Kernel Adaptive Metropolis-Hastings
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1 Kernel Adaptive Metropolis-Hastings Arthur Gretton,?? Gatsby Unit, CSML, University College London NIPS, December 2015 Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
2 Setting: MCMC for intractable non-linear targets Metropolis-Hastings MCMC Unnormalized target (x) / p(x) Generate Markov chain with invariant distribution p Initialize x 0 p 0 At iteration t 0, propose to move to state x 0 q( x t ) Accept/Reject proposals based on ratio x t+1 = What proposal q( x t )? ( x 0, w.p. min n x t, otherwise. 1, (x 0 )q(x t x 0 ) (x t)q(x 0 x t) o, Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
3 Setting: MCMC for intractable non-linear targets Metropolis-Hastings MCMC Unnormalized target (x) / p(x) Generate Markov chain with invariant distribution p Initialize x 0 p 0 At iteration t 0, propose to move to state x 0 q( x t ) Accept/Reject proposals based on ratio x t+1 = ( x 0, w.p. min n x t, otherwise. 1, (x 0 )q(x t x 0 ) (x t)q(x 0 x t) What proposal q( x t )? Too narrow or broad:! slow convergence Does not conform to support of target! slow convergence o, Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
4 Setting: MCMC for intractable non-linear targets Adaptive MCMC Adaptive Metropolis (Haario, Saksman & Tamminen, 2001): Update proposal q t ( x t )=N (x t, 2 ˆ t ),usingestimatesofthetarget covariance Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
5 Setting: MCMC for intractable non-linear targets Adaptive MCMC Adaptive Metropolis (Haario, Saksman & Tamminen, 2001): Update proposal q t ( x t )=N (x t, 2 ˆ t ),usingestimatesofthetarget covariance Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
6 Setting: MCMC for intractable non-linear targets Adaptive MCMC Adaptive Metropolis (Haario, Saksman & Tamminen, 2001): Update proposal q t ( x t )=N (x t, 2 ˆ t ),usingestimatesofthetarget covariance Locally miscalibrated for strongly non-linear targets: directions of large variance depend on the current location Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
7 Setting: MCMC for intractable non-linear targets Motivation: Intractable & Non-linear Targets Previous solutions for non-linear targets: Hamiltonian Monte Carlo (HMC) or Metropolis Adjusted Langevin Algorithms (MALA) (Roberts & Stramer, 2003; Girolami & Calderhead, 2011). Require target gradients and second order information Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
8 Setting: MCMC for intractable non-linear targets Motivation: Intractable & Non-linear Targets Previous solutions for non-linear targets: Hamiltonian Monte Carlo (HMC) or Metropolis Adjusted Langevin Algorithms (MALA) (Roberts & Stramer, 2003; Girolami & Calderhead, 2011). Require target gradients and second order information Our case: not even target ( ) can be computed Pseudo-Marginal MCMC (Beaumont, 2003; Andrieu & Roberts, 2009). Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
9 Setting: MCMC for intractable non-linear targets Bayesian Gaussian Process Classification Example: when is target not computable? GPC model: latent process f, labelsy, (withcovariatematrixx ), and hyperparameters : p(f, y, )=p( )p(f )p(y f) f N(0, K ) GP with covariance K Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
10 Setting: MCMC for intractable non-linear targets Bayesian Gaussian Process Classification Example: when is target not computable? GPC model: latent process f, labelsy, (withcovariatematrixx ), and hyperparameters : p(f, y, )=p( )p(f )p(y f) f N(0, K ) GP with covariance K Automatic Relevance Determination (ARD) covariance: (K ) ij = apple(x i, x 0 j ) =exp 1 2! dx (x i,s xj,s 0 )2 s=1 exp( s ) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
