# Application of Bayesian Approach to Reduce the Uncertainty in Expert Judgments by Using a Posteriori Mean Function

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Application of the Bayesian Approach

#### 3.1. A Priori Probability Density Functions

#### 3.1.1. The Uniform Distribution for Defining the a Priori Information

#### 3.1.2. The Triangular Distribution for Defining the a Priori Information

#### 3.1.3. The Gaussian Distribution for Defining the a Priori Information

#### 3.2. The Conditional Density Function

#### 3.2.1. The Triangular Distribution for Defining the Expert’s Error

#### 3.2.2. Gaussian Distribution Used for Defining the Expert’s Error

## 4. Case Study Using the Bayesian Approach for Expert Judgment

#### 4.1. Case 1: Uniform Distribution with a Triangular Distribution

#### 4.2. Case 2: Uniform Distribution with a Gaussian Distribution

#### 4.3. Case 3: Triangle Distribution That Determines a Priori Information and Expert’s Error

#### 4.4. Case 4: Gaussian Distribution with a Triangle Distribution

#### 4.5. Case 5: Gaussian Distribution That Determines a Priori Information and Expert’s Error

## 5. Experimental Application of the Bayesian Approach for Evaluation of Distance Learning Courses

## 6. Discussion

## 7. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Tredger, E.; Lo, J.; Haria, S.; Lau, H.; Bonello, N.; Hlavka, B.; Scullion, C. Bias, guess and expert judgement in actuarial work. Br. Actuar. J.
**2016**, 21, 545–578. [Google Scholar] [CrossRef] - Hartley, D.; French, S. A Bayesian method for calibration and aggregation of expert judgement. Int. J. Approx. Reason.
**2020**, 130, 192–225. [Google Scholar] [CrossRef] - Vinogradova-Zinkevič, I.; Podvezko, V.; Zavadskas, E.K. Comparative assessment of the stability of AHP and FAHP methods. Symmetry
**2021**, 13, 479. [Google Scholar] [CrossRef] - Klir, G.J. Uncertainty and Information. Foundations of Generalized Information Theory; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2006. [Google Scholar]
- Mockus, J. Experimental Investigation of Distance Graduate Studies of the Open-source Environment by Models of Optimal Sequential Decisions and the Bayesian Approach. In Optimization and Its Applications; Springer: New York, NY, USA, 2007. [Google Scholar] [CrossRef]
- French, S.; Hanea, A.M.; Bedford, T.; Nane, G.F. Introduction and Overview of Structured Expert Judgement. In Expert Judgement in Risk and Decision Analysis; Stanford University: Stanford, CA, USA, 2021; pp. 1–16. [Google Scholar]
- Tversky, A.; Kahneman, D. Extensional Versus Intuitive Reasoning: The Conjunction Fallacy in Probability Judgment. In Heuristics and Biases: The Psychology of Intuitive Judgment; Gilovich, T., Griffin, D., Kahneman, D., Eds.; Cambridge University Press: New York, NY, USA, 2002; pp. 19–49. [Google Scholar]
- Kahneman, D. Thinking Fast and Slow; Farrar, Straus and Giroux: New York, NY, USA, 2011. [Google Scholar]
- Vinogradova, I.; Podvezko, V.; Zavadskas, E.K. The recalculation of the weights of criteria in MCDM methods using the Bayes approach. Symmetry
**2018**, 10, 205. [Google Scholar] [CrossRef] [Green Version] - Cooke, R.; Goossens, L.H.J. Procedures guide for structural expert judgement in accident consequence modelling. Radiat. Prot. Dosim.
**2000**, 90, 303–309. [Google Scholar] [CrossRef] - Brownstein, N.C.; Louis, T.A.; O’Hagan, A.; Pendergast, J. The role of expert judgment in statistical inference and evidence-based decision-making. Am. Stat.
**2019**, 73, 56–68. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Anscombe, F.J.; Aumann, R.J. A definition of subjective probability. Ann. Math. Statist.
**1963**, 34, 199–205. [Google Scholar] [CrossRef] - Mockus, J.; Vinogradova, I. Bayesian approach to evaluation of distance courses. Liet. Matem. Rink. LMD Darb. B.
**2014**, 55, 90–95. [Google Scholar] [CrossRef] - Surowiecki, J. The Wisdom of Crowds; Anchor Books: New York, NY, USA, 2005. [Google Scholar]
- Veen, D.; Stoel, D.; Zondervan-Zwijnenburg, M.; van de Schoot, R. Proposal for a five-step method to elicit expert judgment. Front. Psychol.
