Inverse Problems in the Bayesian Framework

Transcription

1 Inverse Problems in the Bayesian Framework Daniela Calvetti Case Western Reserve University Cleveland, Ohio Raleigh, NC, July 2016

2 Bayes Formula Stochastic model: Two random variables X R n, B R m, where B is the observed quantity, X is the quantity of primary interest. Goal: Find the posterior probability density, π X (x b) = density of X, given the observation B = b, b = b observed.

3 Bayes Formula Prior information: Given the prior density π X (x), encoding our prior information or prior belief about the possible vaues of X. Likelihood: Assuming that X = x, what would the forward model predict for the value distribution of B? Encode this information in π B (b x). Evidence: With prior and likelihood, compute π B (b) = π XB (x, b)dx = π B (b x)π X (x)dx. R n R n Bayes formula for probability densities π X (x b) = π B(b x)π X (x). π B (b)

4 Linear Inverse Problems Consider the problem of estimating x from b = Ax + e, A R m n. Stochastic extension: Write B = AX + E, and assume the noise and prior model X N (0, D), E N (0, C). Usually it is assumed that X and E are independent. In particular, E { XE T} = E { X } E { E } T = 0.

5 Linear Inverse Problems However, this is not necessary, and we may have E { XE T} = R R n m. Define a new random variable [ X Z = B ] R n+m. Covariance matrix of Z: ZZ T = = [ ] X [ X B T B T ] [ XX T XB T ] BX T BB T. Compute the expectation of this matrix.

6 Linear Inverse Problems Expectations: E { XX T} = D, E { XB T} = E { X (AX + E) T} = E { XX T A T + XE T} = E { XX T} A T + E { XE T} = DA T + R. Furthermore, E { BX T} = E { XB T} T = AD + R T,

7 Linear Inverse Problems and, finally, E { BB T} = E { (AX + E)(AX + E) T} = E { AXX T A T + EX T A T + AXE T + EE T} = ADA T + R T A T + AR + C. Conclusion: [ Cov(Z) = D AD + R T DA T + R ADA T + R T A T + AR + C ] [ Γ11 Γ = 12 Γ 21 Γ 22 ].

8 Schur Complements and Conditioning Given a Gaussian random variable [ ] X Z = R n+m B with covariance Γ R (m+n) (m+n), what is the probability density of X, given B = b?

9 Schur Complements and Conditioning Assume that X N (0, Γ), where Γ R n n is a given SPD matrix. Partitioning of X, X = [ X1 X 2 ] R k R n k. Question: Assume that X 2 = x 2 is observed. What is the conditional probability density of X 1, π X1 (x 1 x 2 ) =?

10 Schur Complements and Conditioning Write π X (x) = π X1,X 2 (x 1, x 2 ). Bayes formula: The distribution of unknown part x 1 provided that x 2 is known, is π X1 (x 1 x 2 ) π X1,X 2 (x 1, x 2 ), x 2 = x 2,observed. In terms of the Gaussian density, π X1,X 2 (x 1, x 2 ) exp ( 1 ) 2 x T Γ 1 x. (1)

11 Schur Complements and Conditioning Partitioning of the covariance matrix: [ ] Γ11 Γ Γ = 12 R n n, (2) Γ 21 Γ 22 where and Γ 11 R k k, Γ 22 R (n k) (n k), k < n, Γ 12 = Γ T 21 R k (n k).

