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# MCMC and likelihood-free methods

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### Transcript of "MCMC and likelihood-free methods"

1. 1. MCMC and Likelihood-free Methods MCMC and Likelihood-free Methods Christian P. Robert Universit´e Paris-Dauphine & CREST http://www.ceremade.dauphine.fr/~xian November 2, 2010
2. 2. MCMC and Likelihood-free Methods Outline Computational issues in Bayesian statistics The Metropolis-Hastings Algorithm The Gibbs Sampler Population Monte Carlo Approximate Bayesian computation ABC for model choice
3. 3. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Motivation and leading example Computational issues in Bayesian statistics The Metropolis-Hastings Algorithm The Gibbs Sampler Population Monte Carlo Approximate Bayesian computation ABC for model choice
4. 4. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Latent structures make life harder! Even simple models may lead to computational complications, as in latent variable models f(x|θ) = f (x, x |θ) dx
5. 5. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Latent structures make life harder! Even simple models may lead to computational complications, as in latent variable models f(x|θ) = f (x, x |θ) dx If (x, x ) observed, ﬁne!
6. 6. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Latent structures make life harder! Even simple models may lead to computational complications, as in latent variable models f(x|θ) = f (x, x |θ) dx If (x, x ) observed, ﬁne! If only x observed, trouble!
7. 7. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Example (Mixture models) Models of mixtures of distributions: X ∼ fj with probability pj, for j = 1, 2, . . . , k, with overall density X ∼ p1f1(x) + · · · + pkfk(x) .
8. 8. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Example (Mixture models) Models of mixtures of distributions: X ∼ fj with probability pj, for j = 1, 2, . . . , k, with overall density X ∼ p1f1(x) + · · · + pkfk(x) . For a sample of independent random variables (X1, · · · , Xn), sample density n i=1 {p1f1(xi) + · · · + pkfk(xi)} .
9. 9. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Example (Mixture models) Models of mixtures of distributions: X ∼ fj with probability pj, for j = 1, 2, . . . , k, with overall density X ∼ p1f1(x) + · · · + pkfk(x) . For a sample of independent random variables (X1, · · · , Xn), sample density n i=1 {p1f1(xi) + · · · + pkfk(xi)} . Expanding this product of sums into a sum of products involves kn elementary terms: too prohibitive to compute in large samples.
10. 10. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Simple mixture (1) −1 0 1 2 3 −10123 µ1 µ2 Case of the 0.3N (µ1, 1) + 0.7N (µ2, 1) likelihood
11. 11. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Simple mixture (2) For the mixture of two normal distributions, 0.3N(µ1, 1) + 0.7N(µ2, 1) , likelihood proportional to n i=1 [0.3ϕ (xi − µ1) + 0.7 ϕ (xi − µ2)] containing 2n terms.
12. 12. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Complex maximisation Standard maximization techniques often fail to ﬁnd the global maximum because of multimodality or undesirable behavior (usually at the frontier of the domain) of the likelihood function. Example In the special case f(x|µ, σ) = (1 − ) exp{(−1/2)x2 } + σ exp{(−1/2σ2 )(x − µ)2 } with > 0 known,
13. 13. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Complex maximisation Standard maximization techniques often fail to ﬁnd the global maximum because of multimodality or undesirable behavior (usually at the frontier of the domain) of the likelihood function. Example In the special case f(x|µ, σ) = (1 − ) exp{(−1/2)x2 } + σ exp{(−1/2σ2 )(x − µ)2 } with > 0 known, whatever n, the likelihood is unbounded: lim σ→0 L(x1, . . . , xn|µ = x1, σ) = ∞
14. 14. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Unbounded likelihood −2 0 2 4 6 1234 µ n= 3 −2 0 2 4 6 1234 µ σ n= 6 −2 0 2 4 6 1234 µ n= 12 −2 0 2 4 6 1234 µ σ n= 24 −2 0 2 4 6 1234 µ n= 48 −2 0 2 4 6 1234 µ σ n= 96 Case of the 0.3N (0, 1) + 0.7N (µ, σ) likelihood
15. 15. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Example (Mixture once again) press for MA Observations from x1, . . . , xn ∼ f(x|θ) = pϕ(x; µ1, σ1) + (1 − p)ϕ(x; µ2, σ2)
16. 16. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Example (Mixture once again) press for MA Observations from x1, . . . , xn ∼ f(x|θ) = pϕ(x; µ1, σ1) + (1 − p)ϕ(x; µ2, σ2) Prior µi|σi ∼ N (ξi, σ2 i /ni), σ2 i ∼ I G (νi/2, s2 i /2), p ∼ Be(α, β)
17. 17. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Example (Mixture once again) press for MA Observations from x1, . . . , xn ∼ f(x|θ) = pϕ(x; µ1, σ1) + (1 − p)ϕ(x; µ2, σ2) Prior µi|σi ∼ N (ξi, σ2 i /ni), σ2 i ∼ I G (νi/2, s2 i /2), p ∼ Be(α, β) Posterior π(θ|x1, . . . , xn) ∝ n j=1 {pϕ(xj; µ1, σ1) + (1 − p)ϕ(xj; µ2, σ2)} π(θ) = n =0 (kt) ω(kt)π(θ|(kt)) n
18. 18. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Example (Mixture once again (cont’d)) For a given permutation (kt), conditional posterior distribution π(θ|(kt)) = N ξ1(kt), σ2 1 n1 + × I G ((ν1 + )/2, s1(kt)/2) ×N ξ2(kt), σ2 2 n2 + n − × I G ((ν2 + n − )/2, s2(kt)/2) ×Be(α + , β + n − )
19. 19. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Example (Mixture once again (cont’d)) where ¯x1(kt) = 1 t=1 xkt , ˆs1(kt) = t=1(xkt − ¯x1(kt))2, ¯x2(kt) = 1 n− n t= +1 xkt , ˆs2(kt) = n t= +1(xkt − ¯x2(kt))2 and ξ1(kt) = n1ξ1 + ¯x1(kt) n1 + , ξ2(kt) = n2ξ2 + (n − )¯x2(kt) n2 + n − , s1(kt) = s2 1 + ˆs2 1(kt) + n1 n1 + (ξ1 − ¯x1(kt))2 , s2(kt) = s2 2 + ˆs2 2(kt) + n2(n − ) n2 + n − (ξ2 − ¯x2(kt))2 , posterior updates of the hyperparameters
20. 20. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics Latent variables Example (Mixture once again) Bayes estimator of θ: δπ (x1, . . . , xn) = n =0 (kt) ω(kt)Eπ [θ|x, (kt)] Too costly: 2n terms
21. 21. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The AR(p) model AR(p) model Auto-regressive representation of a time series, xt|xt−1, . . . ∼ N µ + p i=1 i(xt−i − µ), σ2
22. 22. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The AR(p) model AR(p) model Auto-regressive representation of a time series, xt|xt−1, . . . ∼ N µ + p i=1 i(xt−i − µ), σ2 Generalisation of AR(1) Among the most commonly used models in dynamic settings More challenging than the static models (stationarity constraints) Diﬀerent models depending on the processing of the starting value x0
23. 23. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The AR(p) model Unknown stationarity constraints Practical diﬃculty: for complex models, stationarity constraints get quite involved to the point of being unknown in some cases Example (AR(1)) Case of linear Markovian dependence on the last value xt = µ + (xt−1 − µ) + t , t i.i.d. ∼ N (0, σ2 ) If | | < 1, (xt)t∈Z can be written as xt = µ + ∞ j=0 j t−j and this is a stationary representation.
24. 24. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The AR(p) model Stationary but... If | | > 1, alternative stationary representation xt = µ − ∞ j=1 −j t+j .
