**6. Bayesian Approach**

Bayesian inference has gained appeal in a variety of sectors in recent years, including engineering, clinical medicine, biology, and so on. Its capacity to analyze data using prior knowledge makes it valuable in dependability studies, where data availability is a major issue. The model parameters β1,β2,β3 and R Bayesian estimates, as well as the corresponding credible intervals, are derived in this section.

### *6.1. Prior Information and Loss Function*

Because the gamma distribution can take on different shapes based on the parameter values, using various gamma priors is simple and can result in more expressive posterior density estimates. As a result, we investigated gamma density priors, which are more adaptable than other challenging prior distributions and APE distribution under progressive first-failure censoring model parameters. As a result, under progressive first-failure censoring model parameters gamma aj, bj ; j = 1, ... , 4, independent gamma PDFs are assumed for the APE distribution. The joint prior is as follows

$$
\pi(\boldsymbol{\beta}\_{1}, \boldsymbol{\beta}\_{2}, \beta\_{3}, \boldsymbol{\aleph}) \propto \boldsymbol{\beta}\_{1}^{\ \mathbf{a}\_{1}-1} \mathbf{e}^{-\mathbf{b}\_{1}\beta\_{1}} \ \boldsymbol{\beta}\_{2}^{\ \mathbf{a}\_{2}-1} \mathbf{e}^{-\mathbf{b}\_{2}\beta\_{2}} \ \boldsymbol{\beta}\_{3}^{\ \mathbf{a}\_{3}-1} \mathbf{e}^{-\mathbf{b}\_{3}\beta\_{3}} \ \mathbf{R}^{\mathbf{a}\_{4}-1} \mathbf{e}^{-\mathbf{b}\_{4}\mathbf{R}},\tag{20}
$$

where aj, bj; j = 1, ... , 4 indicate prior knowledge of the unknown parameters β1,β2,β3 and R and are anticipated to be non-negative.

According to the literature, choosing the symmetric loss function (SLF), (squared loss function) (SEL) is a critical issue in Bayesian analysis. The SEL function is the most often utilized SLF in this study for estimating the considered unknown values.

$$\mathcal{L}\left(\mathbb{R},\mathbb{R}\right) = \left(\mathbb{R}-\mathbb{R}\right)^2,\\ \mathcal{L}\left(\mathbb{\beta}\_1,\overline{\beta\_1}\right) = \left(\overline{\beta\_1}-\beta\_1\right)^2,\\ \mathcal{L}\left(\mathbb{\beta}\_2,\overline{\beta\_2}\right) = \left(\overline{\beta\_2}-\beta\_2\right)^2,\\ \mathcal{L}\left(\mathbb{\beta}\_3,\overline{\beta\_3}\right) = \left(\overline{\beta\_3}-\beta\_3\right)^2,\\ \mathcal{L}\left(\mathbb{\beta}\_4,\overline{\beta\_4}\right) = \left(\overline{\beta\_4}-\beta\_4\right)^2,\\ \mathcal{L}\left(\mathbb{\beta}\_5,\overline{\beta\_5}\right) = \left(\overline{\beta\_5}-\beta\_5\right)^2$$

where R 5 , β 6 1, β 6 2 and β 6 3 are approximations of R, β1,β2 and β3. The posterior mean of R, θ1, θ2 and θ3 is utilized to compute the objective estimate of 5 R, θ 5 1, θ 5 2, and θ 5 3. In contrast, any other loss function can be easily incorporated.

### *6.2. Posterior Analysis by SLF*

Observing the APE distribution under progressive first-failure censoring sample data from the likelihood function and the prior knowledge given both yield the joint posterior density function.

$$\mathrm{L}(\mathsf{R},\ \beta\_{1},\beta\_{2},\beta\_{3}|\mathtt{f}) \propto \pi(\mathsf{R},\ \beta\_{1},\beta\_{2},\beta\_{3}) \prod\_{i=1}^{3} \prod\_{j=1}^{n\_{\mathtt{i}}} \mathrm{g}\left(\mathsf{t}\_{\mathtt{i}}\right) \left(1-\mathrm{G}\left(\mathsf{t}\_{\mathtt{i}}\right)\right)^{\varepsilon\_{\mathtt{i}}},\tag{21}$$

5

5

5

5

The Bayesian estimator of R, θ1, θ2 and θ3 such as R , θ 1, θ 2, and θ 3, under the SEL function, is the posterior expectation of R, θ1, θ2 and θ3. The marginal posterior distributions for each of the parameters (R, θ1, θ2 and θ3) must be gathered in order to generate

these estimates. However, due to the implied mathematical calculations, precise formulations for the marginal PDFs for each unknown parameter are plainly not realistic. As a result, we would like to generate Bayesian estimates and credible intervals utilizing simulation approaches such as MCMC.

The Metropolis–Hastings (MH) algorithm, which is used to generate random samples using the posterior density distribution and an independent proposal distribution to approximate Bayesian estimates and to create the associated Highest Posterior Density (HPD) credible intervals, is one of the most useful MCMC algorithms. In addition, this method provides a chain version of the Bayesian estimate that is simple to use in practice.
