Bayesian techniques for fatigue life prediction and for inference in linear time dependent PDEs

Transcription

1 Bayesian techniques for fatigue life prediction and for inference in linear time dependent PDEs Marco Scavino SRI Center for Uncertainty Quantification Computer, Electrical and Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology, KSA Universidad de la República, Montevideo, Uruguay SRI UQ Center CEMSE KAUST 4 th Annual Workshop 2016 January 8, KAEC, Kingdom of Saudi Arabia 1 / 54

2 Overview Introduction Statistical models Bayesian approach Model comparison Conclusions References Linear Parabolic PDEs with Noisy Boundary Conditions Introduction Formulation of the problem Statistical setting The marginal likelihood of θ Example - inference for thermal diffusivity 2 / 54

3 Introduction Bayesian inference and model comparison for metallic fatigue data Joint work ( with Ivo Babuška ICES, The University of Texas at Austin, Austin, USA, Zaid Sawlan CEMSE, King Abdullah University of Science and Technology, Thuwal, KSA, Barna Szabó Washington University in St. Louis, St. Louis, USA, Raúl Tempone CEMSE, King Abdullah University of Science and Technology, Thuwal, KSA, 3 / 54

4 Introduction Mechanical and structural components subjected to cyclic loading are susceptible to cumulative damage and eventual failure through an irreversible process called metal fatigue. Goal To predict the fatigue life of materials by providing a systematic approach to model calibration, model selection and model ranking with reference to stress-life (S-N) data drawn from a collection of records of fatigue experiments. 4 / 54

5 Introduction The 75S-T6 aluminum sheet specimens data set Data are available on a collection of records of fatigue tests conducted at the Battelle Memorial Institute on 85 75S-T6 aluminum sheet specimens by means of a Krouse direct repeated-stress testing machine [1, table 3, pp.22 24]. The following data are recorded for each specimen: the maximum stress, S max, measured in ksi units. the test ratio, R, defined as the minimum to maximum stress ratio. the fatigue life, N, defined as the number of load cycles at which fatigue failure occurred. a binary variable (0/1) to denote whether or not the test had been stopped prior to the occurrence of failure (run-out). 5 / 54

6 Introduction The 75S-T6 aluminum sheet specimens data set (cont.) In 12 of the 85 experiments, the specimens remained unbroken when the tests were stopped. Specimen Max. Stress (ksi) Load cycles Test ratio Type B24M1 80,0 2,478, B81M4 75,0 10,538, B91M3 80,5 14, B95M4 80,5 71, B94M1 80,5 68, B93M1 80,5 99, B15M2 79,0 162, B23M4 79,0 181, B19M2 75,0 58, B19M3 70,0 88, B39M4 70,0 432, B16M1 65,0 10,780, B19M1 60,0 10,780, B35M3 65,0 89, / 54...

7 Statistical models Statistical models for metallic fatigue data In the Metallic Materials Properties Development and Standardization (MMPDS-01) Handbook, 2003, Battelle Memorial Institute, it is considered the so-called logarithmic fit using the objective function, µ(s eq (lg) ) = A 1 + A 2 log 10 (S eq (lg) A 3 ), e std = ( n ) i=1 (log 10(n i ) µ(s eq (lg) 1/2 )) 2, n p where n is the number of data points and p is the number of fitting parameters (namely A 1, A 2, A 3 and q). 7 / 54

8 Statistical models Statistical models for metallic fatigue data (cont.) The resulting estimated mean value function, without the run-outs, is given by µ(s eq (lg) ) = log 10 (S eq (lg) 31.53), where S eq (lg) = S max (1 R) and the value of the objective function is e std = The estimated fatigue limit is equal to 31.53/( ) = ksi, since the fatigue limit is the value of the maximum stress when the test ratio, R, is equal to 1 (the fully reversed condition). 8 / 54

9 Statistical models Statistical models for metallic fatigue data (cont.) Estimated 0.95 Quantile Function Estimated 0.05 Quantile Function Estimated Median Function Fatigue Limit Parameter = ksi 80 S eq (ksi) Cycles Figure 1 : Logarithmic fit of the 75S-T6 data set without run-outs. 9 / 54