25. 25. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The AR(p) model Stationary but... If | | > 1, alternative stationary representation xt = µ − ∞ j=1 −j t+j . This stationary solution is criticized as artiﬁcial because xt is correlated with future white noises ( t)s>t, unlike the case when | | < 1. Non-causal representation...
26. 26. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The AR(p) model Stationarity+causality Stationarity constraints in the prior as a restriction on the values of θ. Theorem AR(p) model second-order stationary and causal iﬀ the roots of the polynomial P(x) = 1 − p i=1 ixi are all outside the unit circle
27. 27. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The AR(p) model Stationarity constraints Under stationarity constraints, complex parameter space: each value of needs to be checked for roots of corresponding polynomial with modulus less than 1
28. 28. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The AR(p) model Stationarity constraints Under stationarity constraints, complex parameter space: each value of needs to be checked for roots of corresponding polynomial with modulus less than 1 E.g., for an AR(2) process with autoregressive polynomial P(u) = 1 − 1u − 2u2, constraint is 1 + 2 < 1, 1 − 2 < 1 and | 2| < 1 q −2 −1 0 1 2 −1.0−0.50.00.51.0 θ1 θ2
29. 29. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model The MA(q) model Alternative type of time series xt = µ + t − q j=1 ϑj t−j , t ∼ N (0, σ2 ) Stationary but, for identiﬁability considerations, the polynomial Q(x) = 1 − q j=1 ϑjxj must have all its roots outside the unit circle.
30. 30. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model Identiﬁability Example For the MA(1) model, xt = µ + t − ϑ1 t−1, var(xt) = (1 + ϑ2 1)σ2 can also be written xt = µ + ˜t−1 − 1 ϑ1 ˜t, ˜ ∼ N (0, ϑ2 1σ2 ) , Both pairs (ϑ1, σ) & (1/ϑ1, ϑ1σ) lead to alternative representations of the same model.
31. 31. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model Properties of MA models Non-Markovian model (but special case of hidden Markov) Autocovariance γx(s) is null for |s| > q
32. 32. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model Representations x1:T is a normal random variable with constant mean µ and covariance matrix Σ =      σ2 γ1 γ2 . . . γq 0 . . . 0 0 γ1 σ2 γ1 . . . γq−1 γq . . . 0 0 ... 0 0 0 . . . 0 0 . . . γ1 σ2      , with (|s| ≤ q) γs = σ2 q−|s| i=0 ϑiϑi+|s| Not manageable in practice [large T’s]
33. 33. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model Representations (contd.) Conditional on past ( 0, . . . , −q+1), L(µ, ϑ1, . . . , ϑq, σ|x1:T , 0, . . . , −q+1) ∝ σ−T T t=1 exp    −  xt − µ + q j=1 ϑjˆt−j   2 2σ2    , where (t > 0) ˆt = xt − µ + q j=1 ϑjˆt−j, ˆ0 = 0, . . . , ˆ1−q = 1−q Recursive deﬁnition of the likelihood, still costly O(T × q)
34. 34. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model Representations (contd.) Encompassing approach for general time series models State-space representation xt = Gyt + εt , (1) yt+1 = Fyt + ξt , (2) (1) is the observation equation and (2) is the state equation
35. 35. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model Representations (contd.) Encompassing approach for general time series models State-space representation xt = Gyt + εt , (1) yt+1 = Fyt + ξt , (2) (1) is the observation equation and (2) is the state equation Note This is a special case of hidden Markov model
36. 36. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model MA(q) state-space representation For the MA(q) model, take yt = ( t−q, . . . , t−1, t) and then yt+1 =       0 1 0 . . . 0 0 0 1 . . . 0 . . . 0 0 0 . . . 1 0 0 0 . . . 0       yt + t+1        0 0 ... 0 1        xt = µ − ϑq ϑq−1 . . . ϑ1 −1 yt .
37. 37. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model MA(q) state-space representation (cont’d) Example For the MA(1) model, observation equation xt = (1 0)yt with yt = (y1t y2t) directed by the state equation yt+1 = 0 1 0 0 yt + t+1 1 ϑ1 .
38. 38. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model c A typology of Bayes computational problems (i). latent variable models in general
39. 39. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model c A typology of Bayes computational problems (i). latent variable models in general (ii). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models;
40. 40. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model c A typology of Bayes computational problems (i). latent variable models in general (ii). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models; (iii). use of a complex sampling model with an intractable likelihood, as for instance in some graphical models;
41. 41. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model c A typology of Bayes computational problems (i). latent variable models in general (ii). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models; (iii). use of a complex sampling model with an intractable likelihood, as for instance in some graphical models; (iv). use of a huge dataset;
42. 42. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model c A typology of Bayes computational problems (i). latent variable models in general (ii). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models; (iii). use of a complex sampling model with an intractable likelihood, as for instance in some graphical models; (iv). use of a huge dataset; (v). use of a complex prior distribution (which may be the posterior distribution associated with an earlier sample);
43. 43. MCMC and Likelihood-free Methods Computational issues in Bayesian statistics The MA(q) model c A typology of Bayes computational problems (i). latent variable models in general (ii). use of a complex parameter space, as for instance in constrained parameter sets like those resulting from imposing stationarity constraints in dynamic models; (iii). use of a complex sampling model with an intractable likelihood, as for instance in some graphical models; (iv). use of a huge dataset; (v). use of a complex prior distribution (which may be the posterior distribution associated with an earlier sample); (vi). use of a particular inferential procedure as for instance, Bayes factors Bπ 01(x) = P(θ ∈ Θ0 | x) P(θ ∈ Θ1 | x) π(θ ∈ Θ0) π(θ ∈ Θ1) .
44. 44. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis-Hastings Algorithm Computational issues in Bayesian statistics The Metropolis-Hastings Algorithm Monte Carlo basics Importance Sampling Monte Carlo Methods based on Markov Chains The Metropolis–Hastings algorithm Random-walk Metropolis-Hastings algorithms Extensions The Gibbs Sampler Population Monte Carlo
45. 45. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo basics General purpose Given a density π known up to a normalizing constant, and an integrable function h, compute Π(h) = h(x)π(x)µ(dx) = h(x)˜π(x)µ(dx) ˜π(x)µ(dx) when h(x)˜π(x)µ(dx) is intractable.
46. 46. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo basics Monte Carlo 101 Generate an iid sample x1, . . . , xN from π and estimate Π(h) by ˆΠMC N (h) = N−1 N i=1 h(xi). LLN: ˆΠMC N (h) as −→ Π(h) If Π(h2) = h2(x)π(x)µ(dx) < ∞, CLT: √ N ˆΠMC N (h) − Π(h) L N 0, Π [h − Π(h)]2 .
47. 47. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo basics Monte Carlo 101 Generate an iid sample x1, . . . , xN from π and estimate Π(h) by ˆΠMC N (h) = N−1 N i=1 h(xi). LLN: ˆΠMC N (h) as −→ Π(h) If Π(h2) = h2(x)π(x)µ(dx) < ∞, CLT: √ N ˆΠMC N (h) − Π(h) L N 0, Π [h − Π(h)]2 . Caveat announcing MCMC Often impossible or ineﬃcient to simulate directly from Π
48. 48. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Importance Sampling Importance 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).
49. 49. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Importance Sampling Importance 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 ˆΠIS Q,N (h) = N−1 N i=1 h(xi){π/q}(xi). return to pMC
50. 50. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Importance Sampling Properties of importance Then LLN: ˆΠIS Q,N (h) as −→ Π(h) and if Q((hπ/q)2) < ∞, CLT: √ N(ˆΠIS Q,N (h) − Π(h)) L N 0, Q{(hπ/q − Π(h))2 } .
