Normal data with unknown mean (µ) and variance (σ 2 )

In this section, the parameter updating methodology, considering both mean and variance as unknown parameters, is presented. In a first step, a non-informative distribution, the Jeffrey’s prior, is considered. The main results are presented in Table 3.4. Obtained results for the posterior distribution of the mean are identical for both Normal and Lognormal case.

Simulated population mean values are closer to previous ones and they do not differ from Normal to Lognormal approach. This can be observed at Figure 3.9a.

The posterior values of the mean are closer to previous case, where variance is considered as a known parameter. In this situation the variance is only dependent from experimental tests, namely, from obtained variance and number of specimens. Therefore, due to the low number of specimens, obtained value is higher than the one obtained when variance is

Table 3.4. Posterior estimate for mean value, considering the Jeffrey’s prior.

Parameter Normal distribution Lognormal distribution

µ1 [MPa] 57.97 57.83

σ(µ1) [MPa] 1.74 1.76

σ1 [MPa] 5.34 1.10

σ(σ1) [MPa] 1.44 0.03

95% CI for the mean [MPa] 57.29 – 58.65 57.15 – 58.51

µpop [MPa] 57.97 58.10

σpop [MPa] 5.82 5.90

95% CI for the population mean [MPa] 55.71 – 60.23 55.81 – 60.39

In the case of a conjugate prior, the initial distribution, obtained from bibliography [48, 188] is used to compute both mean and variance priors. In this case, the variance of the mean and of the variance is determined by applying first the inference considering a non-informative prior (Jeffrey’s prior). The conditional posterior distribution for the mean and the marginal posterior for the variance are then obtained through the Bayes theorem. The main results are indicated at Table 3.5.

Table 3.5. Prior and posterior estimates for mean value, considering conjugate prior.

Parameter Normal distribution Lognormal distribution

µ0 [MPa] 58.00 57.76

σ(µ0) [MPa] 1.46 1.45

µ1 [MPa] 57.98 57.78

σ(µ1) [MPa] 1.20 1.20

σ1 [MPa] 5.28 1.10

σ(σ1) [MPa] 1.07 0.02

95% CI for the mean [MPa] 57.51 – 58.45 57.31 – 58.25

µpop [MPa] 57.98 58.03

σpop [MPa] 5.50 5.53

95% CI for the population mean [MPa] 55.82 – 60.13 55.86 – 60.20

An insignificant variation on the mean value from prior to posterior estimates can be observed. In fact, the initial mean value is already close to the results provided by

experimental tests. This confirms that prior assumptions provide a good estimate of the analyzed parameter. Obtained distributions for the mean parameter are presented in Figure 3.9b. A clearly uncertainty reduction can be observed with inference, due to the fact that experimental tests provide lower standard deviation than prior distribution.

Using simulation, it is possible to infer the population parameter values. The updating procedure practically did not change the mean value. However, a reduction in uncertainty is verified due to a reduction in standard deviation value. This point out the impact that the information provided by the used conjugate prior has in posterior results.

a) b)

Figure 3.9. Obtained distributions for: a) simulated values of flc, considering Jeffrey’s prior;

b) mean value of f_lc, considering conjugate prior.

Figure 3.10a compares the posterior distribution for the mean value considering different priors. It is possible to observe that the conjugate prior provides a distribution with a lower uncertainty. Additionally, it is possible to observe that Normal distribution provides a higher mean value than the Lognormal distribution. Figure 3.10b compares the posterior distribution for the population, considering different priors. It is verified that the conjugate prior provides a distribution with a lower uncertainty.

a) b)

Figure 3.10. Posterior distribution, considering both priors, for: a) mean value of f_lc; b) simulated values of f_lc.

In this situation, while the number of experimental tests is known (10 values), the number of initial samples is adopted according to the degree of belief in initial assumptions. For instance, in the previous analysis 10 samples were considered in order to equilibrate the belief in prior distribution and in registered data from experimental tests. A study on the impact of different prior samples (n₀) in posterior distribution is thus developed. Accordingly, and for this situation, this value varies from 1 (no belief in prior model) to 10’000 (total belief in prior model). Obtained results are presented at Table 3.6.

Table 3.6. Posterior population distribution, considering different weights for initial assumptions.

n0

µpop σpop 95% CI for the population mean

[MPa] [MPa] [MPa]

1 57.96 5.61 55.76 – 60.16

10 57.98 5.50 55.82 – 60.13

100 58.00 5.39 55.88 – 60.11

1000 58.00 5.37 55.89 – 60.10

10’000 58.00 5.37 55.89 – 60.10

Accordingly, as the number of prior samples increases, the posterior distribution mean converges from data sample to prior distribution mean value. At same time, the standard deviation decreases with this value. This is essentially due to the fact that as the prior sample increases, its weight in posterior distribution increases too and, at same time, the degree of belief on sampled data. Figure 3.11 shows these distributions.

Figure 3.11. Simulated values of flc considering different weights for initial assumptions.

Bayesian methods present an inherent flexibility due to the incorporation of multiple uncertainty levels, the ability to combine information from different sources and the possibility of considering different degrees of belief in initial assumptions. The major advantage of this

framework is to deal, in a rational way, with uncertainty. Accordingly, random variables are updated through this methodology as new data is acquired.

The first approach considered the mean value as a random unknown variable and the variance known and deterministic. Good results were obtained for updating the mean parameter, providing the conjugate prior a higher reduction in uncertainty. In both situations, the population standard deviation values are lower than the initial ones. Obtained results also showed to be less sensitive to distribution type assumed for the data. The major drawback of this approach is related to the computation of each parameter characteristic value, necessary for design purposes. As the population variance is considered to be constant, the characteristic values are kept almost unchanged.

The second approach considers both mean and variance as unknown variables, even computationally costly, allows overcoming this problem. In this situation, the population variance is also updated. This approach allows a more global treatment of uncertainty.

Obtained probabilistic distributions are almost identical for both Normal and Lognormal case and for both priors. An important note is that the mean value practically did not change with conjugate prior, which means that sampled data and prior assumptions are closer. However, the new data led to a significant decrease in parameter uncertainty.

Concluding, the unknown mean and known variance approach is simpler and can be used when the parameter of interest is the mean value. However, in structural parameters the prior information regarding the variance usually presents a high uncertainty and using it to define the deterministic variance may compromise the updating procedure. The more complex problem of considering both mean and variance as unknowns, allows overcoming this problem. It deals with uncertainty in a global way, reducing it in several dimensions. In both situations, posterior inferences showed stability to the choice of different priors.

The Bayesian framework provides a consistent way of treating data from different sources, in order to increase the reliability of structural parameters. In this situation, this approach revealed to be little sensitive to distribution type assumed for the data. This is essentially due to the fact of not being obtained any negative value for this parameter and thus there is no need to truncate any value when using the Lognormal distribution. However, for parameters in which the probability of obtaining a negative value is higher, Lognormal distributions are recommended.