11 Setting: MCMC for intractable non-linear targets Pseudo-Marginal MCMC Example: when is target not computable? Gaussian process classification, latentprocessf ˆ p( y) / p( )p(y ) = p( ) p(f )p(y f, )df =: ( )... but cannot integrate out f Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
12 Setting: MCMC for intractable non-linear targets Pseudo-Marginal MCMC Example: when is target not computable? Gaussian process classification, latentprocessf ˆ p( y) / p( )p(y ) = p( ) p(f )p(y f, )df =: ( )... but cannot integrate out f MH ratio: (, 0 )=min 1, p( 0 )p(y 0 )q( 0 ) p( )p(y )q( 0 ) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
13 Setting: MCMC for intractable non-linear targets Pseudo-Marginal MCMC Example: when is target not computable? Gaussian process classification, latentprocessf ˆ p( y) / p( )p(y ) = p( ) p(f )p(y f, )df =: ( )... but cannot integrate out f MH ratio: (, 0 )=min 1, p( 0 )p(y 0 )q( 0 ) p( )p(y )q( 0 ) Filippone & Girolami, 2013 use Pseudo-Marginal MCMC: unbiased estimate of p(y ) via importance sampling: ˆp( y) / p( )ˆp(y ) p( ) 1 n imp n imp X i=1 p(y f (i) ) p(f(i) ) Q(f (i) ) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
14 Setting: MCMC for intractable non-linear targets Pseudo-Marginal MCMC Example: when is target not computable? Gaussian process classification, latentprocessf ˆ p( y) / p( )p(y ) = p( ) p(f )p(y f, )df =: ( )... but cannot integrate out f Estimated MH ratio: (, 0 )=min 1, p( 0 )ˆp(y 0 )q( 0 ) p( )ˆp(y )q( 0 ) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
15 Setting: MCMC for intractable non-linear targets Pseudo-Marginal MCMC Example: when is target not computable? Gaussian process classification, latentprocessf ˆ p( y) / p( )p(y ) = p( ) p(f )p(y f, )df =: ( )... but cannot integrate out f Estimated MH ratio: (, 0 )=min 1, p( 0 )ˆp(y 0 )q( 0 ) p( )ˆp(y )q( 0 ) Replacing marginal likelihood p(y ) with unbiased estimate ˆp(y ) still results in correct invariant distribution [Beaumont, 2003; Andrieu & Roberts, 2009] Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
16 Setting: MCMC for intractable non-linear targets Intractable & Non-linear Target in GPC Sliced posterior over hyperparameters of a Gaussian Process classifier on UCI Glass dataset obtained using Pseudo-Marginal MCMC Adaptive sampler that learns the shape of non-linear targets without gradient information? Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
17 Setting: MCMC for intractable non-linear targets Two strategies for adaptive sampling Kameleon (Sejdinovic et al ) Learns covariance in RKHS. Locally aligns to (non-linear) target covariance, gradient free. Kernel Adaptive Hamiltonian Monte Carlo (Strathmann et al ) Learns global estimate of gradient of log target density Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
18 MCMC Kameleon The Kameleon D. Sejdinovic, H. Strathmann, M. Lomeli, C. Andrieu, and A. Gretton, ICML 2014 Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
19 MCMC Kameleon Use feature space covariance Capture non-linearities using linear covariance C z in feature space H Input space X x t Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
20 MCMC Kameleon Use feature space covariance Capture non-linearities using linear covariance C z in feature space H Input space X Feature space H x t (x t ) C z Feature space sample f Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
21 MCMC Kameleon Use feature space covariance Capture non-linearities using linear covariance C z in feature space H Input space X x t x Feature space H (x t ) C z H (X ) Feature space sample f (x ) Find a nearby pre-image in X (gradient descent) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
22 MCMC Kameleon Use feature space covariance Capture non-linearities using linear covariance C z in feature space H Input space X x t x Feature space H (x t ) C z H (X ) Feature space sample f (x ) Find a nearby pre-image in X (gradient descent) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