51. 51. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Importance Sampling Properties of importance Then LLN: ˆΠIS Q,N (h) as −→ Π(h) and if Q((hπ/q)2) < ∞, CLT: √ N(ˆΠIS Q,N (h) − Π(h)) L N 0, Q{(hπ/q − Π(h))2 } . Caveat If normalizing constant of π unknown, impossible to use ˆΠIS Q,N Generic problem in Bayesian Statistics: π(θ|x) ∝ f(x|θ)π(θ).
52. 52. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Importance Sampling Self-Normalised Importance Sampling Self normalized version ˆΠSNIS Q,N (h) = N i=1 {π/q}(xi) −1 N i=1 h(xi){π/q}(xi).
53. 53. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Importance Sampling Self-Normalised Importance Sampling Self normalized version ˆΠSNIS Q,N (h) = N i=1 {π/q}(xi) −1 N i=1 h(xi){π/q}(xi). LLN : ˆΠSNIS Q,N (h) as −→ Π(h) and if Π((1 + h2)(π/q)) < ∞, CLT : √ N(ˆΠSNIS Q,N (h) − Π(h)) L N 0, π {(π/q)(h − Π(h)}2 ) .
54. 54. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Importance Sampling Self-Normalised Importance Sampling Self normalized version ˆΠSNIS Q,N (h) = N i=1 {π/q}(xi) −1 N i=1 h(xi){π/q}(xi). LLN : ˆΠSNIS Q,N (h) as −→ Π(h) and if Π((1 + h2)(π/q)) < ∞, CLT : √ N(ˆΠSNIS Q,N (h) − Π(h)) L N 0, π {(π/q)(h − Π(h)}2 ) . c The quality of the SNIS approximation depends on the choice of Q
55. 55. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov Chains Running 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 ,
56. 56. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov Chains Running 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
57. 57. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov Chains Running 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
58. 58. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov Chains Running 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.
59. 59. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov Chains Running 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. Problem: How can one build a Markov chain with a given stationary distribution?
60. 60. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithm The Metropolis–Hastings algorithm Basics The algorithm uses the objective (target) density f and a conditional density q(y|x) called the instrumental (or proposal) distribution
61. 61. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithm The MH algorithm Algorithm (Metropolis–Hastings) Given x(t), 1. Generate Yt ∼ q(y|x(t)). 2. Take X(t+1) = Yt with prob. ρ(x(t), Yt), x(t) with prob. 1 − ρ(x(t), Yt), where ρ(x, y) = min f(y) f(x) q(x|y) q(y|x) , 1 .
62. 62. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithm Features 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
63. 63. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithm Convergence properties 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)
64. 64. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithm Convergence properties 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
65. 65. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithm Convergence properties 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 3. If Pr f(Yt) q(X(t)|Yt) f(X(t)) q(Yt|X(t)) ≥ 1 < 1. (1) that is, the event {X(t+1) = X(t)} is possible, then the chain is aperiodic
66. 66. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithm Convergence properties (2) 4. If q(y|x) > 0 for every (x, y), (2) the chain is irreducible
67. 67. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithm Convergence 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
68. 68. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm The Metropolis–Hastings algorithm Convergence 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)| < ∞, lim T →∞ 1 T T t=1 h(X(t) ) = h(x)df(x) a.e. f. (ii) and lim n→∞ Kn (x, ·)µ(dx) − f T V = 0 for every initial distribution µ, where Kn (x, ·) denotes the kernel for n transitions.
69. 69. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Random 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)
70. 70. MCMC and Likelihood-free 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 X(t+1) =    Yt with prob. min 1, f(Yt) f(x(t)) , x(t) otherwise.
71. 71. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Example (Random walk and normal target) forget History! Generate N(0, 1) based on the uniform proposal [−δ, δ] [Hastings (1970)] The probability of acceptance is then ρ(x(t) , yt) = exp{(x(t)2 − y2 t )/2} ∧ 1.
72. 72. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Example (Random walk & normal (2)) Sample statistics δ 0.1 0.5 1.0 mean 0.399 -0.111 0.10 variance 0.698 1.11 1.06 c As δ ↑, we get better histograms and a faster exploration of the support of f.
73. 73. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms -1 0 1 2 050100150200250 (a) -1.5-1.0-0.50.00.5 -2 0 2 0100200300400 (b) -1.5-1.0-0.50.00.5 -3 -2 -1 0 1 2 3 0100200300400 (c) -1.5-1.0-0.50.00.5 Three samples based on U[−δ, δ] with (a) δ = 0.1, (b) δ = 0.5 and (c) δ = 1.0, superimposed with the convergence of the means (15, 000 simulations).
74. 74. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Example (Mixture models) π(θ|x) ∝ n j=1 k =1 p f(xj|µ , σ ) π(θ)
75. 75. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Example (Mixture models) π(θ|x) ∝ n j=1 k =1 p f(xj|µ , σ ) π(θ) Metropolis-Hastings proposal: θ(t+1) = θ(t) + ωε(t) if u(t) < ρ(t) θ(t) otherwise where ρ(t) = π(θ(t) + ωε(t)|x) π(θ(t)|x) ∧ 1 and ω scaled for good acceptance rate
76. 76. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms p theta 0.0 0.2 0.4 0.6 0.8 1.0 -1012 tau theta 0.2 0.4 0.6 0.8 1.0 1.2 -1012 p tau 0.0 0.2 0.4 0.6 0.8 1.0 0.20.40.60.81.01.2 -1 0 1 2 0.01.02.0 theta 0.2 0.4 0.6 0.8 024 tau 0.0 0.2 0.4 0.6 0.8 1.0 0123456 p Random walk sampling (50000 iterations) General case of a 3 component normal mixture [Celeux & al., 2000]
77. 77. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms −1 0 1 2 3 −10123 µ1 µ2 X Random walk MCMC output for .7N(µ1, 1) + .3N(µ2, 1)
78. 78. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Convergence properties Uniform ergodicity prohibited by random walk structure
79. 79. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Convergence properties Uniform ergodicity prohibited by random walk structure At best, geometric ergodicity: Theorem (Suﬃcient ergodicity) For a symmetric density f, log-concave in the tails, and a positive and symmetric density g, the chain (X(t)) is geometrically ergodic. [Mengersen & Tweedie, 1996] no tail eﬀect
80. 80. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Example (Comparison of tail eﬀects) Random-walk Metropolis–Hastings algorithms based on a N (0, 1) instrumental for the generation of (a) a N(0, 1) distribution and (b) a distribution with density ψ(x) ∝ (1 + |x|)−3 (a) 0 50 100 150 200 -1.5-1.0-0.50.00.51.01.5 (a) 0 50 100 150 200 -1.5-1.0-0.50.00.51.01.5 0 50 100 150 200 -1.5-1.0-0.50.00.51.01.5 0 50 100 150 200 -1.5-1.0-0.50.00.51.01.5 (b) 0 50 100 150 200 -1.5-1.0-0.50.00.51.01.5 0 50 100 150 200 -1.5-1.0-0.50.00.51.01.5 0 50 100 150 200 -1.5-1.0-0.50.00.51.01.5 90% conﬁdence envelopes of the means, derived from 500 parallel independent chains 1 + ξ2 1 + (ξ )2 ∧ 1 ,
81. 81. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Further convergence properties Under assumptions skip detailed convergence (A1) f is super-exponential, i.e. it is positive with positive continuous ﬁrst derivative such that lim|x|→∞ n(x) log f(x) = −∞ where n(x) := x/|x|. In words : exponential decay of f in every direction with rate tending to ∞ (A2) lim sup|x|→∞ n(x) m(x) < 0, where m(x) = f(x)/| f(x)|. In words: non degeneracy of the countour manifold Cf(y) = {y : f(y) = f(x)} Q is geometrically ergodic, and V (x) ∝ f(x)−1/2 veriﬁes the drift condition [Jarner & Hansen, 2000]
82. 82. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Further [further] convergence properties skip hyperdetailed convergence If P ψ-irreducible and aperiodic, for r = (r(n))n∈N real-valued non decreasing sequence, such that, for all n, m ∈ N, r(n + m) ≤ r(n)r(m), and r(0) = 1, for C a small set, τC = inf{n ≥ 1, Xn ∈ C}, and h ≥ 1, assume sup x∈C Ex τC −1 k=0 r(k)h(Xk) < ∞,
83. 83. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms then, S(f, C, r) := x ∈ X, Ex τC −1 k=0 r(k)h(Xk) < ∞ is full and absorbing and for x ∈ S(f, C, r), lim n→∞ r(n) Pn (x, .) − f h = 0. [Tuominen & Tweedie, 1994]
84. 84. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Comments [CLT, Rosenthal’s inequality...] h-ergodicity implies CLT for additive (possibly unbounded) functionals of the chain, Rosenthal’s inequality and so on... [Control of the moments of the return-time] The condition implies (because h ≥ 1) that sup x∈C Ex[r0(τC)] ≤ sup x∈C Ex τC −1 k=0 r(k)h(Xk) < ∞, where r0(n) = n l=0 r(l) Can be used to derive bounds for the coupling time, an essential step to determine computable bounds, using coupling inequalities [Roberts & Tweedie, 1998; Fort & Moulines, 2000]
85. 85. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms Alternative conditions The condition is not really easy to work with... [Possible alternative conditions] (a) [Tuominen, Tweedie, 1994] There exists a sequence (Vn)n∈N, Vn ≥ r(n)h, such that (i) supC V0 < ∞, (ii) {V0 = ∞} ⊂ {V1 = ∞} and (iii) PVn+1 ≤ Vn − r(n)h + br(n)IC.
