Stability of adaptive random-walk Metropolis algorithms
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Stability of adaptive random-walk Metropolis algorithms

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"Stability of adaptive random-walk Metropolis algorithms", M. Vihola's talk at BigMC, 5th May 2011

"Stability of adaptive random-walk Metropolis algorithms", M. Vihola's talk at BigMC, 5th May 2011

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Stability of adaptive random-walk Metropolis algorithms Stability of adaptive random-walk Metropolis algorithms Presentation Transcript

  • Introduction Some adaptive MCMC algorithms ValidityStability of adaptive random-walk Metropolis algorithms Matti Vihola matti.vihola@iki.fi University of Jyv¨skyl¨, Department of Mathematics and Statistics a a joint work with Eero Saksman (University of Helsinki) Paris, May 5th 2011 Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive MCMC Some adaptive MCMC algorithms Stochastic approximation framework Validity Stability?Adaptive MCMC Adaptive MCMC algorithms have been succesfully applied to ‘automatise’ the tuning of the proposal distributions. In general, they are based on a collection of Markov kernels {Pθ }θ∈Θ , each having π as the unique invariant distribution. The aim is to find a good kernel (a good value of θ) automatically. Stochastic approximation framework is the most common way to construct the algorithms [Andrieu and Robert, 2001]. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive MCMC Some adaptive MCMC algorithms Stochastic approximation framework Validity Stability?Stochastic approximation framework One starts with certain (X1 , Θ1 ) ≡ (x1 , θ1 ) ∈ Rd × Re and for n≥2 Xn ∼ PΘn−1 (Xn−1 , · ) Θn = Θn−1 + γn H(Θn−1 , Xn ). where (γn )n≥2 is a non-negative step size sequence that vanishes and the function H : Re × Rd → Re determines the adaptation. The adaptation aims to Θn → θ∗ so that h(θ∗ ) = 0, where h(θ) := H(θ, x)π(dx). Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive MCMC Some adaptive MCMC algorithms Stochastic approximation framework Validity Stability?Stability? I Usually, one can identify a Lyapunov function w : Re → [0, ∞) such that w(θ), h(θ) < 0 whenever h(θ) = 0. This suggests that, “on average”, there should be a “drift” of Θn towards a θ∗ such that h(θ∗ ) = 0. To have sufficient “averaging”, the sequence of Markov kernels PΘn should have nice properties. To have nice properties on PΘn , the sequence (Θn )n≥1 should behave nicely. . . Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive MCMC Some adaptive MCMC algorithms Stochastic approximation framework Validity Stability?Stability? II Usual way to overcome the stability issue is to enforce stability, or to use adaptive reprojections approach like Andrieu and Moulines [2006], Andrieu, Moulines, and Priouret [2005]. Strong belief (based on overwhelming empirical evidence) that many algorithms are intrinsically stable (and do not require stabilisation). Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmAdaptive Metropolis algorithm I The AM algorithm [Haario, Saksman, and Tamminen, 2001]: Define X1 ≡ x1 ∈ Rd such that π(x1 ) > 0. Choose parameters t > 0 and ≥ 0. For n = 2, 3, . . . 1/2 Yn = Xn−1 + Σn−1 Wn , where Wn ∼ N (0, I), and Yn , with probability α(Xn−1 , Yn ) and Xn = Xn−1 , otherwise. where Σn−1 := t2 Cov(X1 , . . . , Xn−1 ) + I, and I ∈ Rd×d stands for the identity matrix. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmAdaptive Metropolis algorithm II Cov(· · · ) is some consistent covariance estimator (not necessarily the standard sample covariance!). In what follows, consider the definition Cov(X1 , . . . , Xn ) := Sn , where (M1 , S1 ) ≡ (x1 , s1 ) with s1 ∈ Rd×d is symmetric and positive definite, and Mn = (1 − γn )Mn−1 + γn Xn Sn = (1 − γn )Sn−1 + γn (Xn − Mn−1 )(Xn − Mn−1 )T . Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmExample run of AM 0 0 0 −2 −2 −2 −4 −4 −4 −6 −6 −6 −8 −8 −8 −10 −10 −10 −2 0 2 −2 0 2 −2 0 2 Figure: The first 10, 100 and √1000 samples of the AM algorithm started with s1 = (0.01)2 I, t = 2.38/ 2, = 0 and ηn := n−1 . Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmAdaptive scaling Metropolis algorithm I This algorithm, essentially proposed by [Gilks, Roberts, and Sahu, 1998, Andrieu and Robert, 2001], adjusts the size of the proposal jumps, and tries to attain a given mean acceptance rate. Define X1 ≡ x1 ∈ Rd such that π(x1 ) > 0. Let Θ1 ≡ θ1 > 0. Define the desired mean acceptance rate α∗ ∈ (0, 1). (Usually α∗ = 0.44 or α∗ = 0.234. . . ) Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmAdaptive scaling Metropolis algorithm II For n = 2, 3, . . ., iterate Yn = Xn−1 + Θn−1 Wn , where Wn ∼ N (0, I), and Yn , with probability α(Xn−1 , Yn ) and Xn = Xn−1 , otherwise. log Θn = log Θn−1 + γn α(Xn−1 , Yn ) − α∗ . Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmExample run of ASM 0 0 0 −2 −2 −2 −4 −4 −4 −6 −6 −6 −8 −8 −8 −10 −10 −10 −2 0 2 −2 0 2 −2 0 2 Figure: The first 10, 100 and 1000 samples of the ASM algorithm started with θ1 = 0.01 and using γn = n−3/4 . Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmAdaptive scaling within adaptive Metropolis The AM and ASM algorithms can be used simultaneously [Atchad´ e and Fort, 2010, Andrieu and Thoms, 2008]. Yn = Xn−1 + Θn−1 Σn−1 Wn , where Wn ∼ N (0, I), and Yn , with probability α(Xn−1 , Yn ) and Xn = Xn−1 , otherwise. log Θn = log Θn−1 + γn α(Xn−1 , Yn ) − α∗ Mn = (1 − γn )Mn−1 + γn Xn Sn = (1 − γn )Sn−1 + γn (Xn − Mn−1 )(Xn − Mn−1 )T Σn = t2 Sn + I. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmRobust adaptive Metropolis algorithm I The RAM algorithm is a multidimensional ‘extension’ of the ASM adaptation mechanism [Vihola, 2010b]. Define X1 ≡ x1 ∈ Rd such that π(x1 ) > 0. Let Σ1 ≡ s1 ∈ Rd×d be symmetric and positive definite. Define the desired mean acceptance rate α∗ ∈ (0, 1). Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmRobust adaptive Metropolis algorithm II For n = 2, 3, . . ., iterate 1/2 Yn = Xn−1 + Σn−1 Wn , where Wn ∼ N (0, I), and Yn , with probability α(Xn−1 , Yn ) and Xn = Xn−1 , otherwise. T Wn Wn 1/2 1/2 Σn = Σn−1 + γn α(Xn−1 , Yn ) − α∗ Σn−1 Σ Wn 2 n−1 where Σn will be symmetric and positive definite whenever γn α(Xn−1 , Yn ) − α∗ > −1. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmRobust adaptive Metropolis algorithm III Figure: The RAM update with γn α(Xn−1 , Yn ) − α∗ = ±0.8, resp. The ellipsoids defined by Σn−1 (Σn ) are drawn in solid (dashed), and the 1/2 vector Σn−1 Wn / Wn as a dot. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmExample run of RAM 0 0 0 −2 −2 −2 −4 −4 −4 −6 −6 −6 −8 −8 −8 −10 −10 −10 −2 0 2 −2 0 2 −2 0 2 Figure: The first 10, 100 and 1000 samples of the RAM algorithm started with Σ1 = 0.01 × I and using γn = min{1, 2 · n−3/4 }. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Adaptive Metropolis algorithm Some adaptive MCMC algorithms Adaptive scaling Metropolis algorithm Validity Robust adaptive Metropolis algorithmRAM vs. ASwAM Computationally, they are of similar complexity (Cholesky update of order O(d2 )). The RAM adaptation does not require that π has a finite second moment (or any moment at all). In RAM, the adaptation is ‘direct’ ‘while estimating the covariance is ‘indirect’ (as it requires also a mean estimate). There are also situations where ASwAM seems to be more efficient than RAM. . . Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMPrevious conditions on validity I What conditions are sufficient to guarantee that a law of large n→∞ numbers (LLN) holds: n n f (Xk ) − − 1 k=1 − → f (x)π(x)dx? Key conditions due to Roberts and Rosenthal [2007]: Diminishing adaptation (DA) The effect of the adaptation becomes smaller and smaller, n→∞ “ PΘn − PΘn−1 − − 0.” −→ Containment (C) The kernels PΘn have all the time sufficiently good mixing properties: k m→∞ sup sup PΘn (Xn , · ) − π − − → 0. −− n≥1 k≥m Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMPrevious conditions on validity II DA is usually easier to verify. Containment is often tightly related to the stability of the process (Θn )n≥1 . Establishing containment (or stability) is often difficult before showing that a LLN holds. . . Typical solution has been to enforce containment. In the case of AM, ASM and RAM algorithms, this means that the eigenvalues of Σn , Θn are constrained to be within [a, b] for some constants 0 < a ≤ b < ∞. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMPrevious conditions on validity III There are also other results in the literature on the ergodicity of adaptive MCMC given conditions similar to DA and C. The original work on AM [Haario, Saksman, and Tamminen, 2001]. Atchad´ and Rosenthal [2005] analyse the ASM algorithm e following the original mixingale approach. The results on adaptive reprojections approach Andrieu and Moulines [2006], Andrieu, Moulines, and Priouret [2005]. The recent paper by Atchad´ and Fort [2010] is based on a e resolvent kernel approach. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMNew results I Key ideas in the new results: Containment is, in fact, not a necessary condition for LLN. That is, the ergodic properties of PΘn can become more and more unfavourable, and still a (strong) LLN can hold. The adaptation mechanism may include a strong explicit ‘drift’ away from ‘bad values’ of θ. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMGeneral ergodicity result I Assume K1 ⊂ K2 ⊂ · · · ⊂ Re are increasing subsets of the adaptation space. Assume that the adaptation follows the stochastic approximation dynamic Xn ∼ PΘn−1 (Xn−1 , · ) Θn = Θn−1 + γn H(Θn−1 , Xn ). and the adaptation parameter Θn ∈ Kn for all n ≥ 1. One may also enforce Θn ∈ Kn . . . Consider the following conditions for constants c < ∞ and 0≤ 1: Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMGeneral ergodicity result II (A1) Drift and minorisation: There is a drift function V : Rd → [1, ∞) such that for all n ≥ 1 and all θ ∈ Kn Pθ V (x) ≤ λn V (x) + ½Cn (x)bn and (1) Pθ (x, A) ≥ ½Cn (x)δn νn (A) (2) where Cn ⊂ Rd are Borel sets, δn , λn ∈ (0, 1) and bn < ∞ are constants and νn is concentrated on Cn . Furthermore, the constants are polynomially bounded so that (1 − λn )−1 ∨ δn ∨ bn ≤ cn . −1 Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMGeneral ergodicity result III (A2) Continuity: For all n ≥ 1 and any r ∈ (0, 1], there is c = c (r) ≥ 1 such that for all θ, θ ∈ Kn , Pθ f − Pθ f Vr ≤cn f Vr |θ − θ |. (A3) Bound for adaptation function: There is a β ∈ [0, 1/2] such that for all n ≥ 1, θ ∈ Kn and x ∈ Rd |H(θ, x)| ≤ cn V β (x). Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMGeneral ergodicity result IV Theorem (Saksman and Vihola [2010]) Assume (A1)–(A3) hold and let f be a function with f V α < ∞ for some α ∈ (0, 1 − β). Assume < κ−1 [(1/2) ∧ (1 − α − β)], ∗ where κ∗ 1 is an independent constant, and that ∞ κ∗ −1 γ < ∞. Then, k=1 k k n 1 lim f (Xk ) = f (x)π(x)dx almost surely. n→∞ n k=1 Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMVerifiable assumptions I Definition (Strongly super-exponential target) The target density π is continuously differentiable and has regular tails that decay super-exponentially, x π(x) lim sup · < 0 and |x|→∞ |x| | π(x)| x lim · log π(x) = −∞, |x|→∞ |x|ρ with some ρ > 1. (The “super-exponential case”, ρ = 1, is due to Jarner and Hansen [2000] who ensure the geometric ergodicity of a nonadaptive random-walk Metropolis process.) Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMVerifiable assumptions II 0 α π(x) x x π(x) Figure: The condition lim sup|x|→∞ |x| · | π(x)| < 0 implies that there is an ε > 0 such that for any sufficiently large |x|, the angle α < π/2 − ε. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMAM when π has unbounded support Saksman and Vihola [2010]: First ergodicity results for the original AM algorithm, without upper bounding the eigenvalues λ(Σn ). Use the proposal covariance Σn = t2 Sn + I, with > 0. SLLN and CLT for strongly super-exponential π and functions satisfying supx |f (x)|π −p (x) < ∞ for some p ∈ (0, 1). Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMAM without the covariance lower bound I Vihola [2011b] contains partial results on the case Σn = t2 Sn , that is, without lower bounding the eigenvalues λ(Σn ). Analysed first an ‘adaptive random walk’: AM run with ‘flat target π ≡ 1’, 1/2 Xn+1 = Xn + tSn Wn+1 Mn+1 = (1 − γn+1 )Mn + γn+1 Xn+1 Sn+1 = (1 − γn+1 )Sn + γn+1 (Xn+1 − Mn )2 . Ten sample paths of this process (univariate) started at x0 = 0, s0 = 1 and with t = 0.01 and ηn = n−1 : Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMAM without the covariance lower bound II 0.1 0.05 Xn 0 −0.05 −0.1 0 2000 4000 6000 8000 10000 0 10 −2 Sn 10 −4 10 0 2000 4000 6000 8000 10000 Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMAM without the covariance lower bound III 600 400 Xn 200 0 −200 0 0.5 1 1.5 2 6 x 10 5 10 0 Sn 10 −5 10 0 0.5 1 1.5 2 6 x 10 Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMAM without the covariance lower bound IV It is shown that Sn → ∞ almost surely. √ The speed of growth is E[Sn ] ∼ exp t n i=2 ηi . √ (Particularly, E[Sn ] ∼ e2t n when ηn = n−1 ). Using the same techniques, one can show the stability (and ergodicity) of AM run with a univariate Laplace target π. These results have little direct practical value, but they indicate that the AM covariance parameter Sn does not tend to collapse. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMAM with a fixed proposal component I Instead of lower bounding the eigenvalues λ(Σn ), employ a fixed proposal component with a probability β ∈ (0, 1) [Roberts and Rosenthal, 2009]. This corresponds to an algorithm where the proposals Yn are generated by 1/2 Σ0 Wn , with probability β, Yn = Xn−1 + 1/2 Σn−1 Wn , otherwise. with some fixed symm.pos.def. Σ0 . In other words, one employs a mixture of ‘adaptive’ and ‘nonadaptive’ Markov kernels: P ( Xn ∈ A | X1 , . . . , Xn−1 ) = (1 − β)PΣn−1 (A) + βPΣ0 (A) Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMAM with a fixed proposal component II Vihola [2011b] shows that, for example with a bounded and compactly supported π or with a super-exponential π, the eigenvalues λ(Σn ) are bounded away from zero. Having a strongly super-exponential target, SLLN (resp. CLT) hold for functions satisfying supx |f (x)|π −p (x) < ∞ for some p ∈ (0, 1) (p ∈ (0, 1/2)). Explicit upper and lower bounds for Σn unnecessary. Mixture proposal may be better in practice than the lower bound I. Also Bai, Roberts, and Rosenthal [2008] analyse this algorithm. A completely different approach, with different assumptions. Also exponentially decaying π are considered; in this case the fixed covariance Σ0 must be large enough. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMErgodicity of unconstrained ASM algorithm Vihola [2011a] shows two results for the unmodified adaptation, without any (upper or lower) bounds for Θn . Assume: the desired acceptance rate α∗ ∈ (0, 1/2). the adaptation weights satisfy γn < ∞ (e.g. γn = n−γ with 2 γ ∈ (1/2, 1]). Two cases: 1. π is bounded, bounded away from zero on the support, and the support X = {x : π(x) > 0} is compact and has a smooth boundary. Then, SLLN holds for bounded functions. 2. Suppose a strongly super-exponential target having tails with uniformly smooth contours. Then, SLLN holds for functions satisfying supx |f (x)|π −p (x) < ∞ for some p ∈ (0, 1). Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMErgodicity of ASM within AM The results in [Vihola, 2011a] cover also the adaptive scaling within adaptive Metropolis algorithm. However, one has to assume that the covariance factor Σn is bounded within 0 < a ≤ b < ∞ Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMFinal remarks I Current results show that some adaptive MCMC algorithms are intrinsically stable, requiring no additional stabilisation structures. Easier for practitioners; less parameters to ‘tune.’ Showing that the methods are fairly ‘safe’ to apply. The results are related to the more general question of the stability of the Robbins-Monro stochastic approximation with Markovian noise. Manuscript in preparation with C. Andrieu. . . Fort, Moulines, and Priouret [2010] consider also interacting MCMC (e.g. the equi-energy sampler) and extend the approach in [Saksman and Vihola, 2010]. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMFinal remarks II It is not necessary to use Gaussian proposal distributions. All the above results apply to elliptical proposals satisfying a particular tail decay condition. For example, the results apply with multivariate Student distributions having the form d+p qc (z) ∝ (1 + c−1/2 z 2 )− 2 where p > 0 [Vihola, 2011a]. Current results apply only for targets with rapidly decaying tails. It is important to establish similar results with heavy-tailed targets. Overall, there is a need for more general yet practically verifiable conditions to check the validity of the methods. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMFinal remarks III There is also a free software available for testing several adaptive random-walk MCMC algorithms, including the new RAM approach [Vihola, 2010a]: http://iki.fi/mvihola/grapham/ Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • Introduction Previous conditions Some adaptive MCMC algorithms New results Validity Results for AM and ASMFinal remarks III There is also a free software available for testing several adaptive random-walk MCMC algorithms, including the new RAM approach [Vihola, 2010a]: http://iki.fi/mvihola/grapham/ Thank you! Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • ReferencesReferences I ´ C. Andrieu and E. Moulines. On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab., 16(3): 1462–1505, 2006. C. Andrieu and C. P. Robert. Controlled MCMC for optimal sampling. Technical Report Ceremade 0125, Universit´ Paris e Dauphine, 2001. C. Andrieu and J. Thoms. A tutorial on adaptive MCMC. Statist. Comput., 18(4):343–373, Dec. 2008. ´ C. Andrieu, E. Moulines, and P. Priouret. Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim., 44(1):283–312, 2005. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • ReferencesReferences II Y. Atchad´ and G. Fort. Limit theorems for some adaptive MCMC e algorithms with subgeometric kernels. Bernoulli, 16(1):116–154, Feb. 2010. Y. F. Atchad´ and J. S. Rosenthal. On adaptive Markov chain e Monte Carlo algorithms. Bernoulli, 11(5):815–828, 2005. Y. Bai, G. O. Roberts, and J. S. Rosenthal. On the containment condition for adaptive Markov chain Monte Carlo algorithms. Preprint, July 2008. URL http://probability.ca/jeff/research.html. G. Fort, E. Moulines, and P. Priouret. Convergence of adaptive MCMC algorithms: ergodicity and law of large numbers. 2010. technical report. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • ReferencesReferences III W. R. Gilks, G. O. Roberts, and S. K. Sahu. Adaptive Markov chain Monte Carlo through regeneration. J. Amer. Statist. Assoc., 93(443):1045–1054, 1998. H. Haario, E. Saksman, and J. Tamminen. An adaptive Metropolis algorithm. Bernoulli, 7(2):223–242, 2001. S. F. Jarner and E. Hansen. Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl., 85:341–361, 2000. G. O. Roberts and J. S. Rosenthal. Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab., 44(2):458–475, 2007. G. O. Roberts and J. S. Rosenthal. Examples of adaptive MCMC. J. Comput. Graph. Statist., 18(2):349–367, 2009. Matti Vihola Stability of adaptive random-walk Metropolis algorithms
  • ReferencesReferences IV E. Saksman and M. Vihola. On the ergodicity of the adaptive Metropolis algorithm on unbounded domains. Ann. Appl. Probab., 20(6):2178–2203, Dec. 2010. M. Vihola. Grapham: Graphical models with adaptive random walk Metropolis algorithms. Comput. Statist. Data Anal., 54(1): 49–54, Jan. 2010a. M. Vihola. Robust adaptive Metropolis algorithm with coerced acceptance rate. Preprint, arXiv:1011.4381v1, Nov. 2010b. M. Vihola. On the stability and ergodicity of adaptive scaling Metropolis algorithms. Preprint, arXiv:0903.4061v3, Apr. 2011a. M. Vihola. Can the adaptive Metropolis algorithm collapse without the covariance lower bound? Electron. J. Probab., 16:45–75, 2011b. Matti Vihola Stability of adaptive random-walk Metropolis algorithms