Bayesian inference with reliability methods without knowing the maximum of the likelihood function

Transcription

1 Bayesian inference with reliaility methods without knowing the maximum of the likelihood function Wolfgang Betz a,, James L. Beck, Iason Papaioannou a, Daniel Strau a a Engineering Risk Analysis Group, Technische Universität München, 8333 München, Germany Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 95 Astract In the BUS (Bayesian Updating with Structural reliaility methods approach, the uncertain parameter space is augmented y a uniform random variale and the Bayesian inference prolem is interpreted as a structural reliaility prolem. A posterior sample is given y an augmented vector sample within the failure domain of the structural reliaility prolem where the realization of the uniform random variale is smaller than the likelihood function scaled y a constant c. The constant c must e selected such that /c is larger or equal than the maximum of the likelihood function, which, however, is typically unknown a-priori. For BUS comined with sampling ased reliaility methods, choosing c too small has a negative impact on the computational efficiency. To overcome the prolem of selecting c, we propose a post-processing step for BUS that returns an uniased estimate for the evidence and samples from the posterior distriution, even if /c is selected smaller than the maximum of the likelihood function. The applicaility of the proposed post-processing step is demonstrated y means of rejection sampling. However, it can e comined with any structural reliaility method applied within the BUS framework. Keywords: Bayesian updating, Bayesian model class selection, rejection sampling, structural reliaility. Introduction The Bayesian learning process is formalized in Bayes theorem: p( d = c E L( d p( ( The prior proailistic model p( represents the current state of knowledge aout the vector of uncertain model parameters R M. Information (data d that ecomes availale in form of measurements or oservations is expressed in the likelihood function L( d = p(d. The posterior proailistic model p( d descries the comined knowledge from the prior distriution and the likelihood function. Consequently, the posterior uncertainty aout is reduced compared to its prior uncertainty. The constant c E is a normalizing scalar that is called the evidence (marginal likelihood; it measures the plausiility of the assumed stochastic model class [] and is defined as: c E = L( d p( d ( The evidence c E is required for Bayesian model class selection and model averaging [, 3]. Except for some special cases, the posterior distriution usually cannot e derived analytically, and posterior samples have Corresponding author addresses: wolfgang.etz@tum.de (Wolfgang Betz, jimeck@caltech.edu (James L. Beck, iason.papaioannou@tum.de (Iason Papaioannou, strau@tum.de (Daniel Strau Manuscript, pulished in Proailistic Engineering Mechanics 53 (8 4 to e generated numerically. However, it is typically challenging to compute the evidence c E, ecause of the multidimensional integral in Eq. (. The BUS approach [4] transforms the Bayesian inference prolem to a structural reliaility prolem. In structural reliaility, proailities of rare events are estimated [5, 6, 7]. By interpreting the Bayesian updating prolem as a rare event estimation, existing structural reliaility methods can e used to perform the Bayesian analysis (see e.g., [8]. An estimate for the evidence c E is otained as a yproduct of BUS; in BUS, the evidence is directly related to the rare event proaility. In BUS, prior to the analysis, a constant c has to e selected: On the one hand, c should not e smaller than the maximum value that the likelihood function can take; i.e., c L max. On the other hand, selecting c conservatively large decreases the efficiency of the method. Therefore, an appropriate choice of c is crucial. However, in many cases the maximum of the likelihood function is not known in advance. We propose a post-processing step for BUS that returns an uniased estimate for the evidence and samples from the posterior distriution, even if /c is selected smaller than the maximum of the likelihood function. Like the original BUS approach, the proposed extension of BUS can e comined with any structural reliaility method. The applicaility of the proposed post-processing step is demonstrated y means of the standard Monte Carlo method, which corresponds to the rejection sampling approach when comined with BUS. The structure of the paper is as follows: In Section the BUS approach is formally introduced and the role of the scaling constant c in BUS is discussed. In Section 3 the post-processing DOI:.6/j.proengmech.8.3.4