23 MCMC Kameleon Proposal Construction Summary 1 Get a chain subsample z = {z i } n i=1 2 Construct an RKHS sample f N( (x t ), 2 C z ) 3 Propose x such that (x ) is close to f (with an additional exploration term N 0, 2 I d ). Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
24 MCMC Kameleon Proposal Construction Summary 1 Get a chain subsample z = {z i } n i=1 2 Construct an RKHS sample f N( (x t ), 2 C z ) 3 Propose x such that (x ) is close to f (with an additional exploration term N 0, 2 I d ). Integrate out RKHS samples f,gradient step,and to obtain marginal Gaussian proposal on the input space: q z (x x t )=N (x t, 2 I d + 2 M z,xt HM > z,x t ) M z,xt = 2 [r x k(x, z 1 ) x=xt,...,r x k(x, z n ) x=xt ], k(x, x 0 )=h (x), (x 0 )i H. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
25 MCMC Kameleon Examples of Covariance Structure for Standard Kernels Kameleon proposals capture local covariance structure Gaussian kernel: k(x, x 0 )=exp kx x 0 k 2 2 cov[qz( y) ] ij = 2 ij nx [k(y, z a )] 2 (z a,i y i )(z a,j y j )+O a=1 1. n Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
26 MCMC Kameleon Kernel Adaptive Hamiltonian Monte Carlo (KMC) Heiko Strathmann, Dino Sejdinovic, Samuel Livingstone, Zoltan Szabo, and Arthur Gretton, NIPS 2015 Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
27 MCMC Kameleon Hamiltonian Monte Carlo HMC: distant moves, high acceptance probability. Potential energy U(q) = log (q), auxiliary momentum p exp( K(p)), simulate for t 2 R along Hamiltonian flow of H(p, q) =K(p)+U(q), using operator Numerical simulation (i.e. leapfrog) depends on gradient information What if gradient unavailable, e.g. in Bayesian GP classification? Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
28 MCMC Kameleon Infinite dimensional exponential families Proposal is RKHS exponential family model [Fukumizu, 2009; Sriperumbudur et al. 2014], but accept using true Hamiltonian (to correct for both model and leapfrog) const (x) exp (hf, k(x, )i H A(f )) Sufficient statistics: feature map k(, x) 2H, satisfies f (x) =hf, k(x, )i H for any f 2H. Natural parameters: f 2H. The model is dense in continuous densities on compact domains (TV, KL, etc.), relatively robust to increasing dimensions, as opposed to e.g. KDE. How to learn f from samples without access to A(f )? Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
29 MCMC Kameleon Score matching Estimation of unnormalised density models from samples [Sriperumbudur et al. 2014] Minimises Fisher divergence J(f )= 1 ˆ (x) krf (x) 2 r log (x)k 2 2 dx Possible without accessing r log (x) and accessing (x) only through samples x := {x i } t i=1 ( " Ĵ(f )= E x dx`=1 2 f 2` + 1 (x) 2 Expensive: full solution requires solving (td + 1)-dimensional linear system. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
30 MCMC Kameleon Approximate solution: KMC finite f (x) = > Random Fourier Features > x y k(x, y) 2 R m can be computed from ˆ := (C + I ) 1 b x Updates fast, uses all Markov chain history. Caveat: need to initialise correctly. Gradient norm: Gaussian b := 1 tx dx t i =1 `=1 ` x i tx dx C := 1 t i =1 `=1 ` x i ` x i T KMC Finite where x and 2` On-line updates cost O(dm 2 ). x. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
31 MCMC Kameleon Approximate solution: KMC lite f (x) = nx i k(z i, x) i=1 z x sub-sample. from linear system ˆ = 2 (C + I ) 1 b Geometrically ergodic on logconcave targets (fast convergence). Gradient norm: Gaussian where C 2 R n n, b 2 R n depend on kernel matrix Cost O(n 3 + n 2 d) (or cheaper with low-rank approx., conjugate gradient). KMC Lite Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
