Hierarchical Bayesian Inversion

Transcription

1 Hierarchical Bayesian Inversion Andrew M Stuart Computing and Mathematical Sciences, Caltech cw/ S. Agapiou, J. Bardsley and O. Papaspiliopoulos SIAM/ASA JUQ 2(2014), pp cw/ M. Dunlop and M. Iglesias arxiv Funded by: DARPA, EPSRC, ERC, ONR 1

2 Orientation Bayesian inversion for functions requires MCMC on R N with N 1 (approximating N =.) Succesful Bayesian inversion often requires hierarchical thinking. This talk: design efficient algorithms which confront interaction between these two issues. This talk: showcase the ideas in use for inversion in class of piecewise continuous functions with unknown discontinuity sets. 2

3 Table of Contents MCMC in High Dimensions Hierarchical Priors Centering versus Non-Centering Hierarchical Level Set Conclusions 3

4 Table of Contents MCMC in High Dimensions Hierarchical Priors Centering versus Non-Centering Hierarchical Level Set Conclusions 4

5 The Setting Let (X, F) be an arbitrary measurable space. Target Measure P is a probability measure on X defined via its density with respect to another probability measure P 0 on X : P(du) = Z 1 e Φ(u) P 0 (du), Z = e Φ(u) P 0 (du). X AM Stuart. Acta Numerica 19(2010). (Bayesian Inverse Problems: P 0 prior, P posterior.) M Hairer, AM Stuart and J Voss. Oxford Handbook of Nonlinear Filtering (2011). (Conditioned Diffusions: P 0 Wiener measure, P SDE pathspace measure.) 5

6 Metropolis-Hastings Let X = R N and P and P 0 have Lebesgue densities π and π 0. Let q(, ) be a proposal density for Metropolis-Hastings. Accept-Reject The acceptance probabilty for a proposed move from u to v is { π(v)q(v, u) } a(u, v) = min 1,. π(u)q(u, v) WK Hastings. Biometrika(1970). 6

7 Tierney Formulation of Metropolis-Hastings on X Let Q(, dv) be a proposal probability kernel for Metropolis-Hastings. Define probability measures ν and ν T on X X by ν(du, dv) = P(du)Q(u, dv) ν T (du, dv) = P(dv)Q(v, du) Accept-Reject Metropolis-Hastings is well-defined on X if ν T has a density with respect to ν. Then the acceptance probabilty for a proposed move from u to v is a(u, v) = min {1, dνt } dν (u, v). L. Tierney. Annals Appl. Prob. 8(1998). 7

8 Prior-Reversible Proposal Assume that ν 0 (du, dv) = ν0 T (du, dv) (prior-reversible) where ν 0 (du, dv) = P 0 (du)q(u, dv) ν T 0 (du, dv) = P 0 (dv)q(v, du). Accept-Reject Then Metropolis-Hastings is well-defined on X and the acceptance probabilty for a proposed move from u to v is a(u, v) = min {1, e Φ(v) } e Φ(u). Example 1: X = R N, P 0 Lebesgue, proposal is random walk RWM. Example 2: X = Hilbert (H = R ), P 0 Gaussian, proposal is pcn. SL Cotter, GO Roberts, AM Stuart and D White. Stat. Sci. 28(2013). 8

9 Key Theorem: Key Idea P 0 Gaussian on H. Approximate target measure P on H by measure P N on R N. Theorem 1 pcn to sample P N : spectral gap O(1). RWM to sample P N : spectral gap O(N 1 2 ). M. Hairer, AM Stuart and S. Vollmer. Ann. Appl. Prob. 24(2014). Key Idea MCMC will not degenerate under mesh-refinement MCMC is well-defined on H in sense of Tierney. 9

10 Table of Contents MCMC in High Dimensions Hierarchical Priors Centering versus Non-Centering Hierarchical Level Set Conclusions 10