12 Schur Complements and Conditioning Precision matrix B = Γ 1. Partition B: [ B11 B B = 12 B 21 B 22 ] R n n. (3) Quadratic form x T Bx appearing in the exponential: [ ] [ ] [ B11 B Bx = 12 x1 B11 x = 1 + B 12 x 2 B 21 B 22 x 2 B 21 x 1 + B 22 x 2 ],

13 Schur Complements and Conditioning x T Bx = [ x1 T x2 T ] [ ] B 11 x 1 + B 12 x 2 B 21 x 1 + B 22 x 2 = x1 T ( ) ( ) B11 x 1 + B 12 x 2 + x T 2 B21 x 1 + B 22 x 2 = x1 T B 11 x 1 + 2x1 T B 12 x 2 + x2 T B 22 x 2 = ( x 1 + B 1 11 B ) TB11 ( 12x 2 x1 + B 1 11 B ) 12x 2 + x2 T ( B22 B 21 B 1 11 B 12) x2. }{{} independent of x 1

14 Schur Complements and Conditioning From this key equation for conditional densities it follows that ( π X1 (x 1 x 2 ) exp 1 ( x1 + B B ) TB11 ( 12x 2 x1 + B 1 11 B ) ) 12x 2. Thus the conditional density is Gaussian, with mean and covariance matrix x 1 = B 1 11 B 12x 2, C = B Question: How to express these formulas in terms of Γ?

15 Schur Complements and Conditioning Consider a partitioned SPD matrix Γ R n n. For any v R k, x 0 v T Γ 11 v = [ v T 0 ] [ ] [ Γ 11 Γ 12 v T Γ 21 Γ 22 0 ] > 0, showing the positive definiteness of Γ 11. The same holds for Γ 22. In particular, Γ 11 and Γ 22 are invertible.

16 Schur Complements and Conditioning To calculate the inverse of Γ, we solve the equation Γx = y in block form. By partitioning, x = [ x1 x 2 ] R k R n k, y = [ y1 y 2 ] R k R n k. we have Γ 11 x 1 + Γ 12 x 2 = y 1, Γ 21 x 1 + Γ 22 x 2 = y 2.

17 Schur Complements and Conditioning Eliminate x 2 from the second equation, x 2 = Γ 1 ( ) 22 y2 Γ 21 x 1, substitute back into the first equation: Γ 11 x 1 + Γ 12 Γ 1 ( ) 22 y2 Γ 21 x 1 = y1, and by rearranging the terms, ( Γ11 Γ 12 Γ 1 22 Γ ) 21 x1 = y 1 Γ 12 Γ 1 22 y 2. Define the Schur complement of Γ 22 : Γ22 = Γ 11 Γ 12 Γ 1 22 Γ 21

18 Schur Complements and Conditioning It can be shown that Γ 22 must be invertible, and therefore x 1 = 1 Γ 22 y 1 1 Γ 22 Γ 12Γ 1 22 y 2. Similarly, interchanging the roles of x 1 and x 2, where x 2 = is the Schur complement of Γ Γ 11 y 1 2 Γ 11 Γ 21Γ 1 11 y 1, Γ11 = Γ 22 Γ 21 Γ 1 11 Γ 12 In matrix form: [ ] [ x1 = x 2 Γ 1 22 Γ 1 22 Γ 12Γ 1 22 Γ 1 11 Γ 1 11 Γ 21Γ 1 11 ] [ ] y1 y 2

19 Schur Complements and Conditioning Conclusion: [ Γ 1 = Γ 1 22 Γ 1 22 Γ 12Γ 1 22 Γ 1 11 Γ 1 11 Γ 21Γ 1 11 ] [ B11 B = 12 B 21 B 22 ]. Thus The conditional density π X1 (x 1 x 2 ) is a Gaussian π X1 (x 1 x 2 ) N (x 1, C), where and x 1 = B 1 11 B 12x 2 = Γ 12 Γ 1 22 x 2, C = B 1 11 = Γ 22.

20 Linear Inverse Problems For simplicity, let us assume that R = 0. Posterior density π X (x b) is a Gaussian density, with mean and covariance x = Γ 12 Γ 1 22 b = DAT( ADA T + C ) 1 b, Φ = Γ 22 = Γ 11 Γ 12 Γ 1 22 Γ 21 = D DA T( ADA T + C ) 1 AD.