2 step for BUS is proposed that can cope with c L max <. In Section 4 the proposed post-processing step is investigated in comination with rejection sampling for different example prolems. Section 5 riefly summarizes the otained findings.. Bayesian updating with structural reliaility methods.. The idea ehind BUS Strau and Papaioannou show in [4] that a Bayesian updating prolem can e interpreted as a structural reliaility prolem. The principal idea ehind BUS (Bayesian Updating with Structural reliaility methods is to augment the space of random variales spanned y with an additional uniformly distriuted random variale π with support [, ]. The updating prolem is then expressed as a structural reliaility prolem in the so-otained augmented random variale space [, π]. The failure domain of this reliaility prolem is defined as: = {π c L( d} (3 where c is a positive constant chosen such that c L( d is maintained for all. The domain is illustrated in Fig.. Note that denotes oth the failure domain and the corresponding event. The link etween the domain and the actual Bayesian updating prolem is: Samples from the prior distriution of that are in follow the posterior distriution [4]. In reliaility analysis, the limit-state function is defined such that: g(, π if [, π] ; and g(, π > if [, π] is outside of (see Fig.. A limit-state function g(, π that descries the failure domain defined in Eq. (3 is: g(, π = π c L( d (4 Optimally, the constant c should e chosen as the reciprocal of the maximum of the likelihood function, denoted L max [4]. However, L max is not always known in advance. In such cases, it is difficult to select c appropriately. The implications of choosing c too large or too small are discussed in Section.5... Estimating the evidence in BUS An estimate for the evidence c E is otained as a y-product of BUS. Let p e the proaility that samples [, π] from the prior distriution fall into, i.e.: p = Pr [] = Pr [ g(, π ] (5 p is the target quantity of interest in a reliaility analysis and called the proaility of failure. In BUS, p is directly linked to the evidence c E through the constant c [4]: c E = p c Note that some reliaility methods allow evaluating uncertainty ounds for the estimate of p. In this case, the statistical uncertainty in the estimated evidence c E can e quantified directly, as the evidence is directly proportional to p. Uncertainty ounds for the estimated reliaility can e evaluated with (6 /c max. likelihood uniform random variale π/c prior p( likelihood L( d Figure : Illustration of the principle of the rejection sampling algorithm Algorithm (. The highlighted region is the domain defined in Eq. (3. The limit-state function g(, π introduced in Eq. (4 is smaller or equal than zero within (it is zero at the oundary of, and larger than zero outside of. Samples elow the likelihood (i.e., the samples contained in are independent samples from the posterior distriution. In this example, 43 out of 4 samples are accepted. most simulation methods. For Monte Carlo simulation this is straight-forward to do. Moreover, approximate ounds can e derived for any importance sampling method employing a normal approximation of the estimator (e.g. [9]. For Suset Simulation, some ounds were proposed in [, ]..3. Structural reliaility methods in BUS BUS employs structural reliaility methods to perform Bayesian updating. The most straight-forward application of BUS is rejection sampling which corresponds to crude Monte Carlo simulation in the context of structural reliaility. Rejection sampling within BUS is explained in detail in Section.4. As was pointed out in [4], it is possile to use other structural reliaility methods instead of the simple rejection sampling algorithm (i.e., instead of a Monte Carlo simulation. BUS is often comined with Suset Simulation (see for example [4,, 3, 4, 5, 6], ecause this method is efficient for very small failure proailities and its performance does not depend on the dimension M of the vector of uncertain model parameters. Apart from rejection sampling and Suset Simulation, the BUS approach has een comined with the first order reliaility method (FORM and line sampling in [8]. FORM solves the reliaility prolem only approximately y linearizing the limit-state function at the most proale point of failure [7, 8]. The line sampling method computes a correction factor for the linearized solution y performing a specified numer of line searches parallel to an important direction pointing to the failure surface [9,, ]. To generate realizations of the posterior, an additional postprocessing step is required: esides computing the proaility of failure, samples located in have to e returned. In Monte Carlo simulation and Suset Simulation, samples located in are directly generated during the reliaility analysis. Importance sampling ased reliaility methods require an additional