86. 86. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Random-walk Metropolis-Hastings algorithms (b) [Fort 2000] ∃V ≥ f ≥ 1 and b < ∞, such that supC V < ∞ and PV (x) + Ex σC k=0 ∆r(k)f(Xk) ≤ V (x) + bIC(x) where σC is the hitting time on C and ∆r(k) = r(k) − r(k − 1), k ≥ 1 and ∆r(0) = r(0). Result (a) ⇔ (b) ⇔ supx∈C Ex τC −1 k=0 r(k)f(Xk) < ∞.
87. 87. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Langevin Algorithms Proposal based on the Langevin diﬀusion Lt is deﬁned by the stochastic diﬀerential equation dLt = dBt + 1 2 log f(Lt)dt, where Bt is the standard Brownian motion Theorem The Langevin diﬀusion is the only non-explosive diﬀusion which is reversible with respect to f.
88. 88. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Discretization Instead, consider the sequence x(t+1) = x(t) + σ2 2 log f(x(t) ) + σεt, εt ∼ Np(0, Ip) where σ2 corresponds to the discretization step
89. 89. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Discretization Instead, consider the sequence x(t+1) = x(t) + σ2 2 log f(x(t) ) + σεt, εt ∼ Np(0, Ip) where σ2 corresponds to the discretization step Unfortunately, the discretized chain may be be transient, for instance when lim x→±∞ σ2 log f(x)|x|−1 > 1
90. 90. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions MH correction Accept the new value Yt with probability f(Yt) f(x(t)) · exp − Yt − x(t) − σ2 2 log f(x(t)) 2 2σ2 exp − x(t) − Yt − σ2 2 log f(Yt) 2 2σ2 ∧ 1 . Choice of the scaling factor σ Should lead to an acceptance rate of 0.574 to achieve optimal convergence rates (when the components of x are uncorrelated) [Roberts & Rosenthal, 1998; Girolami & Calderhead, 2011]
91. 91. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Optimizing the Acceptance Rate Problem of choice of the transition kernel from a practical point of view Most common alternatives: (a) a fully automated algorithm like ARMS; (b) an instrumental density g which approximates f, such that f/g is bounded for uniform ergodicity to apply; (c) a random walk In both cases (b) and (c), the choice of g is critical,
92. 92. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Case 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.
93. 93. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Case 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 min f(yt) f(x(t)) , 1 1 . For multimodal densities with well separated modes, the negative eﬀect of limited moves on the surface of f clearly shows.
94. 94. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Case 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
95. 95. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Rule of thumb 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]
96. 96. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Rule of thumb 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] This rule is to be taken with a pinch of salt!
97. 97. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Example (Noisy AR(1)) Hidden Markov chain from a regular AR(1) model, xt+1 = ϕxt + t+1 t ∼ N (0, τ2 ) and observables yt|xt ∼ N (x2 t , σ2 )
98. 98. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Example (Noisy AR(1)) Hidden Markov chain from a regular AR(1) model, xt+1 = ϕxt + t+1 t ∼ N (0, τ2 ) and observables yt|xt ∼ N (x2 t , σ2 ) The distribution of xt given xt−1, xt+1 and yt is exp −1 2τ2 (xt − ϕxt−1)2 + (xt+1 − ϕxt)2 + τ2 σ2 (yt − x2 t )2 .
99. 99. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions 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.
100. 100. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Markov chain based on a random walk with scale ω = .1.
101. 101. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Markov chain based on a random walk with scale ω = .5.