21 Example: Numerical differentiation Discretization: f (t) = t 0 g(τ)dτ + noise. b = Ax + e, where 1 A = n

22 Stochastic extension Stochastic model B = AX + E. Model for noise: Independent components, Probability density: π E (e) = E j N (0, σ 2 ), 1 j n. ( ) 1 n/2 ( 2πσ 2 exp 1 ) 2σ 2 e 2. Likelihood: ( π B (b x) exp 1 b Ax 2 2σ2 ).

23 Autoregressive Prior Models Discrete model, x j = g(t j ), t j = j n, 0 j n, Consider two possible prior models: 1. We know that x 0 = 0, and believe that the absolute value of the slope of g is bounded by some m 1 > We know that x 0 = x n = 0 and believe that the curvature of g is bounded by some m 2 > 0.

24 Autoregressive models 1. Slope: g (t j ) x j x j 1, h = 1 h n, Prior information: We believe that x j x j 1 h m 1 with some uncertainty. 2. Curvature: g (t j ) x j 1 2x j + x j+1 h 2. Prior information: We believe that x j 1 2x j + x j+1 h 2 m 2 with some uncertainty.

25 Autoregressive model In both cases, we assume that x j is a realization of a random variable X j. Boundary conditions: 1. X 0 = 0 with certainty. Probabilistic model for X j, 1 j n. 2. X 0 = X n = 0 with certainty. Probabilistic model for X j, 1 j n 1.

26 Autoregressive prior models 1. First order prior: X j = X j 1 + γw j, W j N (0, 1), γ = h m Second order prior: X j = 1 2 (X j 1 + X j+1 ) + γw j, W j N (0, 1), γ = 1 2 h2 m 2. x j x j x j+1 x j 1 STD=γ x j 1 STD=γ

27 Matrix form: first order model System of equations: X 1 = X 1 X 0 = γw 1 X 2 X 1 = γw 2.. X n X n 1 = γw n L 1 = Rn n, X = X 1 X 2. X n, W = W 1 W 2. W n. L 1 X = γw, W N (0, I n ),

28 Matrix form: second order model System of equations: X 2 2X 1 = X 2 2X 1 + X 0 = γw 1 X 3 2X 2 + X 1 = γw 2.. 2X n 1 X n 2 = X n 2X n 1 + X n 2 = γw n 1 L 2 = R(n 1) (n 1), X = X 1 X 2., X n 1 L 2 X = γw, W N (0, I n 1 ),

29 Prior density Given a model that is, LX = γw, W N (0, I), π W (w) exp ( 12 ) w 2, we conclude that π X (x) exp ( 1 ) ( 2γ 2 Lx 2 = exp 1 [ ] ) 1 2 x T γ 2 LT L x. The inverse of the covariance matrix = precision matrix is D 1 = 1 γ 2 LT L.

30 Testing a Prior Question: Given a covariance D, how can we check if the prior corresponds to our expectations? Symmetric decomposition of the precision matrix (let γ = 1 for simplicity): D 1 = L T L. We know that W = LX N (0, I). Sampling of X : 1. Draw a realization w N (0, I n ) 2. Set x = L 1 w.

31 Random draws from priors Generate m draws from the prior using the Matlab command randn. n = 100; t = (0:1/n:1); m = 5; % number of discretization intervals % number of draws % First order model. Boundary condition X_0 = 0 L1 = diag(ones(1,n),0) - diag(ones(1,n-1),-1); gamma = 1/n; % m_1 = 1 W = gamma*randn(n,m); X = L1\W;

32 Plots of the random draws

33 Belief envelopes Diagonal elements of the posterior covariance: Γ jj = ηj 2 = posterior variance of X j. Posterior belief: x j 2η j < X j < x j + 2η j with posterior probability 95%.