3 re-sampling step ased on the importance weights associated with the failed samples to produce samples of equal weights. In case of FORM, samples of the approximated failure domain can e easily generated. Samples from the posterior distriution can e further used for posterior prediction of quantities of interest. A special application is the use of BUS for updating the proaility of rare events ased on oserved system response. In such case, as the target quantity of interest is the posterior proaility of failure, no posterior samples have to e generated and the updating prolem can e directly solved y structural reliaility methods [8, ]. where B denotes the eta function. The most proale value of p a posteriori is K/n. The expectation of p is: E [ p K, n ] = K + ( n + The variance of the distriution in Eq. ( can e derived as: Var [ p K, n ] = (K + (n K + (n + (n + 3 ( For increasing K and n, Eqs. (7 and (, as well as Eqs. (8 and (, approach the same values..4. The rejection sampling algorithm The most trivial application of BUS results in the rejection sampling algorithm [4, 3] (see Algorithm ( elow. This algorithm repeatedly proposes a sample [, π] from the prior distriution and accepts the sample if it is located in the failure domain; i.e., if [, π]. The accepted sample is a sample from the posterior distriution. The algorithm is repeated until K posterior samples are generated. The posterior samples resulting from the rejection sampling algorithm are statistically independent. The quantity p is the proaility that a proposed sample is accepted; p is also referred to as the acceptance proaility. An uniased estimate ˆp of p is [4]: p ˆp = K n where n is the total numer of prior samples that were proposed to generate K posterior samples. Note that the estimator K/n produces a iased estimate for p [4]. An uniased estimate of the variance of ˆp is [5]: Var [ ˆp ] ( ˆp ˆp n The estimates given in Eqs. (7 and (8 are frequentist estimates. To appropriately quantify the uncertainty aout p ased on the outcome of a particular run of rejection sampling, we employ a Bayesian approach. The numer n of prior samples needed to generate K posterior samples in the rejection sampling algorithm follows a negative inomial distriution. Thus, having oserved a certain n for a given K, the likelihood of p is: ( n L(p n, K = (p K ( p n K (9 K where ( n K denotes the inomial coefficient. The eta distriution acts as conjugate prior for the Bayesian updating. If one selects the prior distriution for p ased on the Principle of Maximum Information Entropy [6, 7] with only normalization as constraint, then the resulting prior distriution is the uniform distriution on [, ], which is a special case of the eta distriution. In this case, the posterior is also a eta distriution and can e expressed as: p(p K, n = p K ( p n K B(K +, n K + (7 (8 ( 3 Algorithm (: rejection sampling As input the algorithm requires: K, the total numer of samples to draw from the posterior distriution. c, selected such that c L max. The algorithm evaluates the evidence c E and returns K unweighted and statistically independent posterior samples with k =,..., K.. Initialize counters k = and n =.. while (k K do: (a Propose sample [, π]: i. Draw from the prior distriution p(. ii. Draw π from the uniform distriution that has support [, ]. ( if ( g(, π then: i. Increase the counter k = k +. ii. Accept the proposed sample as a posterior sample, i.e.: set =. (c Increase the counter n = n Estimate p y means of Eq. (7 or Eq. (. 4. Evaluate the evidence c E = p /c On average, the algorithm requires K/p samples from the prior distriution to generate K samples from the posterior distriution. The principle of the rejection sampling algorithm is illustrated in Fig.. Note that Algorithm ( is similar to a Monte Carlo simulation for solving the structural reliaility prolem that has limit-state function g(, π and random variales [, π]. The difference is that in a Monte Carlo simulation typically the total numer of samples n is fixed whereas in Algorithm ( the numer K of samples to e generated in the domain is specified. The main advantage of rejection sampling is that it produces independent samples from the posterior distriution. However, if the posterior distriution does not match the prior distriution well, p ecomes small, which renders the rejection sampling algorithm inefficient. As a consequence, rejection sampling is typically inefficient and usually more advanced reliaility methods are employed see Section.3. However, ecause rejection sampling is straightforward and easy to understand, we use it here to demonstrate the consequences of choosing the constant c such that c < L max.

4 .5. The constant c in BUS An appropriate selection of the constant c is crucial in BUS. The constant c in BUS is motivated y rejection sampling. For the equivalent rejection sampling prolem targeted in BUS, the value of c c E is a scaling factor for the proposal distriution p( that must e chosen such that the scaled proposal distriution is an envelope function for the target distriution [8]; i.e., c c E p( p( d must e maintained for all, which reduces to c L max if p( d is replaced with Eq. (. On the one hand, if a value of c that is larger than the maximum L max of the likelihood function is selected, the efficiency of the approach decreases; the acceptance proaility p of BUS decreases linearly with c, as seen from Eq. (6. On the other hand, if a value of c that is smaller than the maximum L max of the likelihood function is selected, the scaled proposal distriution does not envelope the target distriution for all and, thus, BUS does not produce samples