102. 102. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions MA(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) }
103. 103. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Basic RWHM for MA(2) Algorithm 1 RW-HM-MA(2) sampler set ω and ϑ(1) for i = 2 to T do generate ˜ϑj ∼ U(ϑ (i−1) j − ω, ϑ (i−1) j + 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
104. 104. MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Outcome Result with a simulated sample of 100 points and ϑ1 = 0.6, ϑ2 = 0.2 and scale ω = 0.2 q q −2 −1 0 1 2 −1.0−0.50.00.51.0 θ1 θ2 q qqq q qqq qqqqq q qqq qq qqqq qq qqqqqqqqqq q qqqqqqqq qqq q qqqqqqqq q qq qqqqq qq qqq qqqqqqqqqqq q qqqqq q qq qqqq qqqqqqq qqqqq qqq qqqqqqqqq q q qqqqqqqqqqqq qq qq q qqq qqqqqqqq qqqqqqq qqqqq qqqq q qqqqqq qqqqqqq qqqqq qqq qq q qqq q q qq qqq qqq qq qq qqqq qq q qqq qqq q q qq qqq qqq qq qq qqqqq qqq qqqqq q qqq qqqqq qq qqqq q qqq q q qq qqqqqq qqqqqqqqq qq q qq q q qqqqqqqq qqqqqqqq qqq qqqqqqq qqqqq q qqq qq qqqqqqqqqqqqq qqqq qqq q qq q qq qqqq q qq qq q qqqq q qq q q qqqqq q qqq q qq qqqqqqq q qq q q q q qqq qqqq q qq q q q q qq qq qqqqqqq qqq q qqqq q qqqqq qq qqq qqq q q q qq qq qqqqqqqq qqqqq qqqqqq qq qq qqqqqqq q qqq q qqqqqq qqq qq qq qqq qqq qq qqqqqqq q qqq qqqq q qqq qq q qqqqqqqqqqq q qq qq q qqqq q qqq qqqqq q qqqqqq qq qqqq qq qqq qqq qq qq q qqqq qq q qqq qq qq qqqqqqq q qqqq qqq q q qqqqqqqqqqq q q qqq qqqq q qqqq qq qq qqqqq q qqqqq qqqqqq qq q q q q qqq q qqqq qq qq qqqqqqqqqqq qqqq q qq qqqq qqqqqqqq qq qqq qqq q q qq qq qq qqq qqqqqq q q q q qqqq qq qq qqqq qqq qq q qqqqq qqqqq qqq qq qqq q q qq qqqq qqqq qq qq qqqqqqqqqqq q qqq q q qqq q qq qq qq q qqq q q qqqqqqqqqqqq qqq qqqqq q q qq q qqqqq q q qqqqq q q q qqq qqqqqqqqqqqqqqqqqqq qq qq q q q qq qq q qq qq q q q qqqqq q q qqqqq q qq q qqq q qqq q q q q q qq qqqq q qq q q q qqqqqqqqqq q qqqqqqqqq qqqqqqqqqqqqqq q qq qqqqqqqqqqqqqqqqqqqqqqqqq qq q qq qqqqqqqqq qqqqqqqqqqqqqq qqq qq q q q q qqqqqqqqq q qq q qqqq q q qqq qqqqq q q q q qqqq qq qq qqq q qq qqqq q qqqqqqq qq q qq qqqqqqqqqq q qqqqq qqqqq qqqq qqqq q qqqqq qq qqqqqq q q qq q qq qq qqq qq qqqq qq q qq qq q q q qq qq qqq q qq q q qq q qqqq qqqqqqqqqqqqqqqqq qqq q qq qqq qqq qqqq qqqq qqq qqqqqq qq qqqq qqqq q qqq q qq qqqqqqq q qq qq q qqqq qq qq q qq qqqq qq qq qq qq qq qqqqqqq qqqqqqqqqq qq q qq qqqqq qq qqq qq qqq q qq qq qq qqq qq q q q qqqq qqqqqqqq qqq qq q q qqqqq qq qqqq q qqqqqqqq qqq q q qqqq qq qq q qqqqqqq qqq qq q q qq q qq q qqq q q q qq qq qqqqqq qqqq q qq qqqq qqqq qqqq q q qqq q q qqqqq q qqq qq q qq q q qqq q qqq qqq q qqqqqq q q qq q q qqq q qqqq qq qqq q qq qq q qqqqq q q qqq qqqq qqq qqq qqqqqqqqqq qqqqq q q q qqqq qqqqqqqqqqqqqqqqqq qqq qq qq q qqqqq qqq qqq qqq qqqqq q qqqqqqqqqqqq q qq qqqqqqqqq qqqqq qqq qqqq qq qqqq q qqqq q qq q qq q q q qqq q q qq q qqqqq q q q q qq qqqqq qqq q q qqqqq qq qqqqq q qq qqqqqqqqq qqqqqqqq qq qqqqq qq qqqqqq qqqqqqqqqqqqqq qqq qqqqqqqq qq qqq qqqqqqqqq q q q q q qqq qq qq qqq q q qq q qq q qqqqq qqqq qqqq q q q q qq q qqq q qqqqqq q q q qqqqqqqqq qqq q qqqq qqqqqqqqqq q q q q qqqq q qqqq qq qq qqqqq qq q q q q q qqq qq qqqq qq q q q qqqq qqq q qqqq qq q qq qq qqqqq q qq q q q qqq qqqqqq q q qqqqqqqqqqqqqq qq qq qq qqqq qqqqq qqq qqq q q qq qqqqqqq q qqq q qq q qq q qq qqqqqqqqqqqqq qq q q qqqqqqqqqq qqq qqq qqqqqq qq qq qqqqqqq qqqqqqqqqqqqq q qqqq qq qq q qq q q qqqq qq q q qqqq q qqqq qq qqqq qq qq q q qqqqq q qq qqqqqqqq q qqqqqqqq q qq qqqqqqq qqqqq qqqq qqqqq qq q q qqqqq qqqqqqq qqqqq q qq qq qq qqqqqqq q qqqqqqq qqqq qqq q q qqqq qq q q qq qqq q qqq qq qqqqqqqqqqqqqqqqqq q qqq qqqqqq qqqq qqqqqq qqq q q qqq q q q qqq q q qqqqqqqqq qqqq qq qqq qq qq qqq qq q qq qqqqqqqqq q qq q qqqqqq q qqqqqq qqqqqqqqqqqqqq qqqq q qqqq qqq qqqqqq qqq q qq q q qq q q qq qqqq q q qqq qqq q q qqqqqqq q q qqqqqqqqq q qq qq qqqq qq qqqq qqqqq qq qqq q qqq q qq qqq q qqq q qq q qqq qqq qqq q qqq q qq qq qq q q qqq q qqqqq qq qq qq q q qq qqq qqqqqq qq qqq q qq q q qqqqq q qqqqqqqqqqqqqq qqqqq qqq q qqq qq qqqqq q qq q qq qqq qqq q q qq qq qq qq qq q qqqqqqq qqqq q qqqq qq qqqq q q qq qqq qqq qqq qqqqq qq qq q qq q qqq qq qqq qqq qq q q qq qq q q q qq qqqq q qq q qq q qqqqq q qqqqqqqqq qq q qqq qq qq qqq qqqqqq q qqq qqqq q qqqq qqqqqqqqq q q qq q q qqqq qqqqqq qq qq q qq qqqqqqqqqqq qq qq q q qqqq qq q qqq q q qq qq qq qqq qq qqqq qq q qq q qqqqq qqqqq q q qqq qqq qqq qq qqqqqqqqqqqqqqqqqqqq q q q q qq q qqqqqq qq q q q qqq qq q qq qq q qq qqqq qq qq qqqqqqq qq q q qqqq q qqq qq q qqqq q qqqqqq qqqqqq qqqqq qqqqqqq q qqqq qqqqq qq qq q q qqqqqqq qqqqq q qqq q q qqqqqq qqq qqq q qq qq qq qqqq q qqqqqqqqqq qq q qqq qqq qqq q q qq qqq qq qqqqqqq q qq qqqqqq q qqqqq qq qq q qqqq qqqq q qqqqq q q q qqqq q qq q qqqqq q qqq q qqq qqqqqqq qq qq q q q qqq q q qq q qq qqq q qqqq qqqqqq qqq qqqq qqqqqqqqq qqqq q qqqq qqqq q qqq qqqqq qqqq q q qq qqqqqqq q q qqq q q qqq qqqqqqqqq q qqq qq qqqq qq qqq qqq qqqq qq qqqq q q qqq q qqqqqq qqqqqq qqq qq qqqqqqqqqqq qq q qq qqqqqqqqqqqqq qqqq q q qq qqqq qqq qqqq qqq q qqqq qq qq q qq qq qqq qqqqq q qqqqqqqqqqqqqqq qq q qq qqq q qq qqqqqqqq q qqqqqq qq qq qq q qqqqqqqqqqq q qqqqqqq qq qqqqq q qqqq qq q q qqqqqq qq qqqqq q qqq qq qqqqq qqqqqqq qq qqq q q q qq qq qq qqqqq qqqq q q qqqqq q qqq q q qqq qq qqqq qqqqqqq qqq qqq qq qq qq q qq qqq qqqqqq qq q q q qqqqqqqqqq qqqqqq q q qqq qqqqqq 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MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Outcome Result with a simulated sample of 100 points and ϑ1 = 0.6, ϑ2 = 0.2 and scale ω = 0.5 q q −2 −1 0 1 2 −1.0−0.50.00.51.0 θ1 θ2 qqqqqqq qqqq qqqqq qq qqq q q q qqqqqqqqqqqqqqqqqqq qq qq qqqqqqqqq qqqqqqqqqqqqq qq qqqqqqqqq q qq q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqq qqqqqqqqqqqqqqq qqqqqqq qqqqqqqqq qqqqqq qqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qqqqqqqqqq q qqqqqqqqqq qqqqqq qqqq q qqqqqqqq qqq qqqqqqqqqq qqqqqq qqqqqqqqqqqqqq qqq q qq qqqqqqqqqqqqqqqq qqqq qqqqq qq qqq qqqqqqqq qqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqq qqqqq q qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqqqqq qqqqqqqqqq qqqq qq qqqqq qqqqqq qqqq qqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 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MCMC and Likelihood-free Methods The Metropolis-Hastings Algorithm Extensions Outcome Result with a simulated sample of 100 points and ϑ1 = 0.6, ϑ2 = 0.2 and scale ω = 2.0 q q −2 −1 0 1 2 −1.0−0.50.00.51.0 θ1 θ2 qqqqqqqqqq qqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 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107. 107. MCMC and Likelihood-free Methods The Gibbs Sampler The Gibbs Sampler skip to population Monte Carlo The Gibbs Sampler General Principles Completion Convergence The Hammersley-Cliﬀord theorem
108. 108. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles General Principles A very speciﬁc simulation algorithm based on the target distribution f: 1. Uses the conditional densities f1, . . . , fp from f
109. 109. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles General 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)
110. 110. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles General 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.