34 Bayesian solution Mean solution and 2 STD belief envelope

35 Linear Inverse Problems An alternative (but equivalent) formula for the posterior: Prior, ( π X (x) exp 1 ) 2 x T D 1 x, and likelihood, Posterior density: π B (b x) exp ( 1 ) 2 (b Ax)T C 1 (b Ax). π X (x b) π X (x)π B (b x) exp ( 12 Q(x) ),

36 Linear Inverse Problems Quadratic term in the exponential: Q(x) = (b Ax) T C 1 (b Ax) + x T D 1 x Collect the terms of the same order in x together: Q(x) = x T ( A T C 1 A + D 1) x 2x T A T C 1 b + b T C 1 b. }{{} =M Complete the square: Q(x) = (x T M 1 A T C 1 b) T M(x T M 1 A T C 1 b) +...

37 Linear Inverse Problems Conclusion: The posterior mean and covariance have alternative expressions, x = ( A T C 1 A + D 1) 1 A T C 1 b and Φ = ( A T C 1 A + D 1) 1. The formula for x is also known as Wiener filtered solution.

38 Tikhonov Regularization Revisited Consider the linear model b = Ax + e, e N (0, C). Likelihood: π B (b x) exp Assume a Gaussian prior: or, in terms of densities, π X (x) exp ( 1 ) 2 (b Ax)T C 1 (b Ax). X N (0, D), ( 1 ) 2 x T D 1 x.

39 Tikhonov Regularization Revisited Bayes formula: p X (x b) p X (x)p B (b x) ( = exp 1 2 x T D 1 x 1 ) 2 (b Ax)T C 1 (b Ax). By writing D 1 = L T L, the negative of the exponent is H(x) = 1 ( b Ax 2 2 C + Lx 2), a Tikhonov functional.

40 Tikhonov Regularization Revisited Therefore, x MAP = argmax { π X (x b) } = argmin { b Ax 2 C + Lx 2}. The Tikhonov regularization parameter is absorbed in the prior covariance matrix as well as the noise covariance matrix.

41 Non-Gaussian models The Gaussian models are insufficient when the forward model is non-linear, the prior is non-gaussian, the noise is non-additive, the noise is non-gaussian.

42 Monte Carlo Integration Assume that a probability density π X is given in R n. Problem: Estimate numerically an integral of type E { f (X ) } = f (x)π X (x)dx =? R n Expectation of X : f (x) = x. Covariance of X : f (x) = (x x)(x x) T.

43 Monte Carlo Integration Difficulties with numerical integration using quadrature methods: We may not know the support of µ X (Support: The set in which the function is not vanishing). Where should we put our quadrature points? If n is large, an integration grid becomes huge: K points/direction means K n grid points. Try Monte Carlo integration!

44 Monte Carlo Integration Example: Given a two-dimensional set Ω R 2. Estimate the area of Ω. Raindrop integration: Assume that Ω Q = [0, a] [0, b]. Draw points from uniform density over Q: { x 1, x 2,..., x N}, x j Uniform(Q). Estimate of the area Ω : solve for Ω. Ω Q = Ω ab # of points x j Ω, N

45 Monte Carlo Integration The approximation corresponds to Monte Carlo integral Ω Q = 1 χ Ω (x)dx 1 Q Q N N χ Ω (x j ), j=1 where χ Ω (x) = { 1 if x Ω 0 ifx / Ω and 1/N is the equal weight that every point x j has. 1 Q χ Q(x) = uniform density over Q

46 Monte Carlo Integration Generalize: Given a probability density π X, write f (x)π X (x)dx 1 R n N N f (x j ), j=1 where { x 1, x 2,..., x N} is drawn independently from the probability distribution π X. Problem: How does one draw from a probability density in R n?

47 Sampling and Markov chains: Random walk Random walk is a process of moving around by taking random steps. Most elementary random walk: 1. Start at a point of your choice x 0 R n. 2. Draw a random vector w 1 N (0, I ) and set x 1 = x 0 + σw Repeat the process: Set x k+1 = x k + σw k+1, w k+1 N (0, I ).

