# Uncertainly Analysis of Prior Distribution in Accelerated Degradation Testing Bayesian Evaluation Method Based on Deviance Information Criterion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling of ADT Bayesian Evaluation

#### 2.1. Modeling of Accelerated Degradation Processes

_{ijk}represents the degradation data value at time t

_{ijk}, where i denotes the number of accelerated stress levels, j denotes the number of samples tested under accelerated stress levels, and k denotes the number of inspections. The degradation increment of the product at time t

_{ijk}can be represented as:

^{2}·Λ(t)), where Λ(t) is a non-negative increasing function of time, which is defined as a linear function of time in this paper. The probability density function y(t) is given by

#### 2.2. ADT Bayesian Evaluation Model

**θ**) is the prior distribution of the prior parameter

**θ**in the absence of sample information. f(y|

**θ**), denoted as a likelihood function, represents the probability distribution of the data y given the parameter

**θ**. The marginal density function m(y) contains no information about the parameter

**θ**; it is a “normalizing” constant that is important for determining the posterior distribution. π(

**θ**|y) is the posterior distribution, which is the description of

**θ**given the sample information of y. The posterior distribution is obtained after integrating the three types of information: overall information, sample information, and prior information.

**θ**include {a, b, σ}. Therefore, the likelihood function of the ADT Bayesian evaluation model is:

**θ**|y) = π(a, b, σ|y), the reliability or life expectancy information of the product under normal stress can be evaluated.

^{2}·Λ(t).

## 3. Uncertainty Analysis of Prior Distributions

#### 3.1. Principles for Constructing Prior Distributions

**θ**of the ADT Bayesian evaluation method includes three prior parameters {a, b, σ}. Assuming that the parameters a, b, and σ are independent of each other. To evaluate the impact of different prior distributions on the ADT Bayesian evaluation results, it is possible to consider setting several types of prior distribution schemes with different prior information for each of the three prior parameters. Different forms of prior distributions and different values of parameters can be considered for each type of prior distribution scheme in order to evaluate the uncertainty of different prior distributions.

#### 3.2. Robustness Analysis of Prior Distributions for the ADT Bayesian Method

_{v}} containing different prior information; each set includes different forms of prior distribution {π

_{i}(

**θ**)}.

_{i}(

**θ**) from Π

_{v}and conduct ADT Bayesian evaluation calculations using WinBUGS software to obtain the posterior distribution {π

_{i}(

**θ**|y)}.

#### 3.3. Criteria for Prior Distribution Selection Based on DIC

**θ**, $\mathrm{D}\left(\overline{\mathit{\theta}}\right)$ represents the deviance of the posterior mean of

**θ**. The expression of deviance of

**θ**is given as follows:

_{v}}; each set includes different prior distributions {π

_{i}(

**θ**)}.

_{i}(

**θ**) from Π

_{v}, combine with WinBUGS software to conduct an ADT Bayesian evaluation calculation, and obtain the posterior distribution {π

_{i}(

**θ**|y)}.

## 4. An Illustrative Simulation Case

#### 4.1. Simulation Data Declaration

#### 4.2. Construct Sets of Prior Distributions

_{1}, Π

_{2}, Π

_{3}for prior distribution schemes.

_{1}represents the prior distribution schemes with biased information. All π

_{i}(

**θ**) in Π

_{1}have mean(a)=12, mean(b) = −6000, and mean(σ) = 0.01, which deviates from the hypothesis condition in Table 1. Normal distribution and gamma distribution are selected as prior distribution forms to analyze. In the prior distribution of π

_{1}(

**θ**), the parameters a and b are chosen to follow the normal distribution, and σ is chosen to follow the gamma distribution, for it represents the standard deviation in the Wiener process, which has the characteristic of non-negativity. Then, based on π

_{1}(

**θ**), different distribution forms are sequentially set for the prior parameters a, b, and σ, totaling 4 combinations, i.e., Π

_{1}= {π

_{i}(

**θ**)}, i = 1, 2, 3, 4. Π

_{2}represents the schemes of non-informative prior distributions. Similarly, based on π

