Well-posed Bayesian Inverse Problems: Beyond Gaussian Priors

Transcription

1 Well-posed Bayesian Inverse Problems: Beyond Gaussian Priors Bamdad Hosseini Department of Mathematics Simon Fraser University, Canada The Institute for Computational Engineering and Sciences University of Texas at Austin September 28, / 53

2 The Bayesian approach A model for indirect measurements y Y of a parameter u X X, Y are Banach spaces. G encompasses measurement noise. Simple example, additive noise model G deterministic forward map η independent random variable. Find u given a realization of y. y = G(u). y = G(u) + η. 2 / 53

3 Application 1: atmospheric source inversion ( t L)c = u in D (, T ], c(x, t) = on D (, T ), c(x, ) = y [m] Deposition in mg Advection-diffusion PDE. Estimate u from accumulated deposition measurements x [m] 1 B. Hosseini and J. M. Stockie. Bayesian estimation of airborne fugitive emissions using a Gaussian plume model. In: Atmospheric Environment 141 (216), pp / 53

4 Application 2: high intensity focused ultrasound treatment Attenuation (%) Phase shift (deg) Compensate for phase shift to focus the beam Acoustic waves converge. Ablate diseased tissue. Phase shift due to skull bone. Defocused beam. Estimate phase shift from MR-ARFI data 2. 2 B. Hosseini et al. A Bayesian approach for energy-based estimation of acoustic aberrations in high intensity focused ultrasound treatment. arxiv preprint: / 53

5 Running example ϕ u u Example: Deconvolution S(ϕ u) Let X = L 2 (T) and assume G(u) = S(ϕ u). Here ϕ C (T) and S : C(T) R m collects point values of a function at m distinct points {t k } m k=1. Noise η is additive and Gaussian. We want to find u given noisy pointwise observations of the blurred image. 5 / 53

6 The Bayesian approach Bayes rule 3 in the sense of Radon-Nikodym theorem, dµ y (u) = 1 exp( Φ(u; y)). (1) dµ Z(y) µ prior measure. Φ likelihood potential y = G(u). Z(y) = X exp( Φ(u; y))dµ (u) normalizing constant. µ y posterior measure. µ Likelihood µ y 3 A. M. Stuart. Inverse problems: a Bayesian perspective. In: Acta Numerica 19 (21), pp / 53

7 Why non-gaussian priors? dµ y (u) = 1 exp( Φ(u; y)). dµ Z(y) suppµ y suppµ since µ y µ. The prior has a major influence on the posterior. 7 / 53

8 Application 1: atmospheric source inversion Ω := D (, T ] Measurement operators M i : L 2 (Ω) R, M i (c) = Forward map J i (,T ] c dxdt, i = 1,, m. G : L 2 (Ω) R m, G(u) = (M 1 (c(u)),, M m (c(u)) T, c = ( t L) 1 u Linear in u. c L 2 (Ω) C u L 2 (Ω). G is bounded and linear. y [m] Deposition in mg x [m] 8 / 53

9 Application 1: atmospheric source inversion Assume y = G(u) + η where η N (, σ 2 I). Φ(u; y) = 1 2σ 2 G(u) y 2 2. Positivity constraint on source u. Sources are likely to be localized. y [m] x [m] Deposition in mg 9 / 53

10 Application 2: high intensity focused ultrasound treatment Underlying aberration field u. Pointwise evaluation map for points {t 1,, t d } in T 2 S : C(T 2 ) R m (S(u)) j = u(t j ). (Experiments) A collection of vectors {z j } m j=1 in Rd. Quadratic forward map G : C(T 2 ) R m (G(u)) j := zj T S(u) 2. Phase retrieval in essence Phase shift (deg) Attenuation (%) / 53

11 Application 2: high intensity focused ultrasound treatment Assume y = G(u) + η where η N (, σ 2 I). Φ(u; y) = 1 2σ 2 G(u) y 2 2. G(u) 2 C u 2 C(T 2 ). Nonlinear forward map. Hydrophone experiments show sharp interfaces. Gaussian priors are too smooth. Phase shift (deg) Attenuation (%) / 53

12 We need to go beyond Gaussian priors! 12 / 53

13 Key questions dµ y (u) = 1 exp( Φ(u; y)). dµ Z(y) Is µ y well-defined? What happens if y is perturbed? Easier to address when X = R n. More delicate when X is infinite dimensional. 13 / 53

