 0.2 and scale ω = 2.0
  48. 48. From MCMC to ABC Methods The Gibbs SamplerThe Gibbs Sampler The Gibbs Sampler General Principles Slice sampling Convergence
  49. 49. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesGeneral Principles A very specific simulation algorithm based on the target distribution f : 1. Uses the conditional densities f1 , . . . , fp from f
  50. 50. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesGeneral Principles A very specific simulation algorithm based on the target distribution f : 1. Uses the conditional densities f1 , . . . , fp from f 2. Start with the random variable X = (X1 , . . . , Xp )
  51. 51. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesGeneral Principles A very specific simulation algorithm based on the target distribution f : 1. Uses the conditional densities f1 , . . . , fp from f 2. Start with the random variable X = (X1 , . . . , Xp ) 3. Simulate from the conditional densities, Xi |x1 , x2 , . . . , xi−1 , xi+1 , . . . , xp ∼ fi (xi |x1 , x2 , . . . , xi−1 , xi+1 , . . . , xp ) for i = 1, 2, . . . , p.
  52. 52. From MCMC to ABC Methods The Gibbs Sampler General Principles Algorithm (Gibbs sampler) (t) (t) Given x(t) = (x1 , . . . , xp ), generate (t+1) (t) (t) 1. X1 ∼ f1 (x1 |x2 , . . . , xp ); (t+1) (t+1) (t) (t) 2. X2 ∼ f2 (x2 |x1 , x3 , . . . , xp ), ... (t+1) (t+1) (t+1) p. Xp ∼ fp (xp |x1 , . . . , xp−1 ) X(t+1) → X ∼ f
  53. 53. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesProperties The full conditionals densities f1 , . . . , fp are the only densities used for simulation. Thus, even in a high dimensional problem, all of the simulations may be univariate
  54. 54. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesProperties The full conditionals densities f1 , . . . , fp are the only densities used for simulation. Thus, even in a high dimensional problem, all of the simulations may be univariate The Gibbs sampler is not reversible with respect to f . However, each of its p components is. Besides, it can be turned into a reversible sampler, either using the Random Scan Gibbs sampler see section or running instead the (double) sequence f1 · · · fp−1 fp fp−1 · · · f1
  55. 55. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesLimitations of the Gibbs sampler Formally, a special case of a sequence of 1-D M-H kernels, all with acceptance rate uniformly equal to 1. The Gibbs sampler 1. limits the choice of instrumental distributions
  56. 56. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesLimitations of the Gibbs sampler Formally, a special case of a sequence of 1-D M-H kernels, all with acceptance rate uniformly equal to 1. The Gibbs sampler 1. limits the choice of instrumental distributions 2. requires some knowledge of f
  57. 57. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesLimitations of the Gibbs sampler Formally, a special case of a sequence of 1-D M-H kernels, all with acceptance rate uniformly equal to 1. The Gibbs sampler 1. limits the choice of instrumental distributions 2. requires some knowledge of f 3. is, by construction, multidimensional
  58. 58. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesLimitations of the Gibbs sampler Formally, a special case of a sequence of 1-D M-H kernels, all with acceptance rate uniformly equal to 1. The Gibbs sampler 1. limits the choice of instrumental distributions 2. requires some knowledge of f 3. is, by construction, multidimensional 4. does not apply to problems where the number of parameters varies as the resulting chain is not irreducible.
  59. 59. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesA wee mixture problem 4 3 2 µ2 1 0 −1 −1 0 1 2 3 4 µ1 Gibbs started at random
  60. 60. From MCMC to ABC Methods The Gibbs Sampler General PrinciplesA wee mixture problem Gibbs stuck at the wrong mode 4 3 3 2 2 µ2 1 µ2 1 0 0 −1 −1 −1 0 1 2 3 4 µ1 Gibbs started at random −1 0 1 2 3 µ1
  61. 61. From MCMC to ABC Methods The Gibbs Sampler Slice samplingSlice sampler as generic Gibbs If f (θ) can be written as a product k fi (θ), i=1
  62. 62. From MCMC to ABC Methods The Gibbs Sampler Slice samplingSlice sampler as generic Gibbs If f (θ) can be written as a product k fi (θ), i=1 it can be completed as k I0≤ωi ≤fi (θ) , i=1 leading to the following Gibbs algorithm:
  63. 63. From MCMC to ABC Methods The Gibbs Sampler Slice sampling Algorithm (Slice sampler) Simulate (t+1) 1. ω1 ∼ U[0,f1 (θ(t) )] ; ... (t+1) k. ωk ∼ U[0,fk (θ(t) )] ; k+1. θ(t+1) ∼ UA(t+1) , with (t+1) A(t+1) = {y; fi (y) ≥ ωi , i = 1, . . . , k}.