111. 111. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Algorithm (Gibbs sampler) Given x(t) = (x (t) 1 , . . . , x (t) p ), generate 1. X (t+1) 1 ∼ f1(x1|x (t) 2 , . . . , x (t) p ); 2. X (t+1) 2 ∼ f2(x2|x (t+1) 1 , x (t) 3 , . . . , x (t) p ), . . . p. X (t+1) p ∼ fp(xp|x (t+1) 1 , . . . , x (t+1) p−1 ) X(t+1) → X ∼ f
112. 112. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Properties 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
113. 113. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Properties 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−1fpfp−1 · · · f1
114. 114. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example (Bivariate Gibbs sampler) (X, Y ) ∼ f(x, y) Generate a sequence of observations by Set X0 = x0 For t = 1, 2, . . . , generate Yt ∼ fY |X(·|xt−1) Xt ∼ fX|Y (·|yt) where fY |X and fX|Y are the conditional distributions
115. 115. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles A Very Simple Example: Independent N(µ, σ2 ) Observations When Y1, . . . , Yn iid ∼ N(y|µ, σ2) with both µ and σ unknown, the posterior in (µ, σ2) is conjugate outside a standard familly
116. 116. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles A Very Simple Example: Independent N(µ, σ2 ) Observations When Y1, . . . , Yn iid ∼ N(y|µ, σ2) with both µ and σ unknown, the posterior in (µ, σ2) is conjugate outside a standard familly But... µ|Y 0:n, σ2 ∼ N µ 1 n n i=1 Yi, σ2 n ) σ2|Y 1:n, µ ∼ IG σ2 n 2 − 1, 1 2 n i=1(Yi − µ)2 assuming constant (improper) priors on both µ and σ2 Hence we may use the Gibbs sampler for simulating from the posterior of (µ, σ2)
117. 117. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles R Gibbs Sampler for Gaussian posterior n = length(Y); S = sum(Y); mu = S/n; for (i in 1:500) S2 = sum((Y-mu)^2); sigma2 = 1/rgamma(1,n/2-1,S2/2); mu = S/n + sqrt(sigma2/n)*rnorm(1);
118. 118. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1
119. 119. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1, 2
120. 120. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1, 2, 3
121. 121. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1, 2, 3, 4
122. 122. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1, 2, 3, 4, 5
123. 123. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1, 2, 3, 4, 5, 10
124. 124. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1, 2, 3, 4, 5, 10, 25
125. 125. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50
126. 126. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50, 100
127. 127. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Example of results with n = 10 observations from the N(0, 1) distribution Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50, 100, 500
128. 128. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Limitations 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
129. 129. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Limitations 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
130. 130. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Limitations 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
131. 131. MCMC and Likelihood-free Methods The Gibbs Sampler General Principles Limitations 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.
132. 132. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Latent variables are back The Gibbs sampler can be generalized in much wider generality A density g is a completion of f if Z g(x, z) dz = f(x)
133. 133. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Latent variables are back The Gibbs sampler can be generalized in much wider generality A density g is a completion of f if Z g(x, z) dz = f(x) Note The variable z may be meaningless for the problem
134. 134. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Purpose g should have full conditionals that are easy to simulate for a Gibbs sampler to be implemented with g rather than f For p > 1, write y = (x, z) and denote the conditional densities of g(y) = g(y1, . . . , yp) by Y1|y2, . . . , yp ∼ g1(y1|y2, . . . , yp), Y2|y1, y3, . . . , yp ∼ g2(y2|y1, y3, . . . , yp), . . . , Yp|y1, . . . , yp−1 ∼ gp(yp|y1, . . . , yp−1).
135. 135. MCMC and Likelihood-free Methods The Gibbs Sampler Completion The move from Y (t) to Y (t+1) is deﬁned as follows: Algorithm (Completion Gibbs sampler) Given (y (t) 1 , . . . , y (t) p ), simulate 1. Y (t+1) 1 ∼ g1(y1|y (t) 2 , . . . , y (t) p ), 2. Y (t+1) 2 ∼ g2(y2|y (t+1) 1 , y (t) 3 , . . . , y (t) p ), . . . p. Y (t+1) p ∼ gp(yp|y (t+1) 1 , . . . , y (t+1) p−1 ).
136. 136. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example (Mixtures all over again) Hierarchical missing data structure: If X1, . . . , Xn ∼ k i=1 pif(x|θi), then X|Z ∼ f(x|θZ), Z ∼ p1I(z = 1) + . . . + pkI(z = k), Z is the component indicator associated with observation x
137. 137. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example (Mixtures (2)) Conditionally on (Z1, . . . , Zn) = (z1, . . . , zn) : π(p1, . . . , pk, θ1, . . . , θk|x1, . . . , xn, z1, . . . , zn) ∝ pα1+n1−1 1 . . . pαk+nk−1 k ×π(θ1|y1 + n1¯x1, λ1 + n1) . . . π(θk|yk + nk ¯xk, λk + nk), with ni = j I(zj = i) and ¯xi = j; zj=i xj/ni.
138. 138. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Algorithm (Mixture Gibbs sampler) 1. Simulate θi ∼ π(θi|yi + ni¯xi, λi + ni) (i = 1, . . . , k) (p1, . . . , pk) ∼ D(α1 + n1, . . . , αk + nk) 2. Simulate (j = 1, . . . , n) Zj|xj, p1, . . . , pk, θ1, . . . , θk ∼ k i=1 pijI(zj = i) with (i = 1, . . . , k) pij ∝ pif(xj|θi) and update ni and ¯xi (i = 1, . . . , k).
139. 139. MCMC and Likelihood-free Methods The Gibbs Sampler Completion A wee problem −1 0 1 2 3 4 −101234 µ1 µ2 Gibbs started at random
140. 140. MCMC and Likelihood-free Methods The Gibbs Sampler Completion A wee problem −1 0 1 2 3 4 −101234 µ1 µ2 Gibbs started at random Gibbs stuck at the wrong mode −1 0 1 2 3 −10123 µ1 µ2
141. 141. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Slice sampler as generic Gibbs If f(θ) can be written as a product k i=1 fi(θ),
142. 142. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Slice sampler as generic Gibbs If f(θ) can be written as a product k i=1 fi(θ), it can be completed as k i=1 I0≤ωi≤fi(θ), leading to the following Gibbs algorithm:
143. 143. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Algorithm (Slice sampler) Simulate 1. ω (t+1) 1 ∼ U[0,f1(θ(t))]; . . . k. ω (t+1) k ∼ U[0,fk(θ(t))]; k+1. θ(t+1) ∼ UA(t+1) , with A(t+1) = {y; fi(y) ≥ ω (t+1) i , i = 1, . . . , k}.