that follow the posterior distriution. Instead, in such cases, BUS produces samples that follow a distriution p (c trunc that is proportional to the part of the target distriution that is contained within the envelope of the proposal distriution. The distriution p (c trunc can e derived as: p (c L(c trunc ( d p( trunc ( d = = c L(c trunc ( d p( c (c E p (c (3 (c where c E = p (c /c and p(c are the associated evidence and acceptance proaility that correspond to the selected c < L max. trunc ( d is defined as: trunc ( d = min ( L( d, c (4 Let w (c e a correction factor such that c E = w (c c E (c One can expand the correction factor w (c : w (c = = c E c (c E = = L( d p( d c E (c (5 (6 (7 L( d trunc ( d trunc ( d p( d c (c E (8 L( d p(c trunc ( d d trunc ( d (9 The quantity defined in Eq. (9 can e estimated from the samples { (c } k=,...,k generated with BUS and c < L max, which follow distriution p (c trunc ( d: Using Eq. (5 and the estimate for w (c given in Eq. (, one can correct the evidence computed with c < L max. Let p e the proaility of Eq. (5 for a choice of c = /L max. From Eq. (6, it follows that p = p (c c w(c ( L max where c (c E = p (c /c and c E = p L max is used (Eq. (6 does not hold if c < L max. Note that c L max implies that p p (c, as the relative size of the failure domain increases with decreasing c according to Eq. (3 (c implies p (c. For c > L max, p > p (c implies p = p (c c /L max. Moreover, L( (c d L(c trunc ((c d clearly holds independent of c, due to Eq. (4. Thus, the values that w (c can take are ounded: w (c c L max. For c L max the correction factor is w (c =. For c L max the correction factor must e smaller than c L max in order to maintain p p (c (see Eq. (. Additionally, it is clear that c (c E underestimates the actual evidence c E. The relative error in the evidence associated with selecting c too small (i.e., c < L max is: ε (c c E = = w = c (c E (c c E = p(c p trunc ( d L( d p( L( d trunc ( d = E d L(c L( d c L max (3 c E d (4 (5 For c, the error ε c ( E approaches one and the generated samples follow the prior distriution. For c = L max, it is ε (/L max c E = and the samples follow the posterior distriution. In the following, post-processing steps for BUS are proposed, that allow c in BUS to e chosen smaller than the maximum of the likelihood function. The proposed post-processing steps are applicale to all reliaility methods used within the BUS framework. However, the performance of the proposed approach still depends strongly on the selected c. For the particular comination of BUS with Suset Simulation, the authors have recently proposed a different strategy [], denoted abus, to handle the constant c in BUS. In abus, the constant c does not have to e specified initially and is learned adaptively during the simulation. If Suset Simulation is used as reliaility method in BUS, we recommend to use the abus approach, as preliminary comparative numerical studies of performance indicated abus to e more efficient than the post-processing steps proposed in the following. w (c K = K K k= K k= L( (c d trunc ((c d ( max (, c L( (c d ( 4 3. Proposed post-processing steps for BUS 3.. Evidence As discussed in the previous section, BUS does not return samples from the posterior distriution if the constant c is

5 selected smaller than L max. However, employing Eqs. (6 and (, the evidence of the investigated prolem can e estimated even if c is not selected properly. Note that Eq. ( does not require L max to e known. 3.. Posterior samples Since the aim is to otain samples from the posterior distriution, the distriution of the generated samples needs to e corrected, as the samples produced y BUS with c < L max follow the distriution p (c trunc ( d in Eq. (3. One viale strategy is to consider the samples from distriution p (c trunc ( d as weighted samples from the posterior distriution. In this case, one can directly work with the weighted posterior samples or perform a re-sampling step ased on the sample weights to get unweighted samples that asymptotically follow the posterior distriution. Alternatively, a Metropolis-Hastings [9, 3] step can e added to Algorithm ( to ensure that the generated samples follow the posterior distriution asymptotically. We refer to this algorithm as M-H rejection sampling Algorithm (; it is ased on an algorithm presented in [3]. In this contriution, we focus on generating unweighted posterior samples. More specifically, K unweighted posterior samples are generated from K weighted posterior samples. Using this approach, information is lost: samples with large weights tend to e repeated and samples with small weights tend to e discarded. If applicale, the following two strategies can e more efficient instead: (i Instead of generating unweighted posterior samples, the weighted posterior samples otained with BUS and c < L max are directly used. (ii Based on K weighted posterior samples, consideraly more than K unweighted posterior samples are generated y means of a resampling strategy M-H rejection sampling algorithm The initial part of the proposed