  64. 64. From MCMC to ABC Methods The Gibbs Sampler Slice samplingExample of results with a truncated N (−3, 1) distribution 0.010 0.008 0.006 y 0.004 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x Number of Iterations 2
  65. 65. From MCMC to ABC Methods The Gibbs Sampler Slice samplingExample of results with a truncated N (−3, 1) distribution 0.010 0.008 0.006 y 0.004 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x Number of Iterations 2, 3
  66. 66. From MCMC to ABC Methods The Gibbs Sampler Slice samplingExample of results with a truncated N (−3, 1) distribution 0.010 0.008 0.006 y 0.004 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x Number of Iterations 2, 3, 4
  67. 67. From MCMC to ABC Methods The Gibbs Sampler Slice samplingExample of results with a truncated N (−3, 1) distribution 0.010 0.008 0.006 y 0.004 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x Number of Iterations 2, 3, 4, 5
  68. 68. From MCMC to ABC Methods The Gibbs Sampler Slice samplingExample of results with a truncated N (−3, 1) distribution 0.010 0.008 0.006 y 0.004 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x Number of Iterations 2, 3, 4, 5, 10
  69. 69. From MCMC to ABC Methods The Gibbs Sampler Slice samplingExample of results with a truncated N (−3, 1) distribution 0.010 0.008 0.006 y 0.004 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x Number of Iterations 2, 3, 4, 5, 10, 50
  70. 70. From MCMC to ABC Methods The Gibbs Sampler Slice samplingExample of results with a truncated N (−3, 1) distribution 0.010 0.008 0.006 y 0.004 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x Number of Iterations 2, 3, 4, 5, 10, 50, 100
  71. 71. From MCMC to ABC Methods The Gibbs Sampler Slice samplingGood slices, tough slices The slice sampler usually enjoys good theoretical properties (like geometric ergodicity and even uniform ergodicity under bounded f and bounded X ). As k increases, the determination of the set A(t+1) may get increasingly complex.
  72. 72. From MCMC to ABC Methods The Gibbs Sampler ConvergenceProperties of the Gibbs sampler Theorem (Convergence) For (Y1 , Y2 , · · · , Yp ) ∼ g(y1 , . . . , yp ), if either [Positivity condition] (i) g (i) (y > 0 for every i = 1, · · · , p, implies that i) g(y1 , . . . , yp ) > 0, where g (i) denotes the marginal distribution of Yi , or (ii) the transition kernel is absolutely continuous with respect to g, then the chain is irreducible and positive Harris recurrent.
  73. 73. From MCMC to ABC Methods The Gibbs Sampler ConvergenceProperties of the Gibbs sampler (2) Consequences (i) If h(y)g(y)dy < ∞, then T 1 lim h1 (Y (t) ) = h(y)g(y)dy a.e. g. nT →∞ T t=1 (ii) If, in addition, (Y (t) ) is aperiodic, then lim K n (y, ·)µ(dx) − f =0 n→∞ TV for every initial distribution µ.
  74. 74. From MCMC to ABC Methods The Gibbs Sampler ConvergenceHammersley-Clifford theorem An illustration that conditionals determine the joint distribution Theorem If the joint density g(y1 , y2 ) have conditional distributions g1 (y1 |y2 ) and g2 (y2 |y1 ), then g2 (y2 |y1 ) g(y1 , y2 ) = . g2 (v|y1 )/g1 (y1 |v) dv [Hammersley & Clifford, circa 1970]
  75. 75. From MCMC to ABC Methods The Gibbs Sampler ConvergenceGeneral HC decomposition Under the positivity condition, the joint distribution g satisfies p g j (y j |y 1 , . . . , y j−1 ,y j+1 , . . . , y p) g(y1 , . . . , yp ) ∝ g j (y j |y 1 , . . . , y j−1 ,y , . . . , y p) j=1 j+1 for every permutation on {1, 2, . . . , p} and every y ∈ Y .