11 Gaussian Priors: Covariance Function versus Operator Centred Gaussian probability measure µ 0 on Hilbert space H of real-valued functions is characterized, for u µ 0, by: Covariance Function E µ 0 u(x)u(y) = c(x, y). or by: Covariance Operator E µ 0 u u = C. 11

12 Whittle-Matern: Covariance Function c WM (x, y; θ) = σ ν 1 Γ(ν) (τ x y )ν K ν (τ x y ). σ is an amplitude scale. τ is an inverse length-scale. ν controls smoothness draws from Gaussian fields with this covariance have ν fractional Sobolev and Hölder derivatives. We use θ Θ to denote a subset of ν, τ, σ. L. Roininen, JMJ Huttunen and S. Lasanen. Inv. Prob. Imag. (2014). 12

13 Whittle-Matern: Covariance Operator Covariance operator on unbounded domain R d is C WM (θ) σ 2 τ 2ν (τ 2 I ) ν d 2. Thus we may draw u µ 0 by solving the stochastic PDE (τ 2 I ) ν 2 + d 4 u = στ ν ξ where ξ is Gaussian white noise. F. Lindgren, H. Rue and J. Lindstrom. JRSSB (2011). 13

14 Hierarchical Setting Let P 0 (du, dθ) = P u θ (du; θ)p θ (dθ). P u θ (du θ) = N(0, C WM(θ) ). Hierarchical Target Measure Use data to learn both u and θ. P(du, dθ) = Z 1 e Φ(u) P 0 (du, dθ), Z = e Φ(u) P 0 (du, dθ). X 14

15 Gibbs and Metropolis-within-Gibbs Gibbs sample u k+1 P(du θ k ); sample θ k+1 P(dθ u k+1 ). Q(u, dv θ) is a Metropolis kernel invariant for P(du θ). Q(θ, dφ u) is a Metropolis kernel invariant for P(dθ u). Metropolis-Within-Gibbs (MwG) sample u k+1 Q(u k, dv θ k ); sample θ k+1 Q(θ k, dφ u k+1 ). For N independent behaviour each Metropolis kernel should be well-defined in the sense of Tierney. 15

16 Table of Contents MCMC in High Dimensions Hierarchical Priors Centering versus Non-Centering Hierarchical Level Set Conclusions 16

17 Illustrative Example Find u H from y R J where Centred Linear Inverse Problem y = Ku + η. Prior u θ N(0, θ 1 C 0 ); θ P θ. Likelihood y u, θ y u N(Ku, C 1 ). Posterior on (u, θ). Set w = θ 1 2 u. Non-Centred Linear Inverse Problem Prior w N(0, C 0 ) independent of θ P θ. Likelihood y w, θ N(θ 1 2 Kw, C 1 ). Posterior on (w, θ). O Papaspiliopoulos, GO Roberts, M Sköld. Stat. Sci. (2007). Y. Yu and X.-L. Meng, JCGS(2011). 17

18 Key Theorem: Key Idea Approximate target measure P(du, dθ) on H R by measure P N (du, dθ) on R N R. Theorem 2 Centred method to sample P: θ k = θ 0 for all k 0. ((Reducible). Centred method to sample P N : E(θ k+1 θ k ) = O(N 1 ). S. Agapiou, J. Bardsley, O. Papispiliopoulis and AM Stuart. SIAM/ASA J UQ. 2(2014). Key Idea MwG will not degenerate under mesh-refinement MwG is irreducible on H R. G. Roberts and O. Stramer. Biometrika 88(2001). 18

19 Numerical Results Autocorrelation for centred (left) and non-centred (chains). Key Idea For N = 32 (black), N = 512 (blue) and N = 8192 (red). Non-centering leads to mesh independent mixing. S. Agapiou, J. Bardsley, O. Papispiliopoulis and AM Stuart. SIAM/ASA J UQ. 2(2014). 19