10 Statistical models We consider fatigue-limit models [2] and random fatigue-limit models [3] that are specially designed to allow the treatment of the run-outs (right-censored data). Fatigue data obtained for particular test ratios need to be generalized to arbitrary test ratios. To this purpose the equivalent stress S eq is defined as S eq (q) = S max (1 R) q, where q is a fitting parameter. We distinguish between the fatigue limit, which is a physical notion, and the fatigue limit parameter, which is an unknown parameter, expressed in the same scale as the equivalent stress and calibrated for different models. 10 / 54

11 Statistical models Model Ia Let A 3 be the fatigue limit parameter. At each equivalent stress with S eq > A 3, the fatigue life N is modeled with a lognormal distribution: log 10 (N) N (µ(s eq ), σ(s eq )). Assume that µ(s eq ) = A 1 + A 2 log 10 (S eq A 3 ), if S eq > A 3 σ(s eq ) = τ. Given the sample data, n = (n 1,..., n m ), we consider the likelihood function L(A 1, A 2, A 3, τ, q; n), given by m [ ] g(log10 (n i ) ; µ(s eq ), σ(s eq )) δi [ ( )] log10 (n i ) µ(s eq ) 1 δi 1 Φ, n i log(10) σ(s eq ) i=1 11 / 54

12 Statistical models Model Ia (cont.) } where g(t; µ, σ) = 1 2π σ exp { (t µ)2, Φ is the cumulative 2σ 2 distribution function of the standard normal distribution, and δ i = { 1 if ni is a failure 0 if n i is a run-out. 12 / 54

13 Statistical models Model Ia (cont.) Estimated 0.95 Quantile Function Estimated 0.05 Quantile Function Estimated Median Function Fatigue Limit Parameter = ksi 80 S eq (ksi) Cycles Figure 2 : A 1 = 7.38, A 2 = 2.01, A 3 = 35.04, q = 0.563, τ = / 54

14 Statistical models Model Ib We extend the Model Ia by allowing a non-constant standard deviation: µ(s eq ) = A 1 + A 2 log 10 (S eq A 3 ), if S eq > A 3 σ(s eq ) = 10 (B 1+B 2 log 10 (S eq)), if S eq > A / 54

15 Statistical models Model Ib (cont.) Estimated 0.95 Quantile Function Estimated 0.05 Quantile Function Estimated Median Function Fatigue Limit Parameter = ksi 80 S eq (ksi) Cycles Figure 3 : A 1 = 6.72, A 2 = 1.57, A 3 = 36.21, q = 0.551, B 1 = 4.55, B 2 = / 54

16 Statistical models Profile likelihood To assess the plausibility of a range of values of the fatigue limit parameter, A 3, we construct the profile likelihood [2, p.294]: R(A 3 ) = max θ 0 [ ] L(θ 0, A 3 ), L( θ) where θ 0 denotes all parameters except for the fatigue limit parameter, A 3, and θ is the ML estimate of θ. 16 / 54

17 Statistical models Profile likelihood (cont.) 0.9 Model Ia Model Ib R(A 3 ) Fatigue Limit Parameter (ksi) Figure 4 : Profile likelihood estimates for the fatigue limit parameter, A 3, with Model Ia fit (blue curve) and Model Ib fit (red curve). 17 / 54

18 Statistical models Model II We extend the Model Ia to allow a random fatigue limit parameter as in [3]: µ(s eq ) = A 1 + A 2 log 10 (S eq A 3 ), if S eq > A 3. σ(s eq ) = τ. the density of log 10 (A 3 ) is φ(t; µ f, σ f ). the conditional density of log 10 (N) given S eq > A 3 is φ(t; µ(s eq ), σ(s eq )), where φ(t; µ, σ) = 1 σ exp {( t µ ) ( σ exp t µ )} σ is the smallest extreme value (sev) probability density function with location parameter µ and scale parameter σ [4, Chapter 4]. 18 / 54

19 Statistical models Model II (cont.) Estimated 0.95 Quantile Function Estimated 0.05 Quantile Function Estimated Median Function 80 S eq (ksi) Cycles Figure 5 : A 1 = 6.51, A 2 = 1.47, µ f = 1.60, σ f = , q = 0.489, τ = / 54