144. 144. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example of results with a truncated N(−3, 1) distribution 0.0 0.2 0.4 0.6 0.8 1.0 0.0000.0020.0040.0060.0080.010 x y Number of Iterations 2
145. 145. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example of results with a truncated N(−3, 1) distribution 0.0 0.2 0.4 0.6 0.8 1.0 0.0000.0020.0040.0060.0080.010 x y Number of Iterations 2, 3
146. 146. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example of results with a truncated N(−3, 1) distribution 0.0 0.2 0.4 0.6 0.8 1.0 0.0000.0020.0040.0060.0080.010 x y Number of Iterations 2, 3, 4
147. 147. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example of results with a truncated N(−3, 1) distribution 0.0 0.2 0.4 0.6 0.8 1.0 0.0000.0020.0040.0060.0080.010 x y Number of Iterations 2, 3, 4, 5
148. 148. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example of results with a truncated N(−3, 1) distribution 0.0 0.2 0.4 0.6 0.8 1.0 0.0000.0020.0040.0060.0080.010 x y Number of Iterations 2, 3, 4, 5, 10
149. 149. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example of results with a truncated N(−3, 1) distribution 0.0 0.2 0.4 0.6 0.8 1.0 0.0000.0020.0040.0060.0080.010 x y Number of Iterations 2, 3, 4, 5, 10, 50
150. 150. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example of results with a truncated N(−3, 1) distribution 0.0 0.2 0.4 0.6 0.8 1.0 0.0000.0020.0040.0060.0080.010 x y Number of Iterations 2, 3, 4, 5, 10, 50, 100
151. 151. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Good 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.
152. 152. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example (Stochastic volatility core distribution) Diﬃcult part of the stochastic volatility model π(x) ∝ exp − σ2 (x − µ)2 + β2 exp(−x)y2 + x /2 , simpliﬁed in exp − x2 + α exp(−x)
153. 153. MCMC and Likelihood-free Methods The Gibbs Sampler Completion Example (Stochastic volatility core distribution) Diﬃcult part of the stochastic volatility model π(x) ∝ exp − σ2 (x − µ)2 + β2 exp(−x)y2 + x /2 , simpliﬁed in exp − x2 + α exp(−x) Slice sampling means simulation from a uniform distribution on A = x; exp − x2 + α exp(−x) /2 ≥ u = x; x2 + α exp(−x) ≤ ω if we set ω = −2 log u. Note Inversion of x2 + α exp(−x) = ω needs to be done by trial-and-error.
154. 154. MCMC and Likelihood-free Methods The Gibbs Sampler Completion 0 10 20 30 40 50 60 70 80 90 100 −0.1 −0.05 0 0.05 0.1 Lag Correlation −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 Density Histogram of a Markov chain produced by a slice sampler and target distribution in overlay.
155. 155. MCMC and Likelihood-free Methods The Gibbs Sampler Convergence Properties of the Gibbs sampler Theorem (Convergence) For (Y1, Y2, · · · , Yp) ∼ g(y1, . . . , yp), if either [Positivity condition] (i) g(i)(yi) > 0 for every i = 1, · · · , p, implies that 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.
156. 156. MCMC and Likelihood-free Methods The Gibbs Sampler Convergence Properties of the Gibbs sampler (2) Consequences (i) If h(y)g(y)dy < ∞, then lim nT→∞ 1 T T t=1 h1(Y (t) ) = h(y)g(y)dy a.e. g. (ii) If, in addition, (Y (t)) is aperiodic, then lim n→∞ Kn (y, ·)µ(dx) − f TV = 0 for every initial distribution µ.
157. 157. MCMC and Likelihood-free Methods The Gibbs Sampler Convergence Slice sampler fast on that slice For convergence, the properties of Xt and of f(Xt) are identical Theorem (Uniform ergodicity) If f is bounded and suppf is bounded, the simple slice sampler is uniformly ergodic. [Mira & Tierney, 1997]
158. 158. MCMC and Likelihood-free Methods The Gibbs Sampler Convergence A small set for a slice sampler no slice detail For > , C = {x ∈ X; < f(x) < } is a small set: Pr(x, ·) ≥ µ(·) where µ(A) = 1 0 λ(A ∩ L( )) λ(L( )) d if L( ) = {x ∈ X; f(x) > }‘ [Roberts & Rosenthal, 1998]
159. 159. MCMC and Likelihood-free Methods The Gibbs Sampler Convergence Slice sampler: drift Under diﬀerentiability and monotonicity conditions, the slice sampler also veriﬁes a drift condition with V (x) = f(x)−β, is geometrically ergodic, and there even exist explicit bounds on the total variation distance [Roberts & Rosenthal, 1998]
160. 160. MCMC and Likelihood-free Methods The Gibbs Sampler Convergence Slice sampler: drift Under diﬀerentiability and monotonicity conditions, the slice sampler also veriﬁes a drift condition with V (x) = f(x)−β, is geometrically ergodic, and there even exist explicit bounds on the total variation distance [Roberts & Rosenthal, 1998] Example (Exponential Exp(1)) For n > 23, ||Kn (x, ·) − f(·)||TV ≤ .054865 (0.985015)n (n − 15.7043)
161. 161. MCMC and Likelihood-free Methods The Gibbs Sampler Convergence Slice sampler: convergence no more slice detail Theorem For any density such that ∂ ∂ λ ({x ∈ X; f(x) > }) is non-increasing then ||K523 (x, ·) − f(·)||TV ≤ .0095 [Roberts & Rosenthal, 1998]
162. 162. MCMC and Likelihood-free Methods The Gibbs Sampler Convergence A poor slice sampler Example Consider f(x) = exp {−||x||} x ∈ Rd Slice sampler equivalent to one-dimensional slice sampler on π(z) = zd−1 e−z z > 0 or on π(u) = e−u1/d u > 0 Poor performances when d large (heavy tails) 0 200 400 600 800 1000 -2-101 1 dimensional run correlation 0 10 20 30 40 0.00.20.40.60.81.0 1 dimensional acf 0 200 400 600 800 1000 1015202530 10 dimensional run correlation 0 10 20 30 40 0.00.20.40.60.81.0 10 dimensional acf 0 200 400 600 800 1000 0204060 20 dimensional run correlation 0 10 20 30 40 0.00.20.40.60.81.0 20 dimensional acf 0 200 400 600 800 1000 0100200300400 100 dimensional run correlation 0 10 20 30 40 0.00.20.40.60.81.0 100 dimensional acf Sample runs of log(u) and ACFs for log(u) (Roberts
163. 163. MCMC and Likelihood-free Methods The Gibbs Sampler The Hammersley-Cliﬀord theorem Hammersley-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 g(y1, y2) = g2(y2|y1) g2(v|y1)/g1(y1|v) dv . [Hammersley & Cliﬀord, circa 1970]
164. 164. MCMC and Likelihood-free Methods The Gibbs Sampler The Hammersley-Cliﬀord theorem General HC decomposition Under the positivity condition, the joint distribution g satisﬁes g(y1, . . . , yp) ∝ p j=1 g j (y j |y 1 , . . . , y j−1 , y j+1 , . . . , y p ) g j (y j |y 1 , . . . , y j−1 , y j+1 , . . . , y p ) for every permutation on {1, 2, . . . , p} and every y ∈ Y .