_{1}(

**θ**) in Π

_{1}, prior parameters a, b, and σ are sequentially set to be uniform distribution for non-informative prior, while another combination with all parameters follows uniform distribution, totaling 4 combinations, i.e., Π

_{2}={π

_{i}(

**θ**)}, i = 5, 6, 7, 8. Π

_{3}represents the prior distribution scheme of precise information. All the prior distributions in Π

_{3}are the same as in Π

_{1}, but the parameters’ values of the prior distribution are consistent with the hypothesis condition, which follows mean(a) = 10, mean(b) = −5000, and mean(σ) = 0.005.

#### 4.3. Comparison of Prior and Posterior Distributions

_{i}(

**θ**). After conducting 10,000 iterations, the posterior parameter sampling chain stabilizes and converges. Discard the first 5000 data points from the posterior parameter vector as the aging stage, and extract the subsequent 5000 data sample points for the analysis and calculation of the posterior distribution. Figure 4, Figure 5 and Figure 6 show the comparison of prior and posterior distributions for respective parameters under sets of Π

_{1}, Π

_{2}, and Π

_{3}. Combining the changes from the prior distribution with the posterior distribution, partial conclusions regarding the uncertainty analysis of the prior distribution can be drawn.

- (1)
- From the comparison graphs, it is obvious that there is a difference between the prior distribution and the posterior distribution. This result is consistent with Bayesian theory, which means that by adjusting the prior distribution based on the sample data, a posterior distribution can be obtained. However, the form of posterior distributions is different from each other, which also indicates that the prior distribution settings do have an impact on ADT Bayesian evaluation results.
- (2)
- Under Π
_{1}, the posterior distributions of a and b are divided into two categories: for π_{i}(**θ**), i = 1, 2, 4, the posterior distributions of a and b are adjusted by the sample information normally, but posterior distributions do not exhibit obvious distribution characteristics; for π_{3}(**θ**), when prior parameter b follows a gamma distribution, the prior and posterior distributions of b are almost identical, without the expected adjustment in the posterior distribution, which in turn leads to an unexpected posterior distribution of a. - (3)
- Under Π
_{2}, both posterior distributions of a and b fluctuate around the original simulated hypothetical values. Compared to Π_{1}, setting non-informative prior distributions has a significant impact on the posterior distributions of a and b. Additionally, whether non-informative prior distributions are chosen between a and b will affect the posterior distribution of the other parameter. - (4)
- Under Π
_{3}, the posterior distributions of a and b exhibit more noticeable distribution characteristics, with less fluctuation. However, similar to π_{3}(**θ**), the prior and posterior distributions of parameter b under π_{11}(**θ**) are almost identical. - (5)
- The common feature of π
_{3}(**θ**) and π_{11}(**θ**) is that the prior parameter b is assigned a gamma distribution. It can be seen from the figure that under the gamma prior distribution, the prior data for parameter b is highly concentrated. But the construction of prior distributions should have sufficient support for prior information. An overconcentrated prior distribution may not be conducive to posterior estimation [8]. Since parameter b represents the coefficient of stress level in the Arrhenius acceleration model, when the prior distribution of b is highly concentrated, it may lead to insufficient adjustment capability of small sample data, thereby affecting the calculation of the posterior distribution for a. - (6)
- Under all prior distributions π
_{i}(**θ**), σ exhibits noticeable distribution characteristics with high consistency, indicating that the choice of different prior distributions has little impact on the posterior distribution of σ. - (7)
- According to the posterior data points, Table 3 lists the mean, standard deviation (std), 2.5th percentile, and 97.5th percentile of the posterior parameters under different prior distributions in Π
_{1}, Π_{2}, and Π_{3}after convergence of the iterations, for the purpose of comparative analysis. Combining comparison graphs of prior and posterior distributions, it can be concluded as follows:

- (1)
- Under Π
_{1}, for π_{i}(**θ**|y), i = 1, 2, 4, mean(a) ≈ 10 and mean(b) ≈ −5000, which are close to the values of simulation assumptions. However, for π_{3}(**θ**|y), mean(a) ≈ 12.7 and mean(b) ≈ −6000, showing a significant deviation from the simulated values. - (2)
- In Π
_{2}, the posterior means of a and b are close to the simulated values, and std(a) is in the range of 0.3 to 0.6, and std(b) is in the range of 100 to 250, which shows a clear trend of larger standard deviations compared to those of the posterior distribution in Π_{1}. It can be concluded that under non-informative prior distributions, the posterior parameters are not easily converging and exhibit greater fluctuations in this simulation case. - (3)
- For the posterior means under Π
_{3}, mean(a) ≈ 10 and mean(b) ≈ −5000, with a standard deviation significantly smaller than the posterior parameter standard deviations under Π_{1}and Π_{2}, making the results more accurate and closer to the simulated hypothesis values. - (4)
- Under all prior distributions, the mean of the parameter σ is around 0.005, with a standard deviation of around 0.0003. The 2.5th and 97.5th percentile values are also very close. It can be seen that the different settings of prior distributions have a minor impact on the parameter σ, which is mainly adjusted through the sample data.
- (5)
- The choice of prior distribution for b has a noticeable impact on the posterior distribution. In π
_{3}(**θ**) and π_{11}(**θ**), due to the setting of the gamma distribution for b, the posterior distribution’s mean and variance of b are almost identical to the set prior distribution. In the case of π_{11}(**θ**), although the posterior mean of b is very close to the simulated hypothesis value, its variance still differs significantly from the variances of other posterior distributions under Π_{3}. - (6)
- Under Π
_{2}, when all three parameters a, b, and σ are assigned non-informative prior distributions, the posterior distribution’s mean achieves results close to the simulated hypothesis condition. This indicates that under non-informative prior distributions, the Bayesian model can effectively find posterior parameters that better match the sample data, but the downside is that the posterior data fluctuates significantly.

#### 4.4. Robustness Analysis of Evaluation Results

_{1}, Π

_{2}, and Π

_{3}. From Figure 7, several conclusions can be drawn, as follows:

- (1)
- Reliability curves under different prior distributions in Π
_{1}and Π_{2}have certain differences, even significant disparities, while under different prior distributions in Π_{3}, they are relatively consistent. - (2)
- The reliability curve under π
_{1}(**θ**) is quite close to the correct reliability curve. In π_{1}(**θ**), a and b both use a normal prior distribution, while σ uses a non-negative gamma distribution, which corresponds well to the nature of σ representing the diffusion coefficient of the degradation process. In this setting of prior distribution combination, the reliability curve obtained through the posterior distribution adjusted from the sample data are very accurate. However, the reliability curves under other prior distributions in Π_{1}deviate from the correct reliability curve. - (3)
- In Π
_{2}, reliability curves under different prior distributions lie on both sides of the correct reliability curve, showing a certain deviation from each other. Although using non-informative prior distributions, compared to Π_{1}, it still produces results that are close to the correct reliability curve. Therefore, in the absence of explicit prior information, using non-informative prior distributions is not a terribly bad choice for constructing a prior distribution. - (4)
- Comparing the reliability curves under π
_{1}(**θ**) with other distributions under Π_{2}, it indicates that when conducting ADT Bayesian evaluation with the correct prior distribution model, prior distributions with biased information will still yield better assessment results than non-informative prior distributions. - (5)
- Under Π
_{3}, the reliability curves of all prior distributions are highly consistent and very close to the true curve. This is mainly due to the fact that the posterior distribution of a, b, and σ under Π_{3}is quite consistent and close to the simulated values, with very little fluctuation. This means that when using prior distributions with precise information, the sample data reinforces the prior information, resulting in a smaller variance of the posterior distribution and obtaining a fairly accurate reliability evaluation result.