**2017**, 8, 2011. [Google Scholar] [CrossRef] [Green Version] - Zondervan-Zwijnenburg, M.; van de Schoot-Hubeek, W.; Lek, K.; Hoijtink, H.; van de Schoot, R. Application and evaluation of an expert judgment elicitation procedure for correlations. Front. Psychol.
**2017**, 8, 90. [Google Scholar] [CrossRef] [Green Version] - Wilson, K.J. An investigation of dependence in expert judgement studies with multiple experts. Int. J. Forecast.
**2017**, 33, 325–336. [Google Scholar] [CrossRef] [Green Version] - Arimone, Y.; Begaud, B.; Miremont, G.; Fourrier-Réglat, A.; Molimard, M.; Moore, N.; Haramburu, F. A new method for assessing drug causation provided agreement with experts’ judgment. J. Clin. Epidemiol.
**2006**, 59, 308–314. [Google Scholar] [CrossRef] - Vinogradova, I. Distance Course Selection Optimization. Ph.D. Thesis, Vilnius University, Vilnius, Lithuania, 2015. Available online: https://talpykla.elaba.lt/elaba-fedora/objects/elaba:8264594/datastreams/MAIN/content (accessed on 17 May 2021).
- Podofillini, L.; Pandya, D.; Emert, F.; Lomax, A.J.; Dang, V.N.; Sansavini, G. Bayesian aggregation of expert judgment data for quantification of human failure probabilities for radiotherapy. In Safety and Reliability—Safe Societies in a Changing World; CRC Press: Boca Raton, FL, USA, 2018; pp. 501–508. [Google Scholar]
- Bigün, E.S. Risk analysis of catastrophes using experts’ judgements: An empirical study on risk analysis of major civil aircraft accidents in Europe. Eur. J. Oper. Res.
**1995**, 87, 599–612. [Google Scholar] [CrossRef] - Leden, L.; Gårder, P.; Pulkkinen, U. An expert judgment model applied to estimating the safety effect of a bicycle facility. Accid. Anal. Prev.
**2000**, 32, 589–599. [Google Scholar] [CrossRef] - Zavadskas, E.K.; Vaidogas, E.R. Bayesian reasoning in managerial decisions on the choice of equipment for the prevention of industrial accidents. Eng. Econ.
**2008**, 5, 32–40. [Google Scholar] [CrossRef] - Ramli, N.; Ghani, N.A.; Ahmad, N.; Hashim, I.H.M. Psychological response in fire: A fuzzy Bayesian network approach using expert judgment. Fire Technol.
**2021**, 57, 2305–2338. [Google Scholar] [CrossRef] - Zhou, Y.; Fenton, N.; Neil, M. Bayesian network approach to multinomial parameter learning using data and expert judgments. Int. J. Approx. Reason.
**2014**, 55, 1252–1268. [Google Scholar] [CrossRef] - Sigurdsson, J.; Walls, L.; Quigley, J. Bayesian belief nets for managing expert judgment and modeling reliability. Qual. Reliab. Eng. Int.
**2001**, 17, 181–190. [Google Scholar] [CrossRef] - Varis, O.; Kuikka, S. Bene-Eia: A Bayesian approach to expert judgment elicitation with case studies on climate change impacts on surface waters. Clim. Chang.
**1997**, 37, 539–563. [Google Scholar] [CrossRef] - Rosqvist, T. Bayesian aggregation of experts’ judgements on failure intensity. Reliab. Eng. Syst. Saf.
**2000**, 70, 283–289. [Google Scholar] [CrossRef] - Wisse, B.; Bedford, T.; Quigley, J. Expert judgement combination using moment methods. Reliab. Eng. Syst. Saf.
**2008**, 93, 675–686. [Google Scholar] [CrossRef] - Smets, P. The transferable belief model for expert judgements and reliability problems. Reliab. Eng. Syst. Saf.
**1992**, 38, 59–66. [Google Scholar] [CrossRef] - Mockus, J. Investigation of examples of e-education environment for scientific collaboration and distance graduate studies, Part 1. Informatica
**2006**, 17, 259–278. [Google Scholar] [CrossRef] - Capa Santos, H.; Kratz, M.; Mosquera Munoz, F. Modelling macroeconomic effects and expert judgements in operational risk: A Bayesian approach. J. Oper. Risk
**2012**, 7, 3–23. [Google Scholar] [CrossRef] - Mazzuchi, T.A.; van Dorp, J.R. A Bayesian expert judgement model to determine lifetime distributions for maintenance optimisation. Struct. Infrastruct. Eng.
**2012**, 8, 307–315. [Google Scholar] [CrossRef] - Jiang, P.; Guo, B.; Lim, J.-H.; Zuo, M. Group judgment of relationship between product reliability and quality characteristics based on Bayesian theory and expert’s experience. Expert. Syst. Appl.