algorithm (Steps ( and ( in Algorithm (, which produces samples from distriution p (c trunc ( d, is equivalent to rejection sampling (Algorithm (. These Steps ( and ( can in principle e replaced y any reliaility method. In case the applied reliaility method produces weighted samples from p (c trunc ( d, the weights of Eq. ( need to e multiplied y the weights resulting from the reliaility analysis. In Step (3, the evidence is computed ased on the correction factor w (c derived in Section.5. If (i either c E or p are the quantities of interest, or (ii weighted samples from the posterior distriution are sufficient, the procedure can e terminated at this point (i.e., after Step (3. Assuming that the applied reliaility method resulted in unweighted samples from p (c trunc ( d, the importance weight of the jth generated sample is L( (c ( j d/l(c trunc ((c ( j d. Since the aim is to generate posterior samples with equal weights, the following two additional steps are performed (alternatively, a re-sampling step could e used: In Step (4, an initial seed is selected to initiate the Metropolis-Hastings algorithm. This seed is selected through a re-sampling step such that it asymptotically follows the posterior distriution. Finally, in Step (5, the distriution of the samples is corrected y means of MCMC, employing 5 the Metropolis-Hastings algorithm proposed in [3]. As the set of samples generated in steps ( and ( are used as candidate samples of the MCMC algorithm, the employed proposal distriution is p (c trunc ( d. Therefore, the Metropolis-Hastings accept/reject-ratio of the kth candidate sample (c r k = min, p ( (c d p ( (k d p(c = min, L ( (c d L ( L (k trunc p (c trunc (c trunc trunc is: ( (k d ( (c (6 d ( (k d ( (c d (7 where k {,..., K}. The candidate sample is accepted with proaility r k and rejected with proaility r k. It is a standard result that this Metropolis-Hastings algorithm has stationary distriution p( d and, thus, the generated samples asymptotically follow the posterior distriution. The ratio r k can e simplified to: r k = min, max ( L ( (c d, c L ( (8 (k This can e readily shown y distinguishing four different cases: Case (: L ( (c d c and L ( (k c L ( d = trunc ( d holds and, consequently, oth Eq. (7 and Eq. (8 ecome one. Case (: L ( (c d > c and L ( (k c As in the previous case, Eqs. (7 and (8 ecome one. Case (3: L ( (c d c and L ( (k > c Eqs. (7 and (8 result in c /L ( (k. Case (4: L ( (c d > c and L ( (k > c trunc ( d = c ans so oth Eq. (7 and Eq. (8 translate to min (, L ( (c d /L ( (k. Consequently, Eqs. (7 and (8 are indeed equivalent. The urn-in period of the Markov chain in Step (5 is very short for reasonaly large K, as the initial seed asymptotically follows the stationary distriution of the chain. Note that the likelihood function needs to e evaluated only in Step ( of the algorithm. If the evaluation of the model ehind the likelihood is computationally demanding, the main computational urden of the algorithm is in Step (. All other steps have a small impact on the run-time in this case. Algorithm (: M-H rejection sampling As input the algorithm requires: K, the total numer of samples to draw from the posterior distriution. c, where c may e selected smaller than L max.

6 The algorithm evaluates the evidence c E and returns K unweighted ut dependent posterior samples with k =,..., K.. Initialize counters k = and n =.. while (k K do: (a Propose sample [, π]: i. Draw from the prior distriution p(. ii. Draw π from the uniform distriution that has support [, ]. ( if ( g(, π then: i. Accept the proposed sample as a posterior sample of the truncated posterior distriution, i.e.: set (c =. ii. Increase the counter k = k +. (c Increase the counter n = n Evaluate the evidence c E (a Estimate p (c y means of Eq. (7 or Eq. (. ( Evaluate the quantity w (c, defined in Eq. (. (c An estimate ĉ E,K of c E is otained through Eq. (5, where c (c E = p (c /c. 4. Pick sample ( : (a Resampling step: draw index i randomly from the list {,..., K} where the jth element of the list is associated with proaility ( swap (c (i with (c ( L((c ( j d trunc ((c ( j d/ ( w (c K. (c Set ( = (c ( (d Set k = 5. while (k K do: (a Draw a sample u randomly from a uniform distriution with support [, ]. ( Evaluate the accept/reject-ratio r k defined in Eq. (8; i.e.: r k = min ( ( max L (c d,c, d L ( (k (c if (u r k then accept the candidate sample: Set = (c else; i.e., if (u > r k then reject the candidate sample: Set = (k. (d Increase the counter k = k +. The principle of Algorithm ( is illustrated in Fig.. Contrary to rejection sampling, this algorithm does not produce K independent samples. Instead, the K generated posterior samples are dependent, ecause of the acceptance/rejection step in the Metropolis-Hastings algorithm (steps (5 and (5c in Algorithm (. The smaller the corresponding acceptance rate, the larger is the induced dependency. A particular advantage of Algorithm ( compared to Algorithm ( is that Algorithm ( can e applied without knowing L max. However, the selected c still has a considerale influence