14 Outline (i) General theory of well-posed Bayesian inverse problems. (ii) Convex prior measures. (iii) Models for compressible parameters. (iv) Infinitely divisible prior measures. 14 / 53

15 Well-posedness dµ y (u) = 1 exp( Φ(u; y)) dµ Z(y) Definition: Well-posed Bayesian inverse problem Suppose X is a Banach space and d(, ) R is a probability metric. Given a prior µ and likelihood potential Φ, the problem of finding µ y is well-posed if: (i) (Existence and uniqueness) There exists a unique posterior probability measure µ y µ given by Bayes rule. (ii) (Stability) For every choice of ɛ > there exists a δ > so that d(µ y, µ y ) ɛ for all y, y Y so that y y Y δ. 15 / 53

16 Metrics on probability measures The total variation and Hellinger metrics d T V (µ 1, µ 2 ) := 1 dµ 1 2 X dν dµ 2 dν dν d H (µ 1, µ 2 ) := 1 ( ) 2 dµ1 2 dν dµ2 dν dν Note: X 2d 2 H(µ 1, µ 2 ) d T V (µ 1, µ 2 ) 8d H (µ 1, µ 2 ). Hellinger is more attractive in practice. For h L 2 (X, µ 1 ) L 2 (X, µ 2 ) h(u)dµ 1 (u) h(u)dµ 2 (u) C(h)d H(µ 1, µ 2 ). X X Different convergence rates. 1/2. 16 / 53

17 Well-posedness: analogy The likelihood Φ depends on the map G. Given Φ what classes of priors can be used? PDE analogy A PDE where g H s and L : H p H s is a differential operator. Lu = g Seek a solution u = L 1 g H p. Well-posedness depends on the smoothing behavior of L 1 and regularity of g. In the Bayesian approach we seek µ y that satisfies Pµ y = µ. The mapping P 1 depends on Φ. Well-posedness depends on behavior of P 1 and tail behavior of µ. In a nutshell, if Φ grows at a certain rate we have well-posedness if µ has sufficient tail decay. 17 / 53

18 Assumptions on likelihood Minimal assumptions on Φ (BH, 216) The potential Φ : X Y R satisfies: ab (L1) (Locally bounded from below): There is a positive and non-decreasing function f 1 : R + [1, ) so that Φ(u; y) M log (f 1 ( u X )). (L2) (Locally bounded from above): (L3) (Locally Lipschitz in u): Φ(u; y) K. Φ(u 1 ; y) Φ(u 2, y) L u 1 u 2 X. (L4) (Continuity in y): There is a positive and non-decreasing function f 2 : R + R + so that Φ(u; y 1 ) Φ(u, y 2 ) Cf 2 ( u X ) y 1 y 2 Y. a Stuart, Inverse problems: a Bayesian perspective. b T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors. arxiv preprint: / 53

19 Well-posedness: existence and uniqueness (L1) (Bounded from below) Φ(u; y) M log (f 1 ( u X )). (L2) (Locally bounded from above) Φ(u; y) K. (L3) (Locally Lipschitz) Φ(u 1 ; y) Φ(u 2, y) L u 1 u 2 X. Existence and uniqueness (BH,216) Let Φ satisfy Assumptions L1 L3 with a function f 1 1, then the posterior µ y is well-defined if f 1 ( X ) L 1 (X, µ ). Example: If y = G(u) + η, η N (, Σ) then Φ(u; y) = 1 2 G(u) y 2 Σ and so M = and f 1 = 1 since Φ. 19 / 53

20 Well-posedness: stability (L1) (Lower bound) Φ(u; y) M log (f 1 ( u X )). (L2) (Locally bounded from above) Φ(u; y) K. (L4) (Continuity in y) Φ(u; y 1 ) Φ(u, y 2 ) Cf 2 ( u X ) y 1 y 2 Y. Total variation stability (BH,216) Let Φ satisfy Assumptions L1, L2 and L4 with functions f 1, f 2 and let µ y and µ y be two posterior measures for y and y Y. If f 2 ( X )f 1 ( X ) L 1 (X, µ ) then there is C > such that d T V (µ y, µ y ) C y y Y. Hellinger stability (BH,216) If the stronger condition (f 2 ( X )) 2 f 1 ( X ) L 1 (X, µ ) is satisfied then there is C > so that d H (µ y, µ y ) C y y Y. 2 / 53