20 Bayesian approach Bayesian approach Model Ia Model Ia: A 1 U(5, 13), A 2 U( 5, 0), A 3 U(24, 40), q U(0.1, 0.9), τ U(0.1, 1.5) A 1 A q τ A 3 Figure 6 : Prior densities (red line) and approximate marginal posterior densities (blue line) for A 1, A 2, q, τ and A / 54

21 Bayesian approach Bayesian approach Model Ia (cont.) A A q τ A A A Figure 7 : Contour plots of the approximate bivariate densities of each pair of parameters in Model Ia. q 21 / 54

22 Bayesian approach Bayesian approach Model Ib Model Ib: A 1 U(4, 10), A 2 U( 4, 0), A 3 U(30, 40), q U(0.1, 0.9), B 1 U(2, 7), B 2 U( 5, 1) A 1 A A q B 1 B 2 Figure 8 : Prior densities (red line) and approximate marginal posterior densities (blue line) for A 1, A 2, q, B 1, B 2 and A / 54

23 Bayesian approach Bayesian approach Model Ib (cont.) 1.5 A 2 2 A 3 q B B A A A q B 1 Figure 9 : Contour plots of the approximate bivariate densities of each pair of parameters in Model Ib. 23 / 54

24 Bayesian approach Bayesian approach Model II Model II: A 1 U(4, 10), A 2 U( 4, 0), µ f U(1.4, 1.8), σ f U(0, 0.1), q U(0.1, 0.9), τ U(0, 0.25) A 1 A µ f σ f q Figure 10 : Prior densities (red line) and approximate marginal posterior densities (blue line) for A 1, A 2, q, τ, µ f and σ f. τ 24 / 54

25 Bayesian approach Bayesian approach Model II (cont.) A µ 1.6 f σ f q τ A A µ f σ f Figure 11 : Contour plots of the approximate bivariate densities of each pair of parameters in Model II. q 25 / 54

26 Model comparison Model comparison - classical information criteria Model Ia Model Ib Model II maximum log-likelihood Akaike Information Criterion (AIC) Bayesian Information Criterion (BIC) AIC with correction AIC = 2k 2 log L( θ y) BIC = k log(n) 2 log L( θ y) AIC c = AIC + 2k(k+1) n k 1 26 / 54

27 Model comparison Model comparison - Bayesian predictive criteria Model Ia Model Ib Model II Log marginal likelihood (Laplace) Log marginal likelihood (Laplace-Metropolis) Log pointwise predictive density (lppd) Deviance information criterion (DIC) Watanabe-Akaike information criterion (WAIC) fold cross-validation (elpd) lppd = ( n i=1 log 1 ) S S m=1 ρ(y i θ m ), where θ m, m = 1,..., S, are draws from the posterior of θ. DIC = 2p DIC 2 log p(y θ Bayes ) where p DIC = 2 WAIC = 2(lppd p WAIC ) ( log p(y θ Bayes ) 1 ) S S m=1 log p(y θm ). 27 / 54

28 Model comparison Model comparison - Bayesian predictive criteria (cont.) where p WAIC = 2 ( ( n 1 log S i=1 ) S p(y i θ m ) 1 S m=1 ) S log p(y i θ m ). m=1 K-fold cross-validation (elpd) Data are randomly partitioned into K disjoint subsets: {y k } K k=1. Training set: {y ( k) } = {y 1,..., y k 1, y k+1,..., y K }. For each training set compute the corresponding posterior distribution, p(θ y ( k) ). The log predictive density for y i y k is computed using the training set {y ( k) }: ( ) 1 S lpd i = log p(y i θ k,m ), i k, S m=1 28 / 54

29 Model comparison Model comparison - Bayesian predictive criteria (cont.) where {θ k,m } S m=1 are MCMC samples of the posterior p(θ y ( k)). Finally, compute the expected log predictive density (elpd): elpd = n lpd i. i=1 29 / 54

30 Conclusions Conclusions Models of various complexity that were designed to account for right-censored data are calibrated by means of the maximum likelihood method. Bayesian approach is considered and simulation-based posterior distributions are provided. Classical measures of fit based on information criteria and Bayesian model comparison were applied to determine which model might be preferred under different a priori scenarios. The classical approach and the Bayesian approach for model comparison have provided evidence in favor of Model II given the 75S-T6 data set. 30 / 54