  76. 76. From MCMC to ABC Methods Approximate Bayesian computationApproximate Bayesian computation Computational issues in Bayesian statistics The Metropolis-Hastings Algorithm The Gibbs Sampler Approximate Bayesian computation ABC basics Alphabet soup Calibration of ABC ABC for model choice
  77. 77. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsUntractable likelihoods Cases when the likelihood function f (y|θ) is unavailable and when the completion step f (y|θ) = f (y, z|θ) dz Z is impossible or too costly because of the dimension of z c MCMC cannot be implemented!
  78. 78. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsIllustrations Example Stochastic volatility model: for Highest weight trajectories t = 1, . . . , T, 0.4 0.2 yt = exp(zt ) t , zt = a+bzt−1 +σηt , 0.0 −0.2 T very large makes it difficult to −0.4 include z within the simulated 0 200 400 t 600 800 1000 parameters
  79. 79. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsIllustrations Example Potts model: if y takes values on a grid Y of size k n and f (y|θ) ∝ exp θ Iyl =yi l∼i where l∼i denotes a neighbourhood relation, n moderately large prohibits the computation of the normalising constant
  80. 80. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsIllustrations Example Inference on CMB: in cosmology, study of the Cosmic Microwave Background via likelihoods immensely slow to computate (e.g WMAP, Plank), because of numerically costly spectral transforms [Data is a Fortran program] [Kilbinger et al., 2010, MNRAS]
  81. 81. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsIllustrations Example Phylogenetic tree: in population genetics, reconstitution of a common ancestor from a sample of genes via a phylogenetic tree that is close to impossible to integrate out [100 processor days with 4 parameters] [Cornuet et al., 2009, Bioinformatics]
  82. 82. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsThe ABC method Bayesian setting: target is π(θ)f (x|θ)
  83. 83. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsThe ABC method Bayesian setting: target is π(θ)f (x|θ) When likelihood f (x|θ) not in closed form, likelihood-free rejection technique:
  84. 84. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsThe 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´ et al., 1997] e
  85. 85. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsWhy does it work?! The proof is trivial: f (θi ) ∝ π(θi )f (z|θi )Iy (z) z∈D ∝ π(θi )f (y|θi ) = π(θi |y) . [Accept–Reject 101]
  86. 86. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsA as approximative When y is a continuous random variable, equality z = y is replaced with a tolerance condition, (y, z) ≤ where is a distance
  87. 87. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsA as approximative 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) < )
  88. 88. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsABC algorithm Algorithm 2 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
  89. 89. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsOutput 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)) < }.
  90. 90. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsOutput 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) .