#### 4.5. Prior Distribution Analysis Based on DIC Value

_{1}, Π

_{2}, and Π

_{3}, the DIC is used for analysis to compare DIC values under different prior distributions, which are shown in Figure 8. Combining reliability evaluation results under different prior distributions, some conclusions can be drawn as follows:

- (1)
- Different prior distribution sets exhibit similarity in DIC values: under Π
_{1}, the DIC values are in the range of 200 to 400; under Π_{2}, the DIC values show significant fluctuations, ranging from 300 to 900; under Π_{3}, the DIC values are highly concentrated, ranging from 120 to 150. - (2)
- According to the principle of selecting prior distributions based on DIC, which states that a smaller DIC value indicates better model fitting [35]. The performance of DIC values under prior distribution sets allows us to conclude the following ranking: accurate prior information distribution > biased prior information distribution > non-informative prior distribution. Similar conclusions can be found in reference [38]. This indicates that DIC can effectively screen the quality of prior distributions.
- (3)
- The DIC value of π
_{1}(**θ**) is the smallest in Π_{1,}and the reliability result of π_{1}(**θ**) is also the best in Π_{1,}correspondingly. Additionally, the issue of significant posterior distribution bias caused by the selection of gamma distribution as the prior distribution for b is also reflected in the DIC values, with the DIC value of π_{3}(**θ**) being larger than that of other prior distributions. - (4)
- The DIC values under Π
_{2}are generally large and exhibit significant fluctuations, indicating that non-informative prior distributions generally yield larger DIC values and are not the optimal choice for prior distributions. - (5)
- The relatively smaller DIC values under Π
_{3}indicate that having accurate prior information is significantly beneficial in selecting prior distributions for ADT Bayesian evaluation. - (6)
- It is worth noting that the DIC can serve as a guideline for selecting prior distributions, but it is not an absolute rule. This is because in conducting ADT Bayesian evaluation, there may not always be prior information available, or it may not be certain that the prior information obtained matches the true models. Therefore, when using the DIC as a guideline for selecting prior distributions, judgment based on the actual situation is still necessary.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 7.**(

**a**) Reliability curve under Π

_{1}; (

**b**) Reliability curve under Π

_{2}; (

**c**) Reliability curve under Π

_{3}.

**Figure 8.**Comparison graph of DIC values under different prior distributions. DIC: deviance information criterion.

Content | Values |
---|---|

Degradation process | Wiener |

Accelerated model | Arrhenius |

Simulation parameter θ | a = 10, b = −5000, σ = 0.005 |

Stress levels (Temperature/°C) | 65, 85, 100 |

Normal stress level (Temperature/°C) | 45 |

Sample size under each stress level | 6, 6, 6 |

Monitor times | 10, 10, 10 |

Failure threshold | 30 |

Π_{k} | π_{i}(θ) | a | b | σ |
---|---|---|---|---|

Π_{1} | π_{1}(θ) | N(12,1) ^{1} | N(−6000,100) | Γ(0.1,10) |

π_{2}(θ) | Γ (144,12) | N(−6000,100) | Γ(0.1,10) | |

π_{3}(θ) | N(12,1) | −b~Γ(360,000,60) ^{2} | Γ(0.1,10) | |

π_{4}(θ) | N(12,1) | N(−6000,100) | N(0.01,0.001) | |

Π_{2} | π_{5}(θ) | U(−100,100) ^{3} | N(6000,100) | Γ(0.1,10) |

π_{6}(θ) | N(12,1) | U(−10,000,0) | Γ(0.1,10) | |

π_{7}(θ) | N(12,1) | N(6000,100) | U(0,10) | |

π_{8}(θ) | U(−100,100) | U(−10,000,0) | U(0,1) | |

Π_{3} | π_{9}(θ) | N(10,1) | N(−5000,100) | Γ(0.05,10) |

π_{10}(θ) | Γ(100,10) | N(−5000,100) | Γ(0.05,10) | |

π_{11}(θ) | N(10,1) | −b~Γ(250,000,50) | Γ(0.05,10) | |

π_{12}(θ) | N(10,1) | N(−5000,100) | N(0.005,0.0005) |

^{1}N(p,q) represents normal distribution.

^{2}Γ(p,q) represents the gamma distribution. Due to the non-negativity of the gamma distribution, −b~Γ(p,q) represents that the negative of parameter b follows gamma distribution.