31 Conclusions Conclusions (cont.) An integrated set of computational tools has been developed for model calibration, cross-validation, consistency and model comparison, allowing the user to rank alternative statistical models based on objective criteria. 31 / 54

32 References References [1] H. J. Grover, S. M. Bishop & L. R. Jackson (March 1951). Fatigue Strengths of Aircraft Materials. Axial-Load Fatigue Tests on Unnotched Sheet Specimens of 24S-T3 and 75S-T6 Aluminum Alloys and of SAE 4130 Steel, Battelle Memorial Institute, National Advisory Committee on Aeronautics (NACA), Technical Note 2324, Washington. [2] Francis G. Pascual & William Q. Meeker (1997). Analysis of fatigue data with runouts based on a model with nonconstant standard deviation and a fatigue limit parameter, Journal of Testing and Evaluation, 25, [3] Francis G. Pascual & William Q. Meeker (1999). Estimating fatigue curves with the random fatigue-limit model, Technometrics, 41 (4), / 54

33 References References (cont.) [4] William Q. Meeker & Luis A. Escobar (1998). Statistical Methods for Reliability Data, Wiley. [5] Andrew Gelman, Jessica Hwang & Aki Vehtari (2014). Understanding predictive information criteria for Bayesian models, Statistics and Computing, 24 (6), [6] Sumio Watanabe (2010). Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory, Journal of Machine Learning Research, 11, / 54

34 Linear Parabolic PDEs with Noisy Boundary Conditions Introduction A hierarchical Bayesian setting for an inverse problem in linear parabolic PDEs with noisy boundary conditions Joint work ( with Fabrizio Ruggeri CNR - IMATI, Consiglio Nazionale delle Ricerche, Milano, Italy, fabrizio@mi.imati.cnr.it Zaid Sawlan CEMSE, King Abdullah University of Science and Technology, Thuwal, KSA, zaid.sawlan@kaust.edu.sa Raúl Tempone CEMSE, King Abdullah University of Science and Technology, Thuwal, KSA, raul.tempone@kaust.edu.sa 34 / 54

35 Linear Parabolic PDEs with Noisy Boundary Conditions Introduction We develop a hierarchical Bayesian setting to infer unknown parameters of linear parabolic partial differential equations, as an example of linear time dependent PDEs, under the assumption that noisy measurements are available in the interior of a domain of interest and for the unknown boundary conditions. Goal To solve the inverse problem based on the assumption that the boundary parameters are unknown and modeled by means of adequate probability distributions. 35 / 54

36 Linear Parabolic PDEs with Noisy Boundary Conditions Formulation of the problem Formulation of the problem Consider the deterministic one-dimensional parabolic initial-boundary value problem: t T + L θ T = 0, x (x L, x R ), 0 < t t N < T (x L, t) = T L (t), t [0, t N ] T (x R, t) = T R (t), t [0, t N ] T (x, 0) = g(x), x (x L, x R ), (1) where L θ is a linear second-order partial differential operator that takes the form L θ T = x (a(x) x T ) + b(x) x T + c(x)t, 36 / 54

37 Linear Parabolic PDEs with Noisy Boundary Conditions Formulation of the problem Formulation of the problem (cont.) θ(x) = (a(x), b(x), c(x)) tr, and the partial differential operator t + L θ is parabolic. Our main objective is to provide a Bayesian solution to an inverse problem for θ, where we assume that i θ is unknown, while the initial condition g in the initial-boundary value problem is known; ii θ is allowed to vary with the spatial variable x. Noisy readings of the function T (x, t) at the I + 1 spatial locations, including the boundaries, x L = x 0, x 1, x 2,..., x I 1, x I = x R, at each of the N times t 1, t 2,..., t N, are assumed available. 37 / 54

38 Linear Parabolic PDEs with Noisy Boundary Conditions Statistical setting Statistical setting Let Y n := (Y 0,n,..., Y I,n ) tr denote the vector of observed readings at time t n, and assume a statistical model with an additive Gaussian experimental noise ɛ n : Y n = (I +1) 1 T L (t n ) T (x 1, t n ). T (x I 1, t n ) T R (t n ) + ɛ n, (2) where ɛ n i.i.d. N (0 I +1, σ 2 I I +1 ) for some measurement error variance σ 2 > / 54