  91. 91. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsMA example Back to the MA(q) model q xt = t + ϑi t−i i=1 Simple prior: uniform over the inverse [real and complex] roots in q Q(u) = 1 − ϑi ui i=1 under the identifiability conditions
  92. 92. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsMA example Back to 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)
  93. 93. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsMA 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
  94. 94. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsMA 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 autocorrelations T τj = xt xt−j t=j+1
  95. 95. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsComparison of distance impact Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
  96. 96. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsComparison of distance impact Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
  97. 97. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsComparison of distance impact Evaluation of the tolerance on the ABC sample against both distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
  98. 98. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsABC advances Simulating from the prior is often poor in efficiency
  99. 99. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsABC advances 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]
  100. 100. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsABC advances 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]
  101. 101. From MCMC to ABC Methods Approximate Bayesian computation ABC basicsABC advances 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]
  102. 102. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABC-NP Better usage of [prior] simulations by adjustement: instead of throwing away θ such that ρ(η(z), η(y)) > , replace θs with locally regressed ˆ θ∗ = θ − {η(z) − η(y)}T β [Csill´ry et al., TEE, 2010] e ˆ 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]
  103. 103. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABC-MCMC Markov chain (θ(t) ) created via the transition function  θ ∼ Kω (θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y   π(θ )Kω (t) |θ ) θ (t+1) = and u ∼ U (0, 1) ≤ π(θ(t) )K (θ |θ(t) ) ,   (t) ω (θ θ otherwise,
  104. 104. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABC-MCMC Markov chain (θ(t) ) created via the transition function  θ ∼ Kω (θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y   π(θ )Kω (t) |θ ) θ (t+1) = and u ∼ U (0, 1) ≤ π(θ(t) )K (θ |θ(t) ) ,   (t) ω (θ θ otherwise, has the posterior π(θ|y) as stationary distribution [Marjoram et al, 2003]
  105. 105. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABC-MCMC (2) Algorithm 3 Likelihood-free MCMC sampler Use Algorithm 2 to get (θ(0) , z(0) ) for t = 1 to N do Generate θ from Kω ·|θ(t−1) , Generate z from the likelihood f (·|θ ), Generate u from U[0,1] , π(θ )Kω (θ (t−1) |θ ) if u ≤ I π(θ (t−1) Kω (θ |θ (t−1) ) A ,y (z ) then set (θ(t) , z(t) ) = (θ , z ) else (θ(t) , z(t) )) = (θ(t−1) , z(t−1) ), end if end for
  106. 106. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupWhy does it work? Acceptance probability that does not involve the calculation of the likelihood and π (θ , z |y) Kω (θ(t−1) |θ )f (z(t−1) |θ(t−1) ) × π (θ(t−1) , z(t−1) |y) Kω (θ |θ(t−1) )f (z |θ ) π(θ ) f (z |θ ) IA ,y (z ) = (t−1) ) f (z(t−1) |θ (t−1) )I (t−1) ) π(θ A ,y (z Kω (θ(t−1) |θ ) f (z(t−1) |θ(t−1) ) × Kω (θ |θ(t−1) ) f (z |θ ) π(θ )Kω (θ(t−1) |θ ) = IA (z ) . π(θ(t−1) Kω (θ |θ(t−1) ) ,y
  107. 107. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABCµ [Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS] 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 x when z ∼ f (z|θ)
  108. 108. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABCµ [Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS] 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 x when z ∼ f (z|θ) Warning! Replacement of ξ( |y, θ) with a non-parametric kernel approximation.
  109. 109. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABCµ details Multidimensional distances ρk (k = 1, . . . , K) and errors k = ρk (ηk (z), ηk (y)), with ˆ 1 k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = K[{ k −ρk (ηk (zb ), ηk (y))}/hk ] Bhk b ˆ then used in replacing ξ( |y, θ) with mink ξk ( |y, θ)
  110. 110. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABCµ details Multidimensional distances ρk (k = 1, . . . , K) and errors k = ρk (ηk (z), ηk (y)), with ˆ 1 k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = K[{ k −ρk (ηk (zb ), ηk (y))}/hk ] Bhk b ˆ then used in replacing ξ( |y, θ) with mink ξk ( |y, θ) ABCµ involves acceptance probability ˆ π(θ , ) q(θ , θ)q( , ) mink ξk ( |y, θ ) ˆ π(θ, ) q(θ, θ )q( , ) mink ξk ( |y, θ)
  111. 111. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABCµ multiple errors [ c Ratmann et al., PNAS, 2009]
  112. 112. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABCµ for model choice [ c Ratmann et al., PNAS, 2009]
  113. 113. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupQuestions about ABCµ For each model under comparison, marginal posterior on used to assess the fit of the model (HPD includes 0 or not).