165. 165. MCMC and Likelihood-free Methods Population Monte Carlo Sequential importance sampling Computational issues in Bayesian statistics The Metropolis-Hastings Algorithm The Gibbs Sampler Population Monte Carlo Approximate Bayesian computation ABC for model choice
166. 166. MCMC and Likelihood-free Methods Population Monte Carlo Importance sampling (revisited) basic importance Approximation of integrals I = h(x)π(x)dx by unbiased estimators ˆI = 1 n n i=1 ih(xi) when x1, . . . , xn iid ∼ q(x) and i def = π(xi) q(xi)
167. 167. MCMC and Likelihood-free Methods Population Monte Carlo Iterated importance sampling As in Markov Chain Monte Carlo (MCMC) algorithms, introduction of a temporal dimension : x (t) i ∼ qt(x|x (t−1) i ) i = 1, . . . , n, t = 1, . . . and ˆIt = 1 n n i=1 (t) i h(x (t) i ) is still unbiased for (t) i = πt(x (t) i ) qt(x (t) i |x (t−1) i ) , i = 1, . . . , n
168. 168. MCMC and Likelihood-free Methods Population Monte Carlo Fundamental importance equality Preservation of unbiasedness E h(X(t) ) π(X(t)) qt(X(t)|X(t−1)) = h(x) π(x) qt(x|y) qt(x|y) g(y) dx dy = h(x) π(x) dx for any distribution g on X(t−1)
169. 169. MCMC and Likelihood-free Methods Population Monte Carlo Sequential variance decomposition Furthermore, var ˆIt = 1 n2 n i=1 var (t) i h(x (t) i ) , if var (t) i exists, because the x (t) i ’s are conditionally uncorrelated Note This decomposition is still valid for correlated [in i] x (t) i ’s when incorporating weights (t) i
170. 170. MCMC and Likelihood-free Methods Population Monte Carlo Simulation of a population The importance distribution of the sample (a.k.a. particles) x(t) qt(x(t) |x(t−1) ) can depend on the previous sample x(t−1) in any possible way as long as marginal distributions qit(x) = qt(x(t) ) dx (t) −i can be expressed to build importance weights it = π(x (t) i ) qit(x (t) i )
171. 171. MCMC and Likelihood-free Methods Population Monte Carlo Special case of the product proposal If qt(x(t) |x(t−1) ) = n i=1 qit(x (t) i |x(t−1) ) [Independent proposals] then var ˆIt = 1 n2 n i=1 var (t) i h(x (t) i ) ,
172. 172. MCMC and Likelihood-free Methods Population Monte Carlo Validation skip validation E (t) i h(X (t) i ) (t) j h(X (t) j ) = h(xi) π(xi) qit(xi|x(t−1)) π(xj) qjt(xj|x(t−1)) h(xj) qit(xi|x(t−1) ) qjt(xj|x(t−1) ) dxi dxj g(x(t−1) )dx(t−1) = Eπ [h(X)]2 whatever the distribution g on x(t−1)
173. 173. MCMC and Likelihood-free Methods Population Monte Carlo Self-normalised version In general, π is unscaled and the weight (t) i ∝ π(x (t) i ) qit(x (t) i ) , i = 1, . . . , n , is scaled so that i (t) i = 1
174. 174. MCMC and Likelihood-free Methods Population Monte Carlo Self-normalised version properties Loss of the unbiasedness property and the variance decomposition Normalising constant can be estimated by t = 1 tn t τ=1 n i=1 π(x (τ) i ) qiτ (x (τ) i ) Variance decomposition (approximately) recovered if t−1 is used instead
175. 175. MCMC and Likelihood-free Methods Population Monte Carlo Sampling importance resampling Importance sampling from g can also produce samples from the target π [Rubin, 1987]
176. 176. MCMC and Likelihood-free Methods Population Monte Carlo Sampling importance resampling Importance sampling from g can also produce samples from the target π [Rubin, 1987] Theorem (Bootstraped importance sampling) If a sample (xi )1≤i≤m is derived from the weighted sample (xi, i)1≤i≤n by multinomial sampling with weights i, then xi ∼ π(x)
177. 177. MCMC and Likelihood-free Methods Population Monte Carlo Sampling importance resampling Importance sampling from g can also produce samples from the target π [Rubin, 1987] Theorem (Bootstraped importance sampling) If a sample (xi )1≤i≤m is derived from the weighted sample (xi, i)1≤i≤n by multinomial sampling with weights i, then xi ∼ π(x) Note Obviously, the xi ’s are not iid
178. 178. MCMC and Likelihood-free Methods Population Monte Carlo Iterated sampling importance resampling This principle can be extended to iterated importance sampling: After each iteration, resampling produces a sample from π [Again, not iid!]
179. 179. MCMC and Likelihood-free Methods Population Monte Carlo Iterated sampling importance resampling This principle can be extended to iterated importance sampling: After each iteration, resampling produces a sample from π [Again, not iid!] Incentive Use previous sample(s) to learn about π and q
180. 180. MCMC and Likelihood-free Methods Population Monte Carlo Generic Population Monte Carlo Algorithm (Population Monte Carlo Algorithm) For t = 1, . . . , T For i = 1, . . . , n, 1. Select the generating distribution qit(·) 2. Generate ˜x (t) i ∼ qit(x) 3. Compute (t) i = π(˜x (t) i )/qit(˜x (t) i ) Normalise the (t) i ’s into ¯ (t) i ’s Generate Ji,t ∼ M((¯ (t) i )1≤i≤N ) and set xi,t = ˜x (t) Ji,t
181. 181. MCMC and Likelihood-free Methods Population Monte Carlo D-kernels in competition A general adaptive construction: Construct qi,t as a mixture of D diﬀerent transition kernels depending on x (t−1) i qi,t = D =1 pt, K (x (t−1) i , x), D =1 pt, = 1 , and adapt the weights pt, .
182. 182. MCMC and Likelihood-free Methods Population Monte Carlo D-kernels in competition A general adaptive construction: Construct qi,t as a mixture of D diﬀerent transition kernels depending on x (t−1) i qi,t = D =1 pt, K (x (t−1) i , x), D =1 pt, = 1 , and adapt the weights pt, . Darwinian example Take pt, proportional to the survival rate of the points (a.k.a. particles) x (t) i generated from K
183. 183. MCMC and Likelihood-free Methods Population Monte Carlo Implementation Algorithm (D-kernel PMC) For t = 1, . . . , T generate (Ki,t)1≤i≤N ∼ M ((pt,k)1≤k≤D) for 1 ≤ i ≤ N, generate ˜xi,t ∼ KKi,t (x) compute and renormalize the importance weights ωi,t generate (Ji,t)1≤i≤N ∼ M ((ωi,t)1≤i≤N ) take xi,t = ˜xJi,t,t and pt+1,d = N i=1 ¯ωi,tId(Ki,t)
184. 184. MCMC and Likelihood-free Methods Population Monte Carlo Links with particle ﬁlters Sequential setting where π = πt changes with t: Population Monte Carlo also adapts to this case Can be traced back all the way to Hammersley and Morton (1954) and the self-avoiding random walk problem Gilks and Berzuini (2001) produce iterated samples with (SIR) resampling steps, and add an MCMC step: this step must use a πt invariant kernel Chopin (2001) uses iterated importance sampling to handle large datasets: this is a special case of PMC where the qit’s are the posterior distributions associated with a portion kt of the observed dataset
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