20 Table of Contents MCMC in High Dimensions Hierarchical Priors Centering versus Non-Centering Hierarchical Level Set Conclusions 20

21 Bayesian Level Set Inversion Piecewise constant function v defined through thresholding a continuous level set function u. Let = c0 < c1 < < ck 1 < ck =. v(x) = K X vk χ{ck 1 <u ck } (x); v = F(u). k=1 F : X 7 Z is the level-set map., X cts fns. Z piecewise cts fns. S.Osher and J.Sethian. J. Comp. Phys. 79(1988). 21

22 Issues F is discontinuous. How to impose length scale via regularization? How to choose amplitude scales c k in F? 22

23 Discontinuity of Level Set Map F ( ) is continuous at u F ( ) is discontinuous at u v = F(u) := v + χ {u 0} (x) + v χ {u<0} (x). Causes problems in classical level set inversion. Bayesian formulation: it is a probability zero event. M. Iglesias, Y. Lu and AM Stuart. Interfaces and Free Boundary Problems, to appear. arxiv:

24 Length-Scale Matters Figure demonstrates role of length-scale in level-set function. Suggests Bayesian Hierarchical method to learn length scale. 24

25 Amplitude Matters Prior: P 0 (du, dτ) = P du τ (u τ)p τ (dτ). P(du τ) = N(0, C WM(σ,τ) ). Recall C WM(σ,τ,ν) σ 2 τ 2ν (τ 2 I ) ν d 2. Theorem 3 For fixed (τ, ν) the family of measures N(0, C WM(,τ,ν) ) are mutually singular. Key Idea Hierarchical MwG algorithms to learn amplitude will behave poorly under mesh-refinement N not well-defined in sense of Tierney. 25

26 Hierarchical Priors: Length/Amplitude Coupling Hence choose σ = τ ν and define: C τ,ν (τ 2 I ) ν d 2. Prior: P 0 (du, dτ) = P u τ (u τ)p τ (τ). P 0 (du τ) = N(0, C τ,ν ). Theorem 4 For fixed ν the family of measures N(0, C,ν ) are mutually equivalent. Key Idea Suggests need to scale thresholds in F by τ ν. Let = c 0 < c 1 < < c K 1 < c K =. K v(x) = v k χ {ck 1 <uτ ν c k }(x); k=1 v = F(u, τ). M. Dunlop, M. Iglesias and A. M. Stuart. arxiv:

27 Fixed Levels (Movie) Wrong length-scale gives problems 27

28 Rescale Levels (Movie) Learn length-scale hierarchically (amp. coupled) Centred: invert for (u, θ) : C 1 2 θ u = ξ. 28

29 Non-Centering and Rescaled Levels (Movie) Learn length and regularity hierarchically Non-centred: invert for (ξ, θ) : C 1 2 θ u = ξ. 29

30 Groundwater Flow Application (Movie) Forward Problem: Darcy Flow Let Z := {v Z : essinf x D v > 0}. Given κ Z, find y := G(κ). Here y j = l j (p), V := H 1 0 (D), l j V, j = 1,..., J, f V and κ p = f in D, p = 0 in D.. Let η R J be a realization of an observational noise. Inverse Problem Given that κ = F(u), u X and y R J, find u (and hence κ): y = G(κ) + η. 30

31 Table of Contents MCMC in High Dimensions Hierarchical Priors Centering versus Non-Centering Hierarchical Level Set Conclusions 31

32 Highlights 1 BAYESIAN INVERSION FOR FUNCTIONS Well-posed inverse problems: posterior Lipschitz in parameters. Algorithms defined on function space robust to discretization. 2 HIERARCHICAL BAYESIAN INVERSION Whittle-Matern Priors: amplitude, length-scale and regularity. Metropolis-within-Gibbs. Role of non-centring. 3 LEVEL SET Methodology to invert for discotninuous functions. Learning length-scale is important. Non-centring can help. 32