  114. 114. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupQuestions about ABCµ For each model under comparison, marginal posterior on used to assess the fit of the model (HPD includes 0 or not). Is the data informative about ? [Identifiability] How is the prior π( ) impacting the comparison? How is using both ξ( |x0 , θ) and π ( ) compatible with a standard probability model? [remindful of Wilkinson] Where is the penalisation for complexity in the model comparison? [X, Mengersen & Chen, 2010, PNAS]
  115. 115. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupA PMC version Generate a sample at iteration t by N (t) (t−1) (t−1) πt (θ ) ∝ ˆ ωj Kt (θ(t) |θj ) j=1 modulo acceptance of the associated xt , and use an importance (t) weight associated with an accepted simulation θi (t) (t) (t) ωi ∝ π(θi ) πt (θi ) . ˆ c Still likelihood free [Beaumont et al., 2009]
  116. 116. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupThe ABC-PMC algorithm Given a decreasing sequence of approximation levels 1 ≥ ... ≥ T, 1. At iteration t = 1, For i = 1, ..., N (1) (1) Simulate θi ∼ π(θ) and x ∼ f (x|θi ) until (x, y) < 1 (1) Set ωi = 1/N (1) Take τ 2 as twice the empirical variance of the θi ’s 2. At iteration 2 ≤ t ≤ T , For i = 1, ..., N , repeat (t−1) (t−1) Pick θi from the θj ’s with probabilities ωj (t) 2 (t) generate θi |θi ∼ N (θi , σt ) and x ∼ f (x|θi ) until (x, y) < t (t) (t) N (t−1) −1 (t) (t−1) Set ωi ∝ π(θi )/ j=1 ωj ϕ σt θi − θj ) 2 (t) Take τt+1 as twice the weighted empirical variance of the θi ’s
  117. 117. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupSequential Monte Carlo SMC is a simulation technique to approximate a sequence of related probability distributions πn with π0 “easy” and πT as target. Iterated IS: particles moved from time n to time n via kernel Kn and use of a sequence of extended targets πn˜ n πn (z0:n ) = πn (zn ) ˜ Lj (zj+1 , zj ) j=0 where the Lj ’s are backward Markov kernels [check that πn (zn ) is a marginal] [Del Moral, Doucet & Jasra, Series B, 2006]
  118. 118. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupSequential Monte Carlo (2) Algorithm 4 SMC sampler (0) sample zi ∼ γ0 (x) (i = 1, . . . , N ) (0) (0) (0) compute weights wi = π0 (zi ))/γ0 (zi ) for t = 1 to N do if ESS(w(t−1) ) < NT then resample N particles z (t−1) and set weights to 1 end if (t−1) (t−1) generate zi ∼ Kt (zi , ·) and set weights to (t) (t) (t−1) (t) (t−1) πt (zi ))Lt−1 (zi ), zi )) wi = Wi−1 (t−1) (t−1) (t) πt−1 (zi ))Kt (zi ), zi )) end for
  119. 119. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABC-SMC [Del Moral, Doucet & Jasra, 2009] True derivation of an SMC-ABC algorithm Use of a kernel Kn associated with target π n and derivation of the backward kernel π n (z )Kn (z , z) Ln−1 (z, z ) = πn (z) Update of the weights M m=1 IA (xm ) in win ∝ wi(n−1) M n m=1 IA n−1 (xm i(n−1) ) when xm ∼ K(xi(n−1) , ·) in
  120. 120. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupABC-SMCM Modification: Makes M repeated simulations of the pseudo-data z given the parameter, rather than using a single [M = 1] simulation, leading to weight that is proportional to the number of accepted zi s M 1 ω(θ) = Iρ(η(y),η(zi ))< M i=1 [limit in M means exact simulation from (tempered) target]
  121. 121. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupProperties of ABC-SMC The ABC-SMC method properly uses a backward kernel L(z, z ) to simplify the importance weight and to remove the dependence on the unknown likelihood from this weight. Update of importance weights is reduced to the ratio of the proportions of surviving particles
  122. 122. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupProperties of ABC-SMC The ABC-SMC method properly uses a backward kernel L(z, z ) to simplify the importance weight and to remove the dependence on the unknown likelihood from this weight. Update of importance weights is reduced to the ratio of the proportions of surviving particles Adaptivity in ABC-SMC algorithm only found in on-line construction of the thresholds t , slowly enough to keep a large number of accepted transitions
  123. 123. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupSemi-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.
  124. 124. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupSummary statistics Optimality of the posterior expectations of the parameters of interest as summary statistics!