on the efficiency of the algorithm. General guidelines on how to select c are given in [4]. A potential strategy to select the value of c in Algorithm ( is to 6 max. likelihood /c uniform random variale π/c prior p( likelihood L( d Figure : Illustration of the principle of the M-H rejection sampling algorithm Algorithm (. Note that only the lack samples are independent; the lue samples introduce a dependency (at least one lack sample has the same as a lue sample that decreases the efficiency of the sampling algorithm. For clarity, the ordinate of a sample in is selected randomly etween zero and L( d. perform an initial Monte Carlo simulation from the prior distriution with a small numer of samples, and to select c equal to the largest oserved likelihood value. The initially generated Monte Carlo samples can e re-used in Algorithm (. Note that the performance of the proposed post-processing step does not depend on the dimension of. The actual location of a generated sample is not used, only the associated likelihood is considered. 4. Numerical investigations In this section, the performance of Algorithm ( is investigated y means of three example prolems. 4.. Definitions For the discussion of the example prolems, the following quantities are introduced: acts as a normalized version of c : Let (, ] e defined as = cl max ; i.e., = c = L max and = c =. The performance of the investigated algorithm is assessed for different values of. 3,max represents the largest oserved likelihood multiplied with c in a set of 3 independent posterior samples. Note that 3,max is a stochastic quantity. c E,ref denotes the actual value of the evidence of the example prolem. The quantity p,ref is defined as p,ref = c E,ref /L max. ĉ E,K is the evidence estimated y the investigated algorithm ased on K posterior samples. Moreover, ĉ (c E,K denotes the evidence estimated with Algorithm ( and scaling constant c.

7 a K and s K denote the estimated mean and standard deviation of the first component of the parameter vector in a set of K posterior samples. Note that a K and s K are random variales for finite K. If the investigated algorithm produces posterior samples, we have E[a K ] = E[ d] and E[s K ] = σ[ d]. If the generated posterior samples are independent, then σ[a K ] = K σ[ d]. For dependent samples, σ[a K ] can e expressed as σ[a K ] = + γ K σ[ d] (9 where γ quantifies the dependency of the generated samples. - N eff is the numer of effectively independent samples in the generated set of K posterior samples (of the first component. This quantity specifies how many truly independent posterior samples of would give the same variance in the sample mean as Var[a K ] otained y BUS. N eff = ( E [sk ] = K σ [a K ] + γ (3 Note that N eff can e interpreted as a measure for the dependency of the generated posterior samples; for fixed K, the smaller N eff the stronger the dependency. n K is the total numer of prior samples needed to generate K posterior samples in Algorithm (. ˆ represents a posterior sample otained with the investigated algorithm. 4.. Investigated example prolems Example prolem a: A one dimensional prolem with a standard Gaussian prior. The uncertain parameter is denoted y. The likelihood for is a Gaussian distriution that has mean µ l = 3 and standard deviation σ l =.3. This prolem has an analytical solution: the posterior distriution is Gaussian with mean and standard deviation of µ l / ( σ l + =.75 and / + σ l =.87, respectively. The maximum of the likelihood is L max = / ( σ l π =.33. The evidence of the example prolem is c E,ref = ϕ µ l / / + σ l ( = 6.6 3, + σ l where ϕ( is the PDF of the standard Gaussian distriution. Consequently, p,ref of the rejection sampling algorithm is , if c = /L max. Example prolem : The formulation of this prolem is equivalent to Example prolem a, with the only difference eing that the likelihood function for has mean µ l = 5 and standard deviation σ l =.. The posterior mean and standard deviation are 4.8 and.96, respectively. The evidence for this prolem is c E,ref = Consequently, p,ref of the rejection sampling algorithm is.8 6, if c = /L max and L max = Example prolem : A -dimensional prolem with prior i= ϕ ( i, where ϕ( denotes the standard Gaussian PDF and i is the ith component of the -dimensional parameter vector. The likelihood function of the prolem is i= ϕ ( i µ l σ /σl l, with σ l =.6 where the value µ l is chosen such that the evidence c E,ref ecomes 6 ; i.e., µ l =.46. The posterior mean and standard deviation of each component of are.34 and.5, respectively. The theoretical maximum that the likelihood function can take is L max = (.6 π = Thus, p,ref of the rejection sampling algorithm is.34 4, if c = /L max. Example prolem 3: A two-story frame structure represented as a two-degree-of-freedom shear uilding model is investigated. This example prolem was originally discussed in [3]. BUS is applied in [4, 3] to solve this prolem. The two stiffness coefficients k (first story and k (second story of the model are considered uncertain. The uncertainty in k and k is expressed as k = k n and k = k n, where and are uncertain parameters and k n = N/m. The prior distriutions of and are modeled as independent log-normal distriutions with modes.3 and.8 and standard deviation. The lumped story masses m (first story and m (second story are considered deterministic and have masses m = kg and m = 6. 