^{3}U(p,q) represents uniform distribution within the interval [p,q].

Π_{k} | π_{i}(θ|y) | a | b | σ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std ^{1} | 2.5% | 97.5% | Mean | Std | 2.5% | 97.5% | Mean | Std | 2.5% | 97.5% | ||

Π_{1} | π_{1}(θ|y) | 10.32 | 0.2662 | 9.787 | 10.93 | −5110 | 98.04 | −5336 | −4913 | 0.005 | 0.00029 | 0.0045 | 0.0056 |

π_{2}(θ|y) | 10.69 | 0.3141 | 10.09 | 11.33 | −5247 | 115.7 | −5483 | −5026 | 0.005 | 0.00029 | 0.0045 | 0.0056 | |

π_{3}(θ|y) | 12.73 | 0.0326 | 12.66 | 12.79 | −5999 | 9.981 | −6018 | −5979 | 0.0053 | 0.00031 | 0.0047 | 0.0059 | |

π_{4}(θ|y) | 10.22 | 0.3272 | 9.557 | 10.67 | −5073 | 120.4 | −5243 | −4828 | 0.0049 | 0.00028 | 0.0043 | 0.0054 | |

Π_{2} | π_{5}(θ|y) | 9.833 | 0.5913 | 8.899 | 10.93 | −4931 | 217.4 | −5335 | −4587 | 0.005 | 0.00028 | 0.0045 | 0.0056 |

π_{6}(θ|y) | 10.51 | 0.3716 | 9.899 | 11.05 | −5179 | 137 | −5382 | −4952 | 0.005 | 0.00029 | 0.0045 | 0.0056 | |

π_{7}(θ|y) | 10.91 | 0.3424 | 10.19 | 11.37 | −5329 | 126 | −5496 | −5060 | 0.0049 | 0.00027 | 0.0044 | 0.0055 | |

π_{8}(θ|y) | 9.749 | 0.5756 | 8.649 | 10.61 | −4901 | 211.5 | −5216 | −4497 | 0.0048 | 0.00027 | 0.0044 | 0.0054 | |

Π_{3} | π_{9}(θ|y) | 10.02 | 0.01714 | 9.985 | 10.05 | −5000 | 0.5054 | −5001 | −4999 | 0.005 | 0.00029 | 0.0044 | 0.0056 |

π_{10}(θ|y) | 10.02 | 0.01751 | 9.985 | 10.05 | −5000 | 0.5019 | −5001 | −4999 | 0.005 | 0.00029 | 0.0044 | 0.0056 | |

π_{11}(θ|y) | 10.02 | 0.03202 | 9.954 | 10.08 | −5000 | 9.906 | −5019 | −4980 | 0.005 | 0.0003 | 0.0044 | 0.0056 | |

π_{12}(θ|y) | 10.02 | 0.01692 | 9.985 | 10.05 | −5000 | 0.4979 | −5001 | −4999 | 0.0048 | 0.00027 | 0.0044 | 0.0054 |

^{1}std represents standard deviation.

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**MDPI and ACS Style**

Zou, T.; Liu, K.; Wu, W.; Wang, K.; Lv, C.
Uncertainly Analysis of Prior Distribution in Accelerated Degradation Testing Bayesian Evaluation Method Based on Deviance Information Criterion. *Symmetry* **2024**, *16*, 160.
https://doi.org/10.3390/sym16020160

**AMA Style**

Zou T, Liu K, Wu W, Wang K, Lv C.
Uncertainly Analysis of Prior Distribution in Accelerated Degradation Testing Bayesian Evaluation Method Based on Deviance Information Criterion. *Symmetry*. 2024; 16(2):160.
https://doi.org/10.3390/sym16020160

**Chicago/Turabian Style**

Zou, Tianji, Kai Liu, Wenbo Wu, Ke Wang, and Congmin Lv.
2024. "Uncertainly Analysis of Prior Distribution in Accelerated Degradation Testing Bayesian Evaluation Method Based on Deviance Information Criterion" *Symmetry* 16, no. 2: 160.
https://doi.org/10.3390/sym16020160