  125. 125. From MCMC to ABC Methods Approximate Bayesian computation Alphabet soupSummary statistics Optimality of the posterior expectations of the parameters of interest as summary statistics! Use of the standard quadratic loss function (θ − θ0 )T A(θ − θ0 ) .
  126. 126. From MCMC to ABC Methods Approximate Bayesian computation Calibration of ABCWhich summary? Fundamental difficulty of the choice of the summary statistic when there is no non-trivial sufficient statistics [except when done by the experimenters in the field]
  127. 127. From MCMC to ABC Methods Approximate Bayesian computation Calibration of ABCWhich summary? Fundamental difficulty of the choice of the summary statistic when there is no non-trivial sufficient statistics [except when done by the experimenters in the field] 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.
  128. 128. From MCMC to ABC Methods Approximate Bayesian computation Calibration of ABCWhich summary? Fundamental difficulty of the choice of the summary statistic when there is no non-trivial sufficient statistics [except when done by the experimenters in the field] 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. Does not taking into account the sequential nature of the tests Depends on parameterisation Order of inclusion matters.
  129. 129. From MCMC to ABC Methods ABC for model choiceABC for model choice Computational issues in Bayesian statistics The Metropolis-Hastings Algorithm The Gibbs Sampler Approximate Bayesian computation ABC for model choice Model choice Gibbs random fields Model choice via ABC Illustrations Generic ABC model choice
  130. 130. From MCMC to ABC Methods ABC for 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.
  131. 131. From MCMC to ABC Methods ABC for model choice Model choiceGeneric ABC for model choice Algorithm 5 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
  132. 132. From MCMC to ABC Methods ABC for 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 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?
  133. 133. From MCMC to ABC Methods ABC for 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 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? Extension to a weighted polychotomous logistic regression estimate of π(M = m|y), with non-parametric kernel weights [Cornuet et al., DIYABC, 2009]
  134. 134. From MCMC to ABC Methods ABC for 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 (!)
  135. 135. From MCMC to ABC Methods ABC for 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...
  136. 136. From MCMC to ABC Methods ABC for 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
  137. 137. From MCMC to ABC Methods ABC for 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
  138. 138. From MCMC to ABC Methods ABC for 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
  139. 139. From MCMC to ABC Methods ABC for 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 [Cucala, Marin, CPR & Titterington, 2009]
  140. 140. From MCMC to ABC Methods ABC for model choice Model choice via ABCBayesian Model Choice Comparing a model with potential S0 taking values in Rp0 versus a model with potential 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
  141. 141. From MCMC to ABC Methods ABC for model choice Model choice via ABCBayesian Model Choice Comparing a model with potential S0 taking values in Rp0 versus a model with potential 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 Use of Jeffreys’ scale to select most appropriate model
  142. 142. From MCMC to ABC Methods ABC for model choice Model choice via ABCNeighbourhood 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.
  143. 143. From MCMC to ABC Methods ABC for model choice Model choice via ABCModel index Formalisation via a model index M that appears as a new parameter with prior distribution π(M = m) and π(θ|M = m) = πm (θm )
  144. 144. From MCMC to ABC Methods ABC for model choice Model choice via ABCModel index Formalisation via a model index M that appears as a new parameter with prior distribution π(M = m) and π(θ|M = m) = πm (θm ) Computational target: P(M = m|x) ∝ fm (x|θm )πm (θm ) dθm π(M = m) , Θm
  145. 145. From MCMC to ABC Methods ABC for model choice Model choice via ABCSufficient statistics By definition, if S(x) sufficient statistic for the joint parameters (M, θ0 , . . . , θM −1 ), P(M = m|x) = P(M = m|S(x)) .
  146. 146. From MCMC to ABC Methods ABC for model choice Model choice via ABCSufficient statistics By definition, if S(x) sufficient statistic for the joint parameters (M, θ0 , . . . , θM −1 ), P(M = m|x) = P(M = m|S(x)) . For each model m, own sufficient statistic Sm (·) and S(·) = (S0 (·), . . . , SM −1 (·)) also sufficient.