39 Linear Parabolic PDEs with Noisy Boundary Conditions Statistical setting Statistical setting (cont.) Let Yn I (I 1) 1 := (Y 1,n,..., Y I 1,n ) tr denote the vector of observed data at the interior locations x 1, x 2,..., x I 1 and let Yn B := (Y L,n, Y R,n ) tr be the vector of observed data at the boundary locations x 0, x I at time t n. Consider the time local problem, defined between consecutive measurement times, i.e. t T + L θ T = 0, x (x L, x R ), t n 1 < t t n, T (x L, t) = T L (t), t [t n 1, t n ], T (x R, t) = T R (t), t [t n 1, t n ], T (x, t n 1 ) = T (x, t n 1 ), x (x L, x R ), (3) 39 / 54

40 Linear Parabolic PDEs with Noisy Boundary Conditions Statistical setting Statistical setting (cont.) whose exact solution, denoted by T (, t n ), depends only on θ, T (, t n 1 ) and the boundary values {T L (t), T R (t)} t (tn 1,t n). Lemma The probability density function (pdf) of Y I n is given by = ρ(yn θ, I T (, t n 1 ), {T L (t), T R (t)} t (tn 1,t ( ) n) 1 ( exp 1 ) 2πσ) I 1 2σ 2 R t n 2 l 2, (4) where R tn (I 1) 1 := ( T (x 1, t n ) Y 1,n,..., T (x I 1, t n ) Y I 1,n ) tr denotes the data residual vector at time t = t n. 40 / 54

41 Linear Parabolic PDEs with Noisy Boundary Conditions Statistical setting Statistical setting (cont.) Assume now that the Dirichlet boundary condition functions, T L ( ) and T R ( ), are well approximated by piecewise linear continuous functions in the time partition {t n } n=1,...,n. In this way, only 2N parameters, say T L (t n ) = T L,n, T R (t n ) = T R,n, n = 1, 2,..., N, suffice to determine the boundary conditions that are, in principle, infinite dimensional parameters. Let LBC n denote the time nodes that determine the local boundary conditions {T L,n 1, T L,n, T R,n 1, T R,n }. 41 / 54

42 Linear Parabolic PDEs with Noisy Boundary Conditions Statistical setting Statistical setting (cont.) Lemma The joint likelihood function of θ and the boundary parameters {LBC n } n=1,...,n is given by ρ(y 1,..., Y N θ, {LBC n } n=1,...,n ) = 1 ( 2πσ 2 exp 1 ) 2σ 2 (T L,n Y L,n ) 2 N n=1 exp ( 1 ( exp 1 ) 2πσ) I 1 2σ 2 R t n 2 l 2 ( 1 ) 2σ 2 (T R,n Y R,n ) 2. (5) Notation: T L = (T L,1,..., T L,N ) tr, T R = (T R,1,..., T R,N ) tr, Y L = (Y L,1,..., Y L,N ) tr and Y R = (Y R,1,..., Y R,N ) tr. 42 / 54

43 Linear Parabolic PDEs with Noisy Boundary Conditions Statistical setting Statistical setting (cont.) The joint likelihood function (5) can be written as ρ(y 1,..., Y N θ, T L, T R ) = ( ( ) 2πσ) N(I +1) exp 1 N 2σ 2 R tn 2 l 2 n=1 ( exp 1 [ ] ) 2σ 2 T L Y L 2 l 2 + T R Y R 2 l 2. (6) 43 / 54

44 Linear Parabolic PDEs with Noisy Boundary Conditions The marginal likelihood of θ The marginal likelihood of θ Assume that the nuisance parameters T L and T R are independent Gaussian distributed: T L N (µ L, σ 2 pi N ), T R N (µ R, σ 2 pi N ). (7) Using (6), the marginal likelihood of θ is given by ρ(y 1,..., Y N θ) = ( 2πσ) N(I +1) ( ( ) 2πσ p) 2N exp 1 N R T R T L 2σ 2 tn 2 l 2 n=1 ( exp 1 2σ (T 2 L Y L ) tr (T L Y L ) 1 ) 2σ (T 2 R Y R ) tr (T R Y R ) ( exp 1 (T 2σp 2 L µ L ) tr (T L µ L ) 1 ) (T 2σp 2 R µ R ) tr (T R µ R ) dt L dt R, (8) where R tn is approximated by ( ) R tn = B n T 0 Yn I + A L,n (θ)t L + A R,n (θ)t R. (9) 44 / 54

