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INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Hybrid Monte Carlo:Geometric Integration and Statistics
Andrew Stuart1
1Mathematics Institute andCentre for Scientific Computing
University of Warwick
SCMS2010Heriot-Watt, September 6th 2010
Funded by EPSRC, ONR
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
“Stable Periodic Bifurcations of an Explicit Discretization of aNonlinear Partial Differential Equation in Reaction Diffusion”.D.F. Griffiths and A.R.MitchellIMA J Numerical Analysis 8(1988), 435-454.
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Outline
1 INVARIANT MEASURES AND DYNAMICAL SYSTEMS
2 MARKOV CHAIN MONTE CARLO
3 EXPLICIT DISCRETIZATIONS
4 IMPLICIT DISCRETIZATIONS
5 CONCLUSIONS
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Outline
1 INVARIANT MEASURES AND DYNAMICAL SYSTEMS
2 MARKOV CHAIN MONTE CARLO
3 EXPLICIT DISCRETIZATIONS
4 IMPLICIT DISCRETIZATIONS
5 CONCLUSIONS
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Goals of Work
To find numerical methods to sample a probability densityfunction (pdf) π : Rn → R+.
To analyze and develop methods in the cases of highdimensions n� 1.Baisc building blocks are π−invariant dynamical systemsand MCMC methodology.
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Langevin Stochastic Dynamics
Let A be a positive-definite symmetric matrix.The Langevin SDE is
x = A∇ logπ(x) +√
2AW .
This equation is π− invariant:if x(0) ∼ π then x(t) ∼ π for all t > 0.
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Hybrid Monte Carlo
(Duane et al 1987)
Let A be a positive-definite symmetric matrix.Define the Hamiltonian
H(x ,p) =12〈p,Ap〉 − logπ(x).
Hamiltons equations are
x = Ap,p = ∇
(logπ(x)
).
Assume that p(0) ∼ N (0,A−1).This equation is π− invariant:if x(0) ∼ π, then x(t) ∼ π for all t > 0.
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Outline
1 INVARIANT MEASURES AND DYNAMICAL SYSTEMS
2 MARKOV CHAIN MONTE CARLO
3 EXPLICIT DISCRETIZATIONS
4 IMPLICIT DISCRETIZATIONS
5 CONCLUSIONS
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Metropolis-Hastings Algorithm
Enforcing π−invariant dynamics via accept-reject:
1. Set k = 0 and choose x (0) ∈ Rn.
2. Propose y = G(x (k), ξ(k),∆t), ξ(k) ∼ N (0,1).
3. Set x (k+1) = y with probability α; else x (k+1) = x (k).4. Set k → k + 1 and goto 2.
Step 3. is the proposal: how do we choose it?
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Choice of Parameters
We choose G to be a time-discretization of one of theinvariant dynamical systems.We want largest ∆t compatible with O(1) averageacceptance probability for n� 1.Choose ∆t = n−γ Courant condition.γ0 = minγ≥0
{γ : lim infn→∞ Eα > 0
}.
Number of steps required to adequately sample π is thenM(n) = O(∆t)−1 = O(nγ0).
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Structure of the Target
IID Product in Rn
π0(x) = Πni=1f (xi).
Change of Measure From Gaussian in Rn
π(x) = exp(−Φn(x)
)π0(x)
π0(x) ∝ exp(−1
2〈x , C−1
0 x〉).
The covariance matrix C0 is assumed to have conditionnumber O(n2k ). The resulting Gaussian part is assumed todominate Φn, uniformly in n.
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Outline
1 INVARIANT MEASURES AND DYNAMICAL SYSTEMS
2 MARKOV CHAIN MONTE CARLO
3 EXPLICIT DISCRETIZATIONS
4 IMPLICIT DISCRETIZATIONS
5 CONCLUSIONS
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Langevin 1
π0(x) = Πni=1f (xi).
Recall that the SDE
x = ∇ logπ0(x) +√
2W
is π0− invariant. We use the following discretization asproposal:
Proposaly − x
∆t= β∇ logπ0(x) +
√2
∆tξ, ξ ∼ N (0, I).
Theorem 1. (Roberts et al 97, Roberts/Rosenthal 98)β = 0 then M(n) = O(n1).β = 1 then M(n) = O(n1/3).
Steepest Descents Impacts Cost
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Langevin 2
π(x) = exp(−Φn(x)
)π0(x) = exp
(−Φn(x)− 1
2〈x , C−1
0 x〉).
Proposaly − x
∆t= A∇ logπ0(x) +
√2A∆t
ξ, ξ ∼ N (0, I).
Theorem 2. (Beskos, Roberts, Stuart 2009)
A = I then M(n) = O(n(2k+1/3)).
A = C0 then M(n) = O(n1/3).