3 kg. The influence of damping is neglected. Bayesian updating is performed ased on the measured first two eigen-frequencies of the system: f = 3.3Hz and f = 9.83Hz. The likelihood of the prolem is expressed as L( = exp (.5 J(/σε, ( f j ( where σ ε = /6 and J( = j= λ j f with j λ = λ = and f j ( as the jth eigen-frequency predicted y the model. The posterior distriution of this prolem is imodal [4, 3]. The reference evidence is: c E,ref = p,ref =.5 3 (since L max =. Moreover, E[k d] =. and σ[k d] =.66. The reference solutions of the presented example prolems are summarized in Tale. In addition to the quantities c E,ref, L max, p,ref, E[ d] and σ[ d], some statistics of quantity 3,max are listed in the last four rows. The statistics for 3,max can e computed explicitly for Example prolems a, and, and are evaluated y means of a large numer of repeated runs of the rejection sampling algorithm with K = 3 for Example prolems 3. It is ovious that Example prolem differs from the other prolems with respect to the statistics of 3,max: For Example prolem, E[ 3,max] =.46 and Pr [ 3,max >.8 ] = 4, whereas E[ 3,max] and Pr [ 3,max >.8 ] = for all other example prolems. Consequently, it is extremely unlikely that a 3,max close to one will e oserved in Example prolem in a set of 3 posterior samples Assessment of the performance for < We assess the performance of the introduced example prolems for Algorithm ( and different values of (, ]. The

8 Tale : Reference solution of the investigated example prolems. Example prolem a Example prolem Example prolem Example prolem 3 c E,ref L max p,ref E[ ] σ[ ] E[ 3,max] Pr [ 3,max >.8 ] 4 Pr [ 3,max <.99 ] Pr [ 3,max <.999 ] 5.38 = c Lmax..5.9 a ĉ E, 3 c E,ref..9 ĉ E, 3 c E,ref..9 ĉ E, 3 c E,ref..9 3 ĉ E, 3 c E,ref Figure 3: Statistics of the estimated evidence ĉ E, 3 divided y the reference value of the evidence c E,ref for different values of = L max c. The thick lack line represents the average otained with Algorithm ( as the average is with good approximation, the estimate ĉ E, 3 can e considered uniased. The highlighted area shows the 9% confidence interval of the estimated ĉ E, 3 /c E,ref computed with Algorithm (. The dashed red line shows the average otained with Algorithm (; the ias in the estimated evidence clearly increases for decreasing. The underlying data was generated y solving the updating prolem repeatedly, generating K = 3 posterior samples in each run. numer of posterior samples generated per updating run is K = 3. The smallest value of investigated in the studies is 4. The results are presented in Figs. (3 (5. Fig. 3 shows the statistics of the estimated evidence, Fig. 4 shows the statistics of the generated posterior samples of, and Fig. 5 shows the statistics of oth E[n 3] and σ[a 3]. The data used to plot Figs. (3 (5 was generated y solving the updating prolem repeatedly, generating K = 3 posterior samples in each run. The numer of times the prolem was solved is 5, 4, 9 3 and 8 4 for Example prolems a,, and 3, respectively. The statistics of the estimated evidence ĉ E, 3 as a function of are presented in Fig. 3. Independent of the value of, no. 8 ias in the estimate of the evidence can e oserved. Additional to the mean, the 9% confidence interval of ĉ E, 3/c E,ref is shown. For [.5, ], the influence of on the interval can e considered negligile, independent of the example prolem. The mean of ĉ (c /c E, 3 E,ref computed with Algorithm ( and c < L max is also shown in Fig. 3. The results clearly show that the estimate of the evidence ĉ (c otained with Algorithm ( E, 3 and c < L max underestimates the true evidence c E,ref of the example prolem. Fig. 4 shows that the estimates of oth the mean and the 9% confidence interval of the posterior samples ˆ otained with the M-H rejection sampling algorithm (Algorithm ( are uniased, independent of the choice of. The same cannot e said aout Algorithm ( and c < L max. In this case, the ias in the mean and the deviation from the 9% confidence interval of the samples generated with Algorithm ( increases with decreasing. This effect is more dominant in Example prolems a and than in Example prolems and 3. The samples produced y the standard rejection sampling are independent, whereas the samples generated with Algorithm ( and < are dependent. The dependency of the samples increases with decreasing. This effect is illustrated in the right part of Fig. 5. The effective numer of independent posterior samples N eff decreases with. Thus, the efficiency of Algorithm ( with respect to the generated posterior samples decreases with decreasing. However, the computational cost to generate 3 (dependent posterior samples decreases also with decreasing see left part of Fig. 5. For [.5, ], the influence of on the 9% confidence interval of ĉ E, 3/c E,ref (shown in Fig. 3 was found to e negligile, independent of the example prolem. However, this does not imply that it is irrelevant whether the analysis is performed with = or, for example, with =.5. The reason is that although the discussed confidence interval remains stale, the computational cost decreases with. The computational cost is expressed in terms of the average numer of prior samples E[n 3] required to perform the analysis (see left part of Fig. 5. For [.5, ] the value of E[n 3] is with good approximation proportional to. Thus, working with a reduced is computationally efficient. The plots shown in Fig. 5 demonstrate that the relative size of E[n 3] decreases faster than the relative size of N eff. Thus,