48 Sampling and Markov chains: Random walk In terms of random variables: X k+1 = X k + σw k+1, W k+1 N (0, I n ). The conditional density of X k+1, given X k = x k is π(x k+1 x k ) = ( 1 (2πσ 2 exp 1 ) ) n/2 2σ 2 x k x k+1 2 = q k (x k, x k+1 ). The function q k is called the kth transition kernel.

49 Sampling and Markov chains: Random walk Since q 0 = q 1 = q 2 =..., i.e., the step is always equally distributed, we call the random walk time invariant. (k = time). The chain { Xk, k = 0, 1, } of random variables, is called discrete time stochastic process. Particular feature: the probability distribution X k depends of the past only through the previous member X k 1 : π(x k+1 x 0, x 1,..., x k ) = π(x k+1 x k ). A stochastic process having this property is called a Markov chain.

50 Sampling and Markov chains: Random walk Given: an arbitrary transition kernel q, a random variable X with probability density π X (x) = p(x), generate a new random variable Y by using the kernel q(x, y), that is, π(y x) = q(x, y). Question: What is the probability density of this new variable? The answer is found by marginalization, π Y (y) = π(y x)π X (x)dx = q(x, y)p(x)dx.

51 Sampling and Markov chains: Random walk If the probability density of the new variable is equal to the one of the old one, that is, q(x, y)p(x)dx = p(y), p is called an invariant density of the transition kernel q. The classical problem in the theory of Markov chains is: Given a transition kernel, find the corresponding invariant density.

52 Invariant density and sampling Recall the sampling problem: Given a probability density p = p(x), generate a sample that is distributed according to it. If we had a transition kernel q with invariant density p, generating such sample from p(x) would be easy: Start with some x 0 ; draw x 1 from q(x 0, x 1 ); In general, given x k, draw x k+1 from q(x k, x k+1 ). Rephrasing the sampling problem: Given a probability density p, find a kernel q such that p is its invariant density.

53 Metropolis Hastings algorithm Given a transition density y K(x, y), x R n current point, consider a Markov process: if x R n is the current point, we have two possibilities: 1. Stay at x with probability r(x), 0 r(x) < 1, 2. Move by using a transition kernel K(x, y).

54 Let x and y be realizations of random variables X, Y. π X (x) = p(x), and y is generated according to the algorithm above. Question: What is the probability density of Y?

55 Let A be the event that we opt for moving from x, A be the event of staying put. The probability of Y B R n assuming a move is P { Y B X = x, A } = K(x, y)dy. The kernel K is scaled so that P { X = x, A } = P { Y R n X = x, A } = K(x, y)dy = 1 r(x). R n (4) B

56 On the other hand, if we stay put, Y B happens only if X B, P { Y B X = x, A } = r(x)χ B (x) = where χ B is the characteristic function of B. { r(x), if x B, 0, if x / B,

57 Hence, the total probability of arriving from x to B is P { Y B X = x } = P { Y B X = x, A } + P { Y B X = x, A } = K(x, y)dy + r(x)χ B (x). B

58 Marginalize over x and calculate the probability of Y B P { Y B } = P { Y B X = x } p(x)dx = = = B B ( p(x) ( ( B ) K(x, y)dy dx + χ B (x)r(x)p(x)dx ) p(x)k(x, y)dx dy + r(x)p(x)dx B ) p(x)k(x, y)dx + r(y)p(y) dy.

59 Since P { Y B } = B π(y)dy, we must have π Y (y) = p(x)k(x, y)dx + r(y)p(y). Our goal is then to find a kernel K such that π Y (y) = p(y), that is p(y) = p(x)k(x, y)dx + r(y)p(y), or, equivalently, (1 r(y))p(y) = p(x)k(x, y)dx.

60 Substituting (4) in this formula, with the roles of x and y interchanged, we obtain p(y)k(y, x)dx = p(x)k(x, y)dx This equation is called the balance equation. This holds, in particular, if the integrands are equal, p(y)k(y, x) = p(x)k(x, y). The latter equation is known as detailed balance equation.