Preconditioning Impacts Cost
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Hybrid Monte Carlo 1
The key dynamical system is:
x = Ap,p = ∇
(logπ(x)
).
Volume preserving reversible integration is required toensure that the acceptance probability α is tractable.This can be achieved by operator splitting (eg Verlet)based on the two dynamical systems
x = Ap,∥∥∥ x = 0,
p = 0.∥∥∥ p = ∇
(logπ(x)
).
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Hybrid Monte Carlo 2
π0(x) = Πni=1f (xi).
Theorem 3. (Beskos, Pillai, Roberts, Sanz-Serna and Stuart2010)
For Verlet integration within Hybrid Monte Carlo we haveM(n) = O(n1/4).
Hamiltonian Formulation Impacts Cost
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Outline
1 INVARIANT MEASURES AND DYNAMICAL SYSTEMS
2 MARKOV CHAIN MONTE CARLO
3 EXPLICIT DISCRETIZATIONS
4 IMPLICIT DISCRETIZATIONS
5 CONCLUSIONS
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Langevin
π(x) = exp(−Φn(x)− 1
2〈x , C−1
n x〉).
y − x∆t
+ A(θC−1
n y + (1− θ)C−1n x
)=
√2A∆t
ξ, ξ ∼ N (0, I).
Theorem 4. (Beskos, Roberts, Stuart 2009)
θ 6= 12 and A = Cn then M(n) = O(n1/3).
θ = 12 and A = I, Cn then M(n) = O(1).
Implicitness Impacts Cost
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Hybrid Monte Carlo 1
The key dynamical system is:
x = Ap,
p = −C−1n x −∇Φn(x).
Volume preserving reversible integration is required toensure that the acceptance probability α is tractable.This can be achieved by operator splitting based on thetwo dynamical systems
x = Ap,∥∥∥ x = 0,
p = −C−1n x .
∥∥∥ p = −∇Φn(x).
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Hybrid Monte Carlo 2
If we use a second-order (Strang-splitting) for this operator-splitthen we obtain:
Theorem 4. (Beskos, Pinski, Sanz-Serna and Stuart 2010)If A = Cn then M(n) = O(1).
Implicitness Impacts Cost
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
Outline
1 INVARIANT MEASURES AND DYNAMICAL SYSTEMS
2 MARKOV CHAIN MONTE CARLO
3 EXPLICIT DISCRETIZATIONS
4 IMPLICIT DISCRETIZATIONS
5 CONCLUSIONS
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
What We Have Shown
We have shown that the following ideas from numericalanalysis have direct impact on MCMC based statisticalsampling methods in high dimensions:
Steepest descentsPreconditioningImplicit integration for dissipative systemsGeometric integration
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
References
For all papers see:
http : //www .maths.warwick .ac.uk/ ∼ masdr/sample.html
A. Beskos, N. Pillai, G.O. Roberts, J.-M. Sanz-Serna andA.M. Stuart. “Optimal tuning of hybrid Monte Carlo”.Submitted.A. Beskos, F. Pinski, J.-M. Sanz-Serna and A.M. Stuart.“Hybrid Monte Carlo on Hilbert spaces”. Submitted.A. Beskos, G.O. Roberts and A.M. Stuart. ”Optimalscalings for local Metropolis-Hastings chains onnon-product targets in high dimensions.” Ann. Appl. Prob.19(2009), 863–898.
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
References (Continued)
A. Beskos and A.M. Stuart. ”MCMC Methods for SamplingFunction Space”. To appear, proceedings of ICIAM 2007.M. Hairer, A.M.Stuart and J. Voss. ”Sampling the posterior:an approach to non-Gaussian data assimilation.”PhysicaD, 230(2007), 50–64.S. Duane, A.D. Kennedy, B.J. Pendelton and D. Roweth.“Hybrid Monte Carlo.” Physics Letters B, 195(1987),216-222.
INVARIANT MEASURES AND DYNAMICAL SYSTEMS MARKOV CHAIN MONTE CARLO EXPLICIT DISCRETIZATIONS IMPLICIT DISCRETIZATIONS CONCLUSIONS
References (Continued)
A. Gelman, W.R. Gilks and G.O. Roberts, Weakconvergence and optimal scaling of random walkMetropolis algorithms. Ann. Appl. Prob. 7(1997), 110–120.G.O. Roberts and J. Rosenthal, Optimal scaling of discreteapproximations to Langevin diffusions. JRSSB 60(1998),255–268.M. Bédard, Weak Convergence of Metropolis Algorithmsfor Non-iid Target Distributions. Ann. Appl. Probab.17(2007), 1222-44.M. Bédard and J.S. Rosenthal, Optimal Scaling ofMetropolis Algorithms: Heading Towards General TargetDistributions. To appear Can. J. Stat.