21 The case of additive noise models let Y = R m, η N (, Σ) and suppose y = G(u) + η. Φ(u; y) = 1 2 G(u) y 2 Σ. Φ(u; y) thus (L1) is satisfied with f 1 = 1 and M =. Well-posedness with additive noise models (BH,216) Let the forward map G satisfy: (i) (Bounded) There is a positive and non-decreasing function f 1 so that G(u) Σ C f( u X ) u X. (ii) (Locally Lipschitz) G(u 1 ) G(u 2 ) Σ K u 1 u 2 X. Then the problem of finding µ y is well-posed if f( X ) L 1 (X, µ ). 21 / 53

22 The case of additive noise models Example: polynomially bounded forward map Consider the additive noise model when Y = R m, η N (, I). Then Φ(u; y) = 1 2 G(u) y 2 2. If G is locally Lipschitz, G(u) 2 C max{1, u p X } and p N then we have well-posedness if µ has bounded moments of degree p. In particular, if G is bounded and linear then it suffices for µ to have bounded moment of degree one. Recall the deconvolution example! Example: Gaussian priors In the setting of the above example, if µ is a centered Gaussian then it follows from Fernique s theorem that we have well-posedness if G(u) 2 C exp(α u X ) for any α >. 22 / 53

23 Outline (i) General theory of well-posed Bayesian inverse problems. (ii) Convex prior measures (µ has exponential tails). (iii) Models for compressible parameters. (iv) Infinitely divisible prior measures. 23 / 53

24 From convex regularization to convex priors Let X = R n and Y = R m. Common variational formulation for inverse problems { } 1 u = arg min v R n 2 G(v) y 2 Σ + R(v) R(v) = θ 2 Lv 2 2 (Tikhonov), R(v) = θ Lv 1 (Sparsity). Bayesian analog dµ y dλ Λ Lebesgue measure. ( (v) exp 1 ) 2 G(v) y 2 Σ exp ( R(v)). }{{} prior } {{ } Likelihood A random variable with a log-concave Lebesgue density is convex. 24 / 53

25 Convex priors Gaussian, Laplace, Logistic, etc. l 1 regularization corresponds to Laplace priors. dµ y ( (v) exp 1 ) dλ 2 G(v) y 2 Σ exp ( v 1 ). ( exp 1 ) 2 G(v) n y 2 Σ exp ( v j ) j=1 Definition: Convex measure 4 A Radon probability measure ν on X is called convex whenever it satisfies the following inequality for β [, 1] and Borel sets A, B X. ν(βa + (1 β)b) ν(a) β ν(b) 1 β 4 C. Borell. Convex measures on locally convex spaces. In: Arkiv för Matematik 12.1 (1974), pp / 53

26 Convex priors Convex measures have exponential tails 5 Let ν be a convex measure on X. If X < ν-a.s. then there exists a constant κ > so that X exp(κ u X)dν(u) <. Well-posedness with convex priors (BH & NN, 216) Let the prior µ be a convex measure assume where G is locally Lipschitz and Φ(u; y) = 1 2 G(u) y 2 Σ G(u) Σ C max{1, u p X }, for p N. Then we have a well-posed Bayesian inverse problem. 5 Borell, Convex measures on locally convex spaces. 26 / 53

27 Constructing convex priors Product prior (BH & NN, 216) Suppose X has an unconditional and normalized Schauder basis {x k }. (a) Pick a fixed sequence {γ k } l 2. (b) Pick a sequence of centered, real valued and convex random variables {ξ k } so that Varξ k < uniformly. (c) Take µ to be the law of u γ k ξ k x k. k=1 X <, µ -a.s. and X L 2 (X, µ ). The ξ k are convex then so is µ. Reminiscent of Karhunen-Loève expansion of Gaussians. u γ k ξ k x k, ξ k N (, 1). k=1 {γ k, x k } eigenpairs of covariance operator. 27 / 53

28 Returning to deconvolution ϕ u u t k Example: Deconvolution Let X = L 2 (T) and assume Φ(u; y) = 1 2 G(u) y 2 2 where G(u) = S(ϕ u). Here ϕ C (T) and S : C(T) R m collects point values of a function at m distinct points {t j }. We will construct a convex prior that is supported on B s pp(t) 28 / 53