32 MCMC Kameleon Does kernel HMC work in high dimensions? Challenging Gaussian target (top): Eigenvalues: i Exp(1). Covariance: diag( 1,..., d), randomly rotate. Use Rational Quadratic kernel to account for resulting highly non-singular length-scales. KMC scales up to d 30. An easy, isotropic Gaussian target (bottom): More smoothness allows KMC to scale up to d 100. d d n n Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
33 Experiments Synthetic targets: Banana Banana: B(b, v): takex N(0, ) with =diag(v, 1,...,1), andset Y 2 = X 2 + b(x 2 1 v), andy i = X i for i 6= 2. (Haario et al, 1999; 2001) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
34 Experiments Synthetic targets: Banana Acc. rate n HMC KMC RW KAMH Minimum ESS n KMC behaves like HMC as number n of oracle samples increases. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
35 Experiments Gaussian Process Classification on UCI data Standard GPC model p(f, y, )=p( )p(f )p(y f) where p(f ) is a GP and with a sigmoidal likelihood p(y f). Goal: sample from p( y) / p( )p(y ). Unbiased estimate of ˆp(y ) via importance sampling. No access to likelihood or gradient Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
36 Experiments Gaussian Process Classification on UCI data Standard GPC model p(f, y, )=p( )p(f )p(y f) where p(f ) is a GP and with a sigmoidal likelihood p(y f). Goal: sample from p( y) / p( )p(y ). Unbiased estimate of ˆp(y ) via importance sampling. No access to likelihood or gradient. MMD from ground truth Iterations KMC KAMH RW Significant mixing improvements over state-of-the-art. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
37 Experiments Conclusions Simple, versatile, gradient-free adaptive MCMC samplers: Kameleon: Uses local covariance structure of the target distribution at the current chain state Kernel HMC Derivative of log density fit to samples, use this as proposal in HMC. Outperforms existing adaptive approaches on nonlinear target distributions Future work: For Kameleon, does feature space covariance track high density regions in original space? For kernel HMC, how does convergence rate degrade with increasing dimension? Kameleon code: Kernel HMC code: hmc Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
38 Bayesian Gaussian Process Classification GPC model: latent process f, labelsy, (withcovariatematrixx ), and hyperparameters : p(f, y, )=p( )p(f )p(y f) where f N(0, K ) is a realization of a GP with covariance K (covariance between latent processes evaluated at X ). Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
39 Bayesian Gaussian Process Classification GPC model: latent process f, labelsy, (withcovariatematrixx ), and hyperparameters : p(f, y, )=p( )p(f )p(y f) where f N(0, K ) is a realization of a GP with covariance K (covariance between latent processes evaluated at X ). K : exponentiated quadratic Automatic Relevance Determination (ARD) covariance: (K ) ij = apple(x i, x 0 j ) =exp 1 2! dx (x i,s xj,s 0 )2 s=1 exp( s ) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
40 Bayesian Gaussian Process Classification (2) Fully Bayesian treatment: Interested in the posterior p( y) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
41 Bayesian Gaussian Process Classification (2) Fully Bayesian treatment: Interested in the posterior p( y) Cannot use a Gibbs sampler on p(, f y), whichsamplesfromp(f, y) and p( f, y) in turns, since p( f, y) is extremely sharp Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
42 Bayesian Gaussian Process Classification (2) Fully Bayesian treatment: Interested in the posterior p( y) Cannot use a Gibbs sampler on p(, f y), whichsamplesfromp(f, y) and p( f, y) in turns, since p( f, y) is extremely sharp Filippone & Girolami, 2013 use Pseudo-Marginal MCMC to sample p( y) =p( ) p(, f y)p(f )df. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
