Practical unbiased Monte Carlo for Uncertainty Quantification

Transcription

1 Practical unbiased Monte Carlo for Uncertainty Quantification Sergios Agapiou Department of Statistics, University of Warwick day: Uncertainty in Complex Computer Models, 2nd February 2015, University of Warwick

2 S. Agapiou, G. O. Roberts and S. J. Vollmer, Unbiased Monte Carlo: posterior estimation for intractable/infinite dimensional models, C. H. Rhee, Unbiased estimation with biased samples, PhD thesis, Stanford University, 2013 (supervisor P. W. Glynn).

3 Outline 1 Introduction - General theory 2 UQ example 3 Removing specific sources of bias 4 Performance/Optimization 5 Conclusions

4 Problem Want to estimate expectations of functions f wrt an intractable measure µ, E µ [f ] := E µ [f ( )]. Would like to use Monte Carlo estimator: for X (m) iid µ let Y M := 1 M f (X (m) ). M For all M and Y M E[Y M ] = E µ [f ] m=1 M Eµ [f ], almost surely (Y M unbiased) (Y m consistent) Intractability of µ forces the use of approximations µ i introducing bias.

5 Debiasing idea - John von Neumann, Stanislaw Ulam We study unbiased estimation of E µ [f ] using biased samples. Assume E µi [f ] i E µ [f ]. Let X i µ i and define i := f (X i ) f (X i 1 ). If Fubini applies E µ [f ] = (E µi [f ] E µi 1 [f ]) = E i i=1 i=1 i=1 i is unbiased but requires infinite computing time.? = E i. i=1

6 Debiasing idea - John von Neumann, Stanislaw Ulam Z := N i=0 i P(N i), N integer-valued r.v. independent of i, s.t. P(N i) > 0, i. If Fubini applies then Z unbiased [ ] 1 {N i} i E[Z] = E P(N i) i=0? = i=0 E[1 {N i} i ] P(N i) = E i = E µ [f ]. i=0 To be practical, Z needs to have finite variance and finite expected computing time.

7 Unbiasing theory of Glynn and Rhee Proposition (GR13) Assume Then Z := N i=0 i l i 2 l 2 P (N i) <. i P(N i) is an unbiased estimator for E µ[f ] with finite variance. Can use i copy of i s.t. { i } mutually independent. t i expected cost of generating i. Expected computing time of Z N E(τ) = E t i = t i P(N i). i=0 To be possible to choose P(N i) s.t. Z practical, suffices to generate i s with correct expectation s.t. i 2 2 decays sufficiently faster than t i blows-up. i=0

8 Example - Contamination scenario.v = h, x D p = 0, x D v = u p, x D dz = v(x)dt + z(0) = z init 2εdW - u permeability field - p pressure - v Darcy velocity - p = G(u) Quantity of interest: f (u) = E[ inf { z(t) > R}] t 0

9 Example - UQ in contamination scenario Permeability field u unknown, have prior information u µ 0. Vanilla-UQ: probe µ 0 f 1, e.g. estimate E µ0 [f (u)]. Have noisy indirect measurements of pressure: data model in R J y = G(u) + η, η N(0, Γ). Formulate Bayesian inverse problem (see DS13), µ y posterior on u y dµ y (u; y) exp ( 12 ) dµ y G(u) 2. 0 BIP-UQ: probe µ y f 1, e.g. estimate E µ y[f (u)]. - µ 0 is -dim, needs to be approximated by µ 0,i in R i introducing discretization bias (ARV14). - cannot sample µ y directly, construct Markov chain targeting µ y, use finite-time distributions µ y,k burn-in time issues (GR13, ARV14). - to implement in computer construct Markov chain targeting approximation µ y i in R i, use finite-time distributions µ y,k i introducing discretization bias and burn-in time issues (ARV14).

10 Removing discretization bias X = L 2 [0, 1], {ϕ l } complete orthonormal basis. µ Gaussian measure in X given via the Karhunen-Loeve expansion: ( ) µ = L l a iid 1 ξ l ϕ l, ξ l N(0, 1), a > 2. l=1 To estimate E µ [f ], need to truncate introducing discretization bias in MC estimators. (Vanilla-UQ example) Aim: unbiasedly estimate E µ [f ] in finite time. Approximations µ i = L ( ji i = f (u i ) f (u i 1 ), u i µ i. l=1 l a ξ l ϕ l ), {j i } increasing.