61 Metropolis-Hastings algorithm Start by a selecting a proposal distribution, or candidate generating kernel q(x, y); The kernel should be chosen so that generating a Markov chain with it is easy. A Gaussian kernel is a popular choice.

62 Metropolis-Hastings algorithm If q satisfies the detailed balance equation, i.e., p(y)q(y, x) = p(x)q(x, y), we are done, since p is an invariant density. More likely, the equality does not hold. If p(y)q(y, x) < p(x)q(x, y). (5) force the detailed balance equation to hold, defining K as where α is chosen so that K(x, y) = α(x, y)q(x, y), p(y)α(y, x)q(y, x) = p(x)α(x, y)q(x, y).

63 Metropolis-Hastings algorithm The kernel α need not be symmetric, so let α(y, x) = 1. Now the other factor is uniquely determined. We must have α(x, y) = p(y)q(y, x) p(x)q(x, y) < 1. Observe that if the inequality (5) goes the other way, interchange the roles of x and y, and let α(x, y) = 1. In summary K(x, y) = α(x, y)q(x, y), { } p(y)q(y, x) α(x, y) = min 1,. p(x)q(x, y)

64 Metropolis-Hastings algorithm Draws in two phases: 1. Given x, draw y using the transition kernel q(x, y). 2. Calculate the acceptance ratio, α(x, y) = p(y)q(y, x) p(x)q(x, y). 3. Flip the α coin: draw t Uniform([0, 1]); if α > t, accept y, otherwise stay where you are.

65 Metropolis-Hastings algorithm: Random walk proposal If q(x, y) = q(y, x), the algorithm simplifies: 1. Given x, draw y using the transition kernel q(x, y). 2. Calculate the acceptance ratio, α(x, y) = p(y) p(x). 3. Flip the α coin: draw t Uniform([0, 1]); if α > t, accept y, otherwise stay where you are.

66 Random walk Metropolis-Hastings 1. Set sample size N. Pick initial point x 1. Set k = Propose a new point, y = x k + δw, w N (0, I). 3. Compute the acceptance ratio, α = π(y) π(x k ). 4. Flip α-coin: Draw ξ Uniform([0, 1]), 4.1 If α ξ, accept: x k+1 = y, 4.2 If α < ξ, stay put: x k+1 = x k. 5. If k < N, increase k k + 1 and continue from 2., else stop.

67 Two steps Build the program in two steps: 1. Random walk sampling, no rejections 2. Add the rejection step

68 Sampling, no rejections nsample = 10000; % Sample size x = [0;0]; % Initial point step = 0.1; % Step size of the random walk Sample = NaN(2,nsample); % For memory allocation Sample(:,1) = x; for j = 2:N y = x + step*randn(2,1); end % Accept unconditionally x = y; Sample(:,j) = x;

69 Add the α-coin Write the condition in logarithmic form: π(y) π(x) > t log π(y) log π(x) > log t,

70 x = [0;0]; % Initial point step = 0.1; % Step size of the random walk Sample = NaN(2,nsample); % For memory allocation Sample(:,1) = x; logpx =... for j = 2:N y = x + step*randn(2,1); logpy =... t = rand; if logpy - logpx > log(t) % accept x = y; logpx = logpy; end Sample(:,j) = x; end

71 If a move is not accepted, the previous point has to be repeated:..., x k 1, x k, x k, x k, x k+1, x k+2,... }{{} rejections The acceptance rate tells the relative rate of acceptances. Too low acceptance rate: The chain does not move Too high acceptance rate: The chain is essentially a random walk and learns nothing of the underlying distribution (cf. raising kids). What is a good acceptance rate? Rule of thumb: 15%-35%.

72 Example: Inverse problem in chemical kinetics Reversible single reaction pair, A k 1 B, k 1 Data: With known initial values, measure [A](t j ), 1 j n for The noisy observation model is t min = t 1 < t 2 < < t n = t max. b j = [A](t j ) + e j, e j = additive noise, noise level =σ. Inverse Problem: Estimate k 1 and k 1.