43 Bayesian Gaussian Process Classification (2) Fully Bayesian treatment: Interested in the posterior p( y) Cannot use a Gibbs sampler on p(, f y), whichsamplesfromp(f, y) and p( f, y) in turns, since p( f, y) is extremely sharp Filippone & Girolami, 2013 use Pseudo-Marginal MCMC to sample p( y) =p( ) p(, f y)p(f )df. Unbiased estimate of ˆp(y ) via importance sampling: ˆp( y) / p( )ˆp(y ) p( ) 1 n imp n imp X i=1 p(y f (i) ) p(f(i) ) Q(f (i) ) Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
44 Bayesian Gaussian Process Classification (2) Fully Bayesian treatment: Interested in the posterior p( y) Cannot use a Gibbs sampler on p(, f y), whichsamplesfromp(f, y) and p( f, y) in turns, since p( f, y) is extremely sharp Filippone & Girolami, 2013 use Pseudo-Marginal MCMC to sample p( y) =p( ) p(, f y)p(f )df. Unbiased estimate of ˆp(y ) via importance sampling: ˆp( y) / p( )ˆp(y ) p( ) 1 n imp n imp X i=1 p(y f (i) ) p(f(i) ) Q(f (i) ) No access to likelihood, gradient, or Hessian of the target. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
45 RKHS and Kernel Embedding For any positive semidefinite function k, there is a unique RKHS H k. Can consider x 7! k(, x) as a feature map. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
46 RKHS and Kernel Embedding For any positive semidefinite function k, there is a unique RKHS H k. Can consider x 7! k(, x) as a feature map. Definition (Kernel embedding) Let k be a kernel on X,andP aprobabilitymeasureonx.thekernel embedding of P into the RKHS H k is µ k (P) 2H k such that E P f (X )=hf,µ k (P)i Hk for all f 2H k. Alternatively, can be defined by the Bochner integral µ k (P) = k(, x) dp(x) (expected canonical feature) For many kernels k, includingthegaussian,laplacianandinverse multi-quadratics, the kernel embedding P 7! µ P is injective: characteristic (Sriperumbudur et al, 2010), captures all moments (similarly to the characteristic function). Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
47 Covariance operator Definition The covariance operator of P is C P : H k!h k such that 8f, g 2H k, hf, C P gi Hk = Cov P [f (X )g(x )]. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
48 Covariance operator Definition The covariance operator of P is C P : H k!h k such that 8f, g 2H k, hf, C P gi Hk = Cov P [f (X )g(x )]. Covariance operator: C P : H k!h k is given by C P = k(, x) k(, x) dp(x) µ P µ P (covariance of canonical features) Empirical versions of embedding and the covariance operator: µ z = 1 n nx k(, z i ) i=1 C z = 1 n nx k(, z i ) k(, z i ) i=1 µ z µ z The empirical covariance captures non-linear features of the underlying distribution, e.g. Kernel PCA Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
49 Kernel distance gradient g(x) =k(x, x) 2k(x, y) 2 nx i=1 i [k(x, z i ) r x g(x) x=y = r x k(x, x) x=y 2r x k(x, y) x=y {z } =0 M z,y H µ z (x)] where M z,y = 2 [r x k(x, z 1 ) x=y,...,r x k(x, z n ) x=y ] and H = I n 1 n 1 n n Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
50 Cost function g Samples {z i} 200 i=1 Current position y g varies most along the high density regions of the target Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
51 Synthetic targets: Flower Flower: F(r 0, A,!, ), ad-dimensional target with: F(x; r 0, A,!, ) / p! x 2 exp 1 + x2 2 r 0 A cos (!atan2 (x 2, x 1 )) dy N (x j ; 0, 1). j=3 2 2 Concentrates on r 0 -circle with a periodic perturbation (with amplitude A and frequency!) inthefirst two dimensions. Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
52 Synthetic targets: convergence statistics SM AM-FS AM-LS KAMH-LS Accept [0, 1] Ê[X] / dimensional F(10, 6, 6, 1) target; iterations: , burn-in: Arthur Gretton (Gatsby Unit, UCL) Kernel Adaptive Metropolis-Hastings 12/12/ / 24
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