11 Removing discretization bias Theorem 1 (ARV14) i P(N i) is Assume a > 1 and f Lipschitz. Then choices j i and P(N i), s.t. Z = N i=1 unbiased estimator of E µ [f ] with finite variance and finite expected computing time. Proof. - Consider j i = 2 i. Use Proposition. - Cost of i, t i = O(j i ) = O(2 i ) (# N(0, 1) draws). - Bound i 2 2 = E( f (u i ) f (u i 1 ) 2 ) f 2 E( u i u i 1 2 ) = O(2 i(1 2a) ). - i 2 2 decays sufficiently faster than t i blows-up. - Can choose P(N i) s.t. E(τ), Var(Z) <.

12 Removing burn-in time bias X general state space, d distance in X, f d-lipschitz. Measure µ intractable, cannot be sampled directly but can construct X = (X n ) n N Markov chain with stationary distribution µ. {a i } increasing sequence of positive integers. To estimate E µ [f ], use finite-time distributions µ i = L(X ai ) introducing burn-in issues. Aim: unbiasedly estimate E µ [f ] in finite time.

13 Removing burn-in time bias Weak convergence of µ i not enough to get convergence of i. Contracting coupling assumption: we can simultaneously generate chains started at different states s.t. they come together in d geometrically quickly. Use top level chain T i running for a i steps and bottom level chain B i running for a i 1 steps, coupled as follows: x 0 = B a i i 1... B a i 0... B0 i } i = f (T0 i) f (Bi 0 ) x 0 = T a i i... T a i i T0 i

14 Removing burn-in time bias Theorem 2 (ARV14) choices a i and P(N i), s.t. Z = N i i=1 P(N i) is unbiased estimator of E µ[f ] with finite variance and finite expected computing time. Proof. - Use Proposition. - Using assumptions, can show i 2 2 f 2 Ed 2 ( T i 0, Bi 0) cr a i. - Cost of i, t i = O(a i ) (# steps). - i 2 2 decays sufficiently faster than t i blows-up. - Can choose P(N i) s.t. E(τ), Var(Z) <.

15 UE for BIP-UQ in function space Combining can perform UE of E µ [f ] for µ both -dim and only accessible in the limit of a Markov chain (BIP-UQ example). Approximation using finite-time distributions and discretizing space: top chain T i more steps and higher discretization level than bottom chain B i j i 1 : x 0 = B a i i 1... B a i 0... B0 i } i = f (T0 i) f (Bi 0 ) j i : x 0 = T a i i... T a i i T0 i In ARV14, achieve this: 1. in non-linear BIPs (e.g. groundwater flow example) with uniform priors, using independence sampler; 2. for targets µ which have Lipschitz log-density wrt Gaussian, using pcn algorithm. (MH with proposal X k+1 = λx k + 1 λ 2 ξ)

16 Comparison of Unbiased Estimator vs Ergodic Average 1d Gaussian autoregression ρ (0, 1), ξ n i.i.d. N(0, 1). X n+1 = ρ X n + 1 ρ 2 ξ n+1, Ergodic with invariant distribution µ = N(0, 1). Estimate E µ [Id] = 0. Compare MSE-work product of Monte Carlo estimator based on UE vs EA. For EA lim MSE-work = 1 + ρ n 1 ρ T step. For UE have non-asymptotic expression for MSE-work product, depending on a i and P(N i). Optimize by minimizing wrt a i and P(N i): hard!

17 Comparison of Unbiased Estimator vs Ergodic Average Figure: 1) Left: optimized P(N i) for fixed a i = 4(i + 1) (as in GR13), 2) Right: optimized P(N i) and a i over subclass a i = m(i + 1).

18 Conclusions - further work UE is often feasible. Optimization wrt parameters is crucial especially in function space setting. UE easily parallelizable: a) use independent copies of Z, b) i s independent. UE seems competitive. Looking forward to comparisons in problems of higher complexity (e.g. BIP-UQ example).

19 S. Agapiou, G. O. Roberts and S. J. Vollmer, Unbiased Monte Carlo: posterior estimation for intractable/infinite dimensional models, arxiv: C. H. Rhee, Unbiased estimation with biased samples, PhD thesis, Stanford University, 2013, (supervisor P. W. Glynn). J. G. Propp and D. B. Wilson, Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures and Algorithms, M. Dashti and A. M. Stuart, The Bayesian approach to inverse problems, arxiv: M. Hairer, A. M. Stuart and S. J. Vollmer Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions, The Annals of Applied Probability, 2014.