29 Example: deconvolution with a Besov type prior Let {x k } be an r-regular wavelet basis for L 2 (T). For s < r, p 1 define the Besov space B s pp(t) B s pp(t) := { w L 2 (T) : } k (sp 1/2) w, x k p < k=1 The prior µ is the law of u k=1 γ kξ k x k. ξ k are Laplace random variables with Lebesgue density 1 2 exp( t ). γ k = k ( 1 2p +s). 6 M. Lassas, E. Saksman, and S. Siltanen. Discretization-invariant Bayesian inversion and Besov space priors. In: Inverse Problems and Imaging 3.1 (29), pp T. Bui-Thanh and O. Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. In: Inverse Problems and Imaging 9.1 (215), pp / 53

30 ξ k γ k k u(t) t.1.1 ξ k γ k k u(t) t Example: deconvolution with a Besov type prior B s pp (T) < µ -a.s. and µ is a convex measure. Forward map is bounded and linear. Problem is well-posed. 8 8 M. Dashti, S. Harris, and A. M. Stuart. Besov priors for Bayesian inverse problems. In: Inverse Problems and Imaging 6.2 (212), pp / 53

31 Outline (i) General theory of well-posed Bayesian inverse problems. (ii) Convex prior measures. (iii) Models for compressible parameters. (iv) Infinitely divisible prior measures. 31 / 53

32 Models for compressibility A common problem in compressed sensing p = 1, problem is convex. u 1 = arg min v R n 2 Av y θ v p p. p < 1, no longer convex but a good model for compressibility. Bayesian analog dµ y ( (v) exp 1 ) dλ 2 Av n y 2 2 exp ( θ v j p ). j=1 32 / 53

33 Models for compressibility p = 1. p = 1/2. 33 / 53

34 Models for compressibility Symmetric generalized gamma prior for < p, q 1 dµ n dλ (v) v j p 1 exp ( v j q ). j=1 Corresponding posterior dµ y (v) exp 1 dλ 2 Av y 2 2 v q q + Maximizer is no longer well-defined. Perturbed variational analog for ɛ > u 1 ɛ = arg min v R n 2 Av y v q q n (p 1) ln( v j ) j=1 n (p 1) ln(ɛ + v j ) j=1 34 / 53

35 Models for compressibility p = 1/2, q = 1 p = q = 1/2 35 / 53

36 Models for compressibility SG(p, q, α) density on the real line. p 2αΓ(q/p) t α p 1 Has bounded moments of all order. ( exp t q) α dλ(t). SG(p, q, α) prior: extension to infinite dimensions (BH,216) Suppose X has an unconditional and normalized Schauder basis {x k }. (a) Pick a fixed sequence {γ k } l 2. (b) {ξ k } is an i.i.d sequence of SG(p, q, α) random variables. (c) Take µ to be the law of u k=1 γ kξ k x k. 36 / 53

37 Returning to deconvolution Example: deconvolution with a SG(p, q, α) prior Let {x k } be the Fourier basis in L 2 (T). Define the Sobolev space H 1 (T) { H 1 (T) := w L 2 (T) : } (1 + k 2 ) w, x k 2 < k=1 The prior µ is the law of u k=1 γ kξ k x k. ξ k are i.i.d. SG(p, q, α) random variables. γ k = (1 + k 2 ) 3/4. 37 / 53

38 ξ k γ k u(t) k t ξ k γ k u(t) k t Example: deconvolution with a SG(p, q, α) prior H 1 (T) < µ -a.s. Forward map is bounded and linear. Problem is well-posed. 38 / 53

39 Outline (i) General theory of well-posed Bayesian inverse problems. (ii) Convex prior measures. (iii) Models for compressible parameters. (iv) Infinitely divisible prior measures. 39 / 53

40 Infinitely divisible priors Definition: infinitely divisible measure (ID) A Radon probability measure ν on X is infinitely divisible (ID) if for each n N there exists a Radon probability measure ν 1/n so that ν = (ν 1/n ) n. ξ is ID if for any n N there exist i.i.d random variables {ξ 1/n k } n k=1 so that ξ = d n k=1 ξ1/n k. SG(p, q, α) priors are ID. Gaussian, Laplace, compound Poisson, Cauchy, student s-t, etc. ID measures have an interesting compressible behavior 9. 9 M. Unser and P. Tafti. An introduction to sparse stochastic processes. Cambridge University Press, Cambridge, / 53