45 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Consider the heat equation (one dim diffusion equation for T (x, t)): t T x (θ(x) x T ) = 0, x (x L, x R ), 0 < t t N < T (0, t) = T L (t), t [0, t N ] T (1, t) = T R (t), t [0, t N ] T (x, 0) = g(x), x (x L, x R ). (10) Goal To infer the thermal diffusivity, θ(x), an unknown parameter that measures the rapidity of the heat propagation through a material, using a Bayesian approach, when the temperature is measured at I + 1 locations, x 0 = x L, x 1, x 2,..., x I 1, x I = x R, at each of the N times, t 1, t 2,..., t N. Clearly, this problem is a special case of (1) where L θ = x (θ(x) x T ) and θ(x) > / 54

46 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Consider a lognormal prior log θ N (ν, τ), where ν R and τ > 0. Assume noisy boundary measurements and a Gaussian distribution for the nuisance boundary parameters as in (7). The non-normalized posterior density for θ is given by ρ ν,τ (θ Y 1,..., Y N ) ) 1 (log θ ν)2 exp ( 2πθτ 2τ 2 ρ(y 1,..., Y N θ), where ρ(y 1,..., Y N θ) is the marginal likelihood of θ. To assess the performance of our method we use a synthetic dataset, and assume that θ is a lognormal random variable with ν = τ = / 54

47 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Bayesian inference for thermal diffusivity Figure 12 : Example: Comparison between log-likelihoods (on the left) and log-posteriors (on the right) for θ using different numbers of observations, N, and different values of σ. 47 / 54

48 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Bayesian inference for thermal diffusivity (cont.) Figure 13 : Example: Comparison between log-likelihoods (on the left) and log-posteriors (on the right) for θ using different values of σ and σ p, with N = / 54

49 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Bayesian inference for thermal diffusivity (cont.) Figure 14 : Example: Lognormal prior and approximated Gaussian posterior densities for θ where σ p = σ = 0.5 and N = / 54

50 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Bayesian inference for thermal diffusivity (cont.) Introduce three experimental setups (es s) of interest: es1) ξ consists of three non-overlapping time intervals, with the same length, which cover the entire observational period [0, 1] ; es2) ξ consists of the five inner thermocouples; es3) ξ is the combination of the two previous experimental setups, es1 and es2. 50 / 54

51 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Bayesian inference for thermal diffusivity (cont.) Figure 15 : Example: The expected information gain compared with the information divergence for the synthetic dataset, for the three time intervals experimental setup (es1). 51 / 54

52 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Bayesian inference for thermal diffusivity (cont.) Figure 16 : Example: The expected information gain compared with the information divergence for the synthetic datase, for the five inner thermocouples experimental setup (es2). 52 / 54

53 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Bayesian inference for thermal diffusivity (cont.) Figure 17 : Example: The expected information gain computed for the combination (es3) of the three time intervals and three out of five inner thermocouples experimental setups. 53 / 54

54 Linear Parabolic PDEs with Noisy Boundary Conditions Example - inference for thermal diffusivity Many thanks for your attention! 54 / 54

55 Table 1 : Maximum posterior estimates for Model Ia. Estimator A 1 A 2 A 3 q τ Laplace Laplace-Metropolis Table 2 : MCMC posterior empirical mean estimates with their standard deviations. A 1 A 2 A 3 q τ Mean SD / 3

56 Table 3 : Maximum posterior estimates for Model Ib. Estimator A 1 A 2 A 3 q B 1 B 2 Laplace Laplace-Metropolis Table 4 : MCMC posterior empirical mean estimates with their standard deviations. A 1 A 2 A 3 q B 1 B 2 Mean SD / 3

57 Table 5 : Maximum posterior estimates for Model IIb. Estimator A 1 A 2 µ f σ f q τ Laplace Laplace-Metropolis Table 6 : MCMC posterior empirical mean estimates with their standard deviations. A 1 A 2 µ f σ f q τ Mean SD / 3