9 = c Lmax..5 a.9. ˆ E[ ].95 5 ˆ E[ ] - ˆ E[ ] ˆ E[ ] Figure 4: Statistics of the estimated posterior samples ˆ divided y the reference mean of for different values of = L max c. The thick lack line represents the average otained with Algorithm ( as the average is with good approximation, the sample mean can e considered uniased. The highlighted area shows the 9% confidence interval otained with Algorithm (. The dashed red lines show the average and confidence intervals otained with Algorithm (. The underlying data was generated y solving the updating prolem repeatedly, generating K = 3 posterior samples in each run. Algorithm ( is more efficient for smaller than for larger. In general, the results presented in this section demonstrate that neither Algorithm ( nor Algorithm ( is a particularly favorale strategy. Instead, if a total numer n of model calls that one is willing to invest can e specified, it is etter to generate n samples from the prior distriution and regard the prior samples as weighted samples from the posterior distriution (i.e.,. In this case, the sample weights are proportional to the likelihood function, and the evidence can e estimated as the sample mean of the likelihood function otained with the samples from the prior distriution (see e.g., [33]. However, the advantage of Algorithm ( (and the BUS approach in general is that the rejection sampling step (step ( can e replaced y any structural reliaility method. A reliaility method that is particularly well suited for BUS is Suset Simulation, it can handle prolems with many uncertain parameters and can efficiently estimate very small proailities that may arise within BUS (see e.g. [4]. Moreover, an estimate for the evidence c E is otained as a y-product of BUS. 5. Concluding remarks Bayesian Updating with Structural reliaility methods (BUS reinterprets the Bayesian updating prolem as a structural reliaility prolem. The evidence of the inference prolem is otained as a y-product of BUS. In this paper, we propose a post-processing step for BUS that does not require the scaling constant c to e chosen such that c L max. By means of the proposed post-processing step, an uniased estimate for the evidence as well as samples from the posterior distriution E[n 3 ] 3 p a 3.5 N eff 3. Figure 5: The average numer of model calls (left side of plot, x-axis and the standard deviation of the estimate a 3 (right side of plot, x-axis are shown for [, ] (y-axis of su-plots and Algorithm (. The quantity is plotted on the y-axis of the su-plots and ranges from zero to one. Horizontally, the figure is split in two parts: (left-part On the left-hand side of the figure the average numer E[n 3] of prior samples required to solve the prolem is divided y the average numer of prior samples needed in standard rejection sampling; i.e., y 3 /p. For [, ], this quantity is within [, ]; it measures how many prior samples were needed to solve the prolem compared to standard rejection sampling. (right-part On the right-hand side of the figure the effective numer of independent posterior samples (N eff divided y the total numer ( 3 of posterior samples is shown. can e otained even if the maximum of the likelihood function L max is not known. The performance of the proposed postprocessing step was demonstrated y means of different example prolems and rejection sampling. In practice, the proposed post-processing step for BUS can e comined with any structural reliaility method. Acknowledgments The authors acknowledge the support of the Technische Universität München Institute for Advanced Study, funded y the German Excellence Initiative. The first author also would like to thank the California Institute of Technology (Caltech for his research stay at Caltech, which provided the asis for this work. References [] J. L. Beck, Bayesian system identification ased on proaility logic, Structural Control and Health Monitoring, vol. 7, no. 7, pp ,. [] D. J. MacKay, Bayesian interpolation, Neural computation, vol. 4, no. 3, pp , 99. [3] L. Wasserman, Bayesian model selection and model averaging, Journal of mathematical psychology, vol. 44, no., pp. 9 7,. [4] D. Strau and I. Papaioannou, Bayesian updating with structural reliaility methods, Journal of Engineering Mechanics, 5.

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