41 Deconvolution Example: deconvolution with a compound Poisson prior Let {x k } be the Fourier basis in L 2 (T). µ is the law of u k=1 γ kξ k x k. ξ k are i.i.d. compound Poisson random variables ξ k ν k j= η jk. ν k are i.i.d Poisson random variables with rate b >. η jk are i.i.d unit normals. γ k = (1 + k 2 ) 3/4. ξ k = with probability e b. 41 / 53

42 Deconvolution ξ k γ k u(t) k t ξ k γ k u(t) k t Example: deconvolution with a compound Poisson prior Truncations are sparse in the strict sense. H1 (T) < a.s. We have well-posedness. 42 / 53

43 Lévy-Khintchine Recall the characteristic function of a measure µ on X ˆµ(ϱ) := exp(iϱ(u))dµ(u) ϱ X. Lévy-Khintchine representation of ID measures X A Radon probability measure on X is infinitely divisible if and only if there exists an element m X, a (positive definite) covariance operator Q : X X and a Lévy measure λ, so that ˆµ(ϱ) = exp(ψ(ϱ)) ψ(ϱ) = iϱ(m) }{{} point mass 1 2 ϱ(q(ϱ)) }{{} Gaussian + X ID(m, Q, λ). If λ is a symmetric probability measure on X exp(i(ϱ(u)) 1 iϱ(u)1 BX (u)dλ(u). }{{} compound Poisson ID(m, Q, λ) = δ m N (, Q) compound Poisson. 43 / 53

44 Tail behavior of ID measures and well-posedness Tail behavior of ID is tied to the tail behavior of the Lévy measure λ Moments of ID measures Suppose µ = ID(m, Q, λ). If < λ(x) < and X < µ-a.s. then X L p (X, µ) whenever X L p (X, λ) for p [1, ). Well-posedness with ID priors (BH,216) Suppose µ = ID(m, Q, λ), < λ(x) < and take Φ(u; y) = 1 2 G(u) y 2 Σ. If max{1, p X } L1 (X, λ) for p N and G is locally Lipschitz so that G(u) X C max{1, u p X }, then we have a well-posed Bayesian inverse problem. 44 / 53

45 Deconvolution once more Example: deconvolution with a BV prior Consider the deconvolution problem on T. Stochastic process u(t) for t (, 1) defined via ( ) u() =, û t (s) = exp t exp(iξs) 1 dν(ξ). R ν is a symmetric measure and ξ dν(ξ) <. ξ 1 Pure jump Lévy process. Similar to the Cauchy difference prior 1. 1 M. Markkanen et al. Cauchy difference priors for edge-preserving Bayesian inversion with an application to X-ray tomography. arxiv preprint: / 53

46 Deconvolution once more 5 5 u(t) u(t) t t Example: deconvolution with a BV prior u has countably many jump discontinuities. u BV (T) < a.s. 11 µ is the measure induced by u(t). BV is non-separable. Forward map is bounded and linear. Well-posed problem. 11 R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall/CRC Financial mathematics series. CRC press LLC, New York, / 53

47 Closing remarks Well-posedness can be achieved with relaxed conditions. Gaussians have serious limitations in terms of modelling. Many different priors to choose from. 47 / 53

48 True signal Measurements Posterior standard deviation Posterior mean Closing remarks Sampling. Random walk Metropolis-Hastings for self-decomposable priors. Randomize-then-optimize 12. Fast Gibbs sampler n=32 n=64 n=128 n=256 n= x (a) Posterior mean n=32 n=64 n=128 n=256 n= x (b) Posterior standard deviation x Fig Example B: Variation in the posterior mean and posterior standard deviation with parameter dimension 12 n. Hyperparameter λ is fixed to 32. Z. WangFig. et4.5. al. Example Bayesian B: True signal and inverse noisy measurements. problems with l 1 priors: a Randomize-then-Optimize 4.2. Two-dimensional elliptic PDE inverse problem. Our next numerical example is an approach. arxiv preprint: slightly. 13 elliptic PDE coefficient inverse problem on a two-dimensional domain. The forward model maps the This is an important and encouraging result, as it is evidence of discretization log-conductivity invariancefield of the Poisson equation to observations of the potential field, not only in F. thelucka. problem formulation, Fast but Gibbs in the performance sampling of the for transformed high-dimensional RTO-MH sampling Bayesian inversion. (exp{θ(x)} s(x)) = h(x), x [, 1] scheme. Finally, as we increase the hyperparameter λ, the CM becomes smoother and the posterior 2, arxiv: standard deviation decreases, as in shown Figure 4.7. The sampling efficiency of our where algorithm θ is the also log-conductivity, s is the potential, and h is the forcing function. Neumann boundary deteriorates with increasing λ, as shown in Table 4.3. Overall, the results from these conditions parameter 48 / 53