**2010**, 37, 6844–6849. [Google Scholar] [CrossRef] - Koh, D.H.; Park, J.H.; Lee, S.G.; Kim, H.C.; Choi, S.; Jung, H.; Park, D.U. Combining lead exposure measurements and experts’ judgment through a Bayesian framework. Ann. Work. Expo. Health
**2017**, 61, 1054–1075. [Google Scholar] [CrossRef] [PubMed] - Åström, J.; Pettersson, T.; Reischer, G.; Norberg, T.; Hermansson, M. Incorporating expert judgments in utility evaluation of bacteroidales qPCR assays for microbial source tracking in a drinking water source. Environ. Sci. Technol.
**2014**, 49, 1311–1318. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Parent, E.; Bernier, J. Encoding prior experts judgments to improve risk analysis of extreme hydrological events via POT modeling. J. Hydrol.
**2003**, 283, 1–18. [Google Scholar] [CrossRef] - Washington, S.; Oh, J. Bayesian methodology incorporating expert judgment for ranking countermeasure effectiveness under uncertainty: Example applied to at grade railroad crossings in Korea. Accid. Anal. Prev.
**2006**, 38, 234–247. [Google Scholar] [CrossRef] [PubMed] - Ramachandran, G.; Banerjee, S.; Vincent, J.H. Expert judgment and occupational hygiene: Application to aerosol speciation in the nickel primary production industry. Ann. Occup. Hyg.
**2003**, 47, 461–475. [Google Scholar] [CrossRef] [Green Version] - Werner, C.; Bedford, T.; Cooke, R.M.; Hanea, A.M.; Morales-Nápoles, O. Expert judgement for dependence in probabilistic modelling: A systematic literature review and future research directions. Eur. J. Oper. Res.
**2017**, 258, 801–819. [Google Scholar] [CrossRef] [Green Version] - Vinogradova, I. Multi-attribute decision-making methods as a part of mathematical optimization. Mathematic
**2019**, 7, 915. [Google Scholar] [CrossRef] [Green Version] - Vinogradova, I.; Kliukas, R. Methodology for evaluating the quality of distance learning courses in consecutive stages. Proc. Soc. Behav. Sci.
**2015**, 191, 1583–1589. [Google Scholar] [CrossRef] [Green Version] - Ziemba, P. Multi-Criteria Stochastic Selection of Electric Vehicles for the Sustainable Development of Local Government and State Administration Units in Poland. Energies
**2020**, 13, 6299. [Google Scholar] [CrossRef] - Lam, W.S.; Lam, W.H.; Jaaman, S.H.; Liew, K.F. Performance Evaluation of Construction Companies Using Integrated Entropy–Fuzzy VIKOR Model. Entropy
**2021**, 23, 320. [Google Scholar] [CrossRef] - Riaz, M.; Farid, H.M.A.; Aslam, M.; Pamucar, D.; Bozanić, D. Novel Approach for Third-Party Reverse Logistic Provider Selection Process under Linear Diophantine Fuzzy Prioritized Aggregation Operators. Symmetry
**2021**, 13, 1152. [Google Scholar] [CrossRef] - Narayanamoorthy, S.; Ramya, L.; Kalaiselvan, S.; Kureethara, J.V.; Kang, D. Use of DEMATEL and COPRAS method to select best alternative fuel for control of impact of greenhouse gas emissions. Socio-Econ. Plan. Sci.
**2021**, 76, 100996. [Google Scholar] [CrossRef] - Narayanamoorthy, S.; Annapoorani, V.; Kalaiselvan, S.; Kang, D. Hybrid Hesitant Fuzzy Multi-Criteria Decision Making Method: A Symmetric Analysis of the Selection of the Best Water Distribution System. Symmetry
**2020**, 12, 2096. [Google Scholar] [CrossRef] - Geetha, S.; Narayanamoorthy, S.; Kang, D. Extended hesitant fuzzy SWARA Techniques to Examine the Criteria Weights and VIKOR Method for Ranking Alternatives. Available online: https://aip.scitation.org/doi/abs/10.1063/5.0017049?journalCode=apc (accessed on 17 May 2021).
- Narayanamoorthy, S.; Ramya, L.; Kang, D. Normal wiggly hesitant fuzzy set with multi-criteria decision-making problem. In Proceedings of the AIP Conference Proceedings; Institute of Physics: London, UK, 2020; Volume 2261, p. 030023. [Google Scholar] [CrossRef]
- Narayanamoorthy, S.; Annapoorani, V.; Kang, D.; Ramya, L. Sustainable Assessment for Selecting the Best Alternative of Reclaimed Water Use Under Hesitant Fuzzy Multi-Criteria Decision Making. IEEE Access
**2019**, 7, 137217–137231. [Google Scholar] [CrossRef] - Zavadskas, E.K.; Bausys, R.; Lescauskiene, I.; Usovaite, A. MULTIMOORA under Interval-Valued Neutrosophic Sets as the Basis for the Quantitative Heuristic Evaluation Methodology HEBIN. Mathematics
**2021**, 9, 66. [Google Scholar] [CrossRef] - Roest, I. Expert Opinion. Use in Practice; Vrije Universiteit: Amsterdam, The Netherlands, 2002. [Google Scholar]