  147. 147. From MCMC to ABC Methods ABC for model choice Model choice via ABCSufficient statistics By definition, if S(x) sufficient statistic for the joint parameters (M, θ0 , . . . , θM −1 ), P(M = m|x) = P(M = m|S(x)) . For each model m, own sufficient statistic Sm (·) and S(·) = (S0 (·), . . . , SM −1 (·)) also 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 [Specific to Gibbs random fields!]
  148. 148. From MCMC to ABC Methods ABC for model choice Model choice via ABCABC 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
  149. 149. From MCMC to ABC Methods ABC for model choice Model choice via ABCABC approximation to the Bayes factor Frequency ratio: ˆ P(M = m0 |x0 ) π(M = m1 ) BF m0 /m1 (x0 ) = × ˆ P(M = m1 |x0 ) π(M = m0 ) {mi∗ = m0 } π(M = m1 ) = × , {mi∗ = m1 } π(M = m0 )
  150. 150. From MCMC to ABC Methods ABC for model choice Model choice via ABCABC approximation to the Bayes factor Frequency ratio: ˆ P(M = m0 |x0 ) π(M = m1 ) BF m0 /m1 (x0 ) = × ˆ P(M = m1 |x0 ) π(M = m0 ) {mi∗ = m0 } π(M = m1 ) = × , {mi∗ = m1 } π(M = m0 ) replaced with 1 + {mi∗ = m0 } π(M = m1 ) BF m0 /m1 (x0 ) = × 1 + {mi∗ = m1 } π(M = m0 ) to avoid indeterminacy (also Bayes estimate).
  151. 151. From MCMC to ABC Methods ABC for model choice IllustrationsToy 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).
  152. 152. From MCMC to ABC Methods ABC for model choice IllustrationsToy example (2) (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.
  153. 153. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceBack to sufficiency 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 )
  154. 154. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceBack to sufficiency 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 [X, Cornuet, Marin, and Pillai, 2011]
  155. 155. From MCMC to ABC Methods ABC for 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
  156. 156. From MCMC to ABC Methods ABC for 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)
  157. 157. From MCMC to ABC Methods ABC for 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
  158. 158. From MCMC to ABC Methods ABC for 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)
  159. 159. From MCMC to ABC Methods ABC for 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 g1 (y) η = η = B (y) . g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12 [Didelot, Everitt, Johansen & Lawson, 2011]
  160. 160. From MCMC to ABC Methods ABC for 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 g1 (y) η = η = B (y) . g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12 [Didelot, Everitt, Johansen & Lawson, 2011] c No discrepancy only when cross-model sufficiency
  161. 161. From MCMC to ABC Methods ABC for model choice Generic ABC model choicePoisson/geometric example Sample x = (x1 , . . . , xn ) from either a Poisson P(λ) or from a geometric G(p) Then 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
  162. 162. From MCMC to ABC Methods ABC for 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.
  163. 163. From MCMC to ABC Methods ABC for 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]
  164. 164. From MCMC to ABC Methods ABC for 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
  165. 165. From MCMC to ABC Methods ABC for 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
  166. 166. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceMeaning of the ABC-Bayes factor In the Poisson/geometric case, if E[yi ] = θ0 > 0, η (θ0 + 1)2 −θ0 lim B12 (y) = e n→∞ θ0
  167. 167. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceMA(q) divergence 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.
  168. 168. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceMA(q) divergence 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.
  169. 169. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceA population genetics evaluation Population genetics example with 3 populations 2 scenari 15 individuals 5 loci single mutation parameter
  170. 170. From MCMC to ABC Methods ABC for 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
  171. 171. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceStability of importance sampling
  172. 172. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceComparison with ABC Use of 24 summary statistics and DIY-ABC logistic correction
  173. 173. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceComparison with ABC Use of 15 summary statistics and DIY-ABC logistic correction
  174. 174. From MCMC to ABC Methods ABC for model choice Generic ABC model choiceComparison with ABC Use of 24 summary statistics and DIY-ABC logistic correction
  175. 175. From MCMC to ABC Methods ABC for 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]
  176. 176. From MCMC to ABC Methods ABC for 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 et al., 2009, 2011]

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