49 Closing remarks Analysis of priors: What constitutes compressibility? What is the support of the prior? Hierarchical priors. Modelling constraints. 49 / 53

50 Thank you B. Hosseini. Well-posed Bayesian inverse problems with infinitely-divisible and heavy-tailed prior measures. arxiv preprint: B. Hosseini and N. Nigam. Well-posed Bayesian inverse problems: priors with exponential tails. arxiv preprint: / 53

51 References [1] C. Borell. Convex measures on locally convex spaces. In: Arkiv för Matematik 12.1 (1974), pp [2] T. Bui-Thanh and O. Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. In: Inverse Problems and Imaging 9.1 (215), pp [3] R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall/CRC Financial mathematics series. CRC press LLC, New York, 24. [4] M. Dashti, S. Harris, and A. M. Stuart. Besov priors for Bayesian inverse problems. In: Inverse Problems and Imaging 6.2 (212), pp [5] B. Hosseini. Well-posed Bayesian inverse problems with infinitely-divisible and heavy-tailed prior measures. arxiv preprint: [6] B. Hosseini and N. Nigam. Well-posed Bayesian inverse problems: priors with exponential tails. arxiv preprint: [7] B. Hosseini and J. M. Stockie. Bayesian estimation of airborne fugitive emissions using a Gaussian plume model. In: Atmospheric Environment 141 (216), pp [8] B. Hosseini et al. A Bayesian approach for energy-based estimation of acoustic aberrations in high intensity focused ultrasound treatment. arxiv preprint: [9] M. Lassas, E. Saksman, and S. Siltanen. Discretization-invariant Bayesian inversion and Besov space priors. In: Inverse Problems and Imaging 3.1 (29), pp [1] F. Lucka. Fast Gibbs sampling for high-dimensional Bayesian inversion. arxiv: [11] M. Markkanen et al. Cauchy difference priors for edge-preserving Bayesian inversion with an application to X-ray tomography. arxiv preprint: [12] A. M. Stuart. Inverse problems: a Bayesian perspective. In: Acta Numerica 19 (21), pp [13] T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors. arxiv preprint: [14] M. Unser and P. Tafti. An introduction to sparse stochastic processes. Cambridge University Press, Cambridge, 213. [15] Z. Wang et al. Bayesian inverse problems with l 1 priors: a Randomize-then-Optimize approach. arxiv preprint: / 53

52 Well-posedness Minimal assumptions on Φ (BH, 216) The potential Φ : X Y R satisfies: 1415 (L1) (Lower bound in u): There is a positive and non-decreasing function f 1 : R + [1, ) so that r >, there is a constant M(r) R such that u X and y Y with y Y < r, Φ(u; y) M log (f 1 ( u X )). (L2) (Boundedness above): r > there is a constant K(r) > such that u X and y Y with max{ u X, y Y } < r, Φ(u; y) K. (L3) (Continuity in u): r > there exists a constant L(r) > such that u 1, u 2 X and y Y with max{ u 1 X, u 2 X, y Y } < r, Φ(u 1 ; y) Φ(u 2, y) L u 1 u 2 X. (L4) (Continuity in y): There is a positive and non-decreasing function f 2 : R + R + so that r >, there is a constant C(r) R such that y 1, y 2 Y with max{ y 1 Y, y 2 Y } < r and u X, Φ(u; y 1 ) Φ(u, y 2 ) Cf 2 ( u X ) y 1 y 2 Y. 15 Stuart, Inverse problems: a Bayesian perspective. 52 / 53

53 The case of additive noise models Well-posedness with additive noise models Consider the above additive noise model. In addition, let the forward map G satisfy the following conditions with a positive, non-decreasing and locally bounded function f 1: (i) (Bounded) There is a constant C > for which G(u) Σ C f( u X ) u X. (ii) (Locally Lipschitz) r > there is a constant K(r) > so that for all u 1, u 2 X and max{ u 1 X, u 2 X } < r G(u 1 ) G(u 2 ) Σ K u 1 u 2 X. Then the problem of finding µ y is well-posed if µ is a Radon probability measure on X such that f( X ) L 1 (X, µ ). 53 / 53