*3.3. Probabilistic Model Development*

Since the results show that changes in certain loading conditions do not result in any significant influences on the linear model of AE count rate versus crack growth rate, all the experimental data from different tests could be used to develop the crack length vs. AE count rate model. To do this, the experimental data set for Al7075-T6 was divided into three different sets. One set was used for parameter estimation when defining the model, one set was used for evaluating the error term in the model, and the final data set was used for model validation. The first set was of four experiments that were selected from different loading conditions to arrive at a more generic model in the phase of model development. Those experiments were CT1, CT3, CT5 and CT7 from Table 1. The second set of data used to estimate the uncertainties and capture the error term in the model were CT2, CT4, CT6, CT10 and CT12 from Table 1. The last step was to validate the developed model. The experiments used for validation were CT9 and CT11 from Table 1. The procedure and results of model development and validation are reported in the remainder of this section.

#### 3.3.1. Bayesian Data Analysis

A Bayesian regression approach was implemented to estimate the model parameters and error. This technique is used to estimate and update the posterior distribution of the unknown model parameters [28]. In the Bayesian inference, a prior probability distribution function (pdf) of the model parameters is combined with observed data (evidence) in the form of a likelihood function of an unknown parameter (*θ*). The result is an updated state of knowledge in the form of the posterior joint distribution of the model parameters, *f*(*θ*|*x*). The posterior pdf of the model parameter can be assessed as described according to the Bayes' Theorem [29] as:

$$f(\theta|\mathbf{x}) = \frac{f(\mathbf{x}|\theta)}{f(\mathbf{x})} \approx f(\mathbf{x}|\theta)f(\theta) \tag{5}$$

where, *f*(*θ*|*x*) is the posterior pdf of the vector of parameters (*θ*) *with θ<sup>T</sup>* = [*α*1, *α*2, *σ*] given the observed data (*x*); *f*(*x*) is the marginal pdf of the random variable *x*; *f*(*θ*) is the prior pdf of the model parameters; and *f*(*x*|*θ*) is the likelihood of the model and contains the available information provided by the observed data: *n*

$$f(\boldsymbol{\theta}|\boldsymbol{x}) = \prod\_{i=1}^{n} f(\boldsymbol{x}\_i|\boldsymbol{\theta})\tag{6}$$

The results of the analyzed experimental data were used to develop a probabilistic linear model for the estimation of the crack growth rate as the dependent variable, and using the AE count rate as the independent variable. So, based on the observed correlation, the unknown parameters of the linear model described in Equation (2) are updated. An error term was added later to assess the model error.

After the first steps of analyzing the experimental data (crack growth measurement, AE data filtration, and AE count rate calculations) were completed, the results were used to estimate the marginal and joint distribution of the unknown parameters in Equations (2) and (3) using the Bayesian regression using Equations (5) and (6). The software package WinBUGS [28] was used to perform the Bayesian inference. In this Bayesian inference, the likelihood function in Equation (6) for the observed independent crack data points (*xi*, *yi*) can be expressed as a normal distribution:

$$p(D|\mathbf{a}\_1, \mathbf{a}\_2, \mathbf{s}) = \prod\_{i=1}^n \frac{1}{s\sqrt{2\pi}} \exp\left(-\frac{1}{2} \left(\frac{y\_i - (a\_1 \mathbf{x}\_i + a\_2)}{s}\right)^2\right) \tag{7}$$

where *p*(*.*) is the likelihood of all data in form of *xi* = *log*(*dc*/*dN*)*i*, *yi* = *log*(*da*/*dN*)*i*, and *D* is the data set of all pairs (*xi*, *yi*) for *i* = 1 to *n* of the tests.