73 Forward model: Mass Balance Equations Denote Assuming unit volume, c 1 (t) = [A](t), c 2 (t) = [B](t). or where dc 1 dt dc 2 dt = k 1 c 1 + k 1 c 2, c 1 (0) = c 01 = k 1 c 1 k 1 c 2, c 2 (0) = c 02 [ dc dt = Kc, c = c1 c 2 [ k1 k K = 1 k 1 k 1 ], ].

74 Eigenvalues of K are Time constant Eigenvectors: v 1 = λ 1 = 0, λ 2 = k 1 k 1. [ 1 δ τ = ] [, v 2 = 1 k 1 + k ], δ = k 1 k 1. Solution: c = αv 1 + βv 2 e t/τ, α, β R.

75 Initial conditions imply In particular, α = c 01 + c δ, β = δc 01 c δ where c 1 (t) = f (t; k) = c 01 + c δ k = [ k1 k 1 δc 01 c 02 e t/τ, 1 + δ ].

76 Observation model: Data consists of n measurements of c 1 (t) corrupted by additive noise, b j = f (t j, k) + e j, 1 j n. Observation errors e j mutually independent, zero mean normally distributed, e j N (0, σ 2 ).

77 Likelihood density Assume that the noise is Gaussian white noise: ( π noise (e) exp 1 ) 2σ 2 e 2. Likelihood density is π(b k) exp 1 2σ 2 n (b j f (t j, k)) 2. j=1

78 Posterior density Flat prior over an interval: We believe that 0 < k 1 K 1, 0 < k 1 K 1, with some reasonable upper bounds. Write π prior (k) χ [0,K1 ](k 1 )χ [0,K 1 ](k 1 ). Posterior density by Bayes formula, π(k b) π prior (k)π(b k). Contour plots of the posterior density?

79 Data K 1 = 6, K 1 = 2, five uniformly distributed measurements of c 1 (t) over the interval [t min, t max ], t min = 0.1 τ, t max = 4.1 τ (left), t min = 5 τ, t max = 9 τ (right)

80 Posterior densities K 1 = 6, K 1 = 2, five uniformly distributed measurements of c 1 (t) over the interval [t min, t max ], t min = 0.1 τ, t max = 4.1 τ (left), t min = 5 τ, t max = 9 τ (right) The hair cross indicates the value used for data generation.

81 MCMC exploration Generate the data: Define t 1 t 2 t =. t n, b = b 1 b 2. b n, where b j = c 01 + c δ true δ truec 01 c δ true e t/τtrue + e j. δ true = k 1,true k 1,true, τ true = 1 k 1,true + k 1,true.

82 Random walk Metropolis-Hastings Start with the transient measurements. White noise proposal, k prop = k + δw, w N (0, I ). Choose first δ = 0.1, different initial points k 0 = (1, 2) or k 0 = (5, 0.1). Acceptance rates with these values are of the order 45%.

83 Scatter plots Transient measurements, t min = 0.1 τ, t max = 4.1 τ

84 Sample histories:first component Initial value k 1 = 1 (left) and k 1 = 5 (right)

85 Sample histories: Second component Initial value k 2 = 2 (left) and k 2 = 0.2 (right)

86 Burn-in: first component Initial value k 1 = 1 (left) and k 1 = 5 (right)

87 Burn-in: Second component Initial value k 2 = 2 (left) and k 2 = 0.2 (right)

88 Scatter plots: Steady state measurement t min = 5 τ, t max = 9 τ. Use the same step size as before. Initial point (k 1, k 2 ) = (1, 2).

89 Sample histories, a.k.a. fuzzy worms Increase the step size Initial point (k 1, k 2 ) = (1, 2). Acceptance remains high, about 55%

90 Scatter plots, steady state data Increase the step size Initial point (k 1, k 2 ) = (1, 2). Acceptance remains high, about 55%

91 Fuzzy worms