**Figure 2.**The probability density functions of uniform (a priori information) and triangle (expert error) distributions.

**Figure 5.**Dependence of the function ${f}_{m}\left(X\right)$ in case 5 on different $\mathsf{\sigma}$, when $\mu =5,k=1$.

**Table 1.**Dependence of the function ${f}_{m}\left(X\right)$ value on the parameter $\mu $ and the grade $X$ in case 3.

${\mathit{f}}_{\mathit{m}}\left(\mathit{X}\right)$ | $\mathit{\mu}=5$ | $\mathit{\mu}=6.5$ | $\mathit{\mu}=7$ | $\mathit{\mu}=8.5$ | $\mathit{\mu}=9$ | $\mathit{\mu}=9.5$ |
---|---|---|---|---|---|---|

$X=1$ | 1.3687 | 1.3477 | 1.3431 | 1.3333 | 1.3309 | 1.3288 |

$X=2$ | 2.2696 | 2.2959 | 2.3021 | 2.3162 | 2.3199 | 2.3232 |

$X=3$ | 3.1762 | 3.2392 | 3.2556 | 3.2952 | 3.3061 | 3.3159 |

$X=4$ | 4.0880 | 4.1770 | 4.2024 | 4.2688 | 4.2882 | 4.3064 |

$X=5$ | 5.0045 | 5.1082 | 5.1410 | 5.2346 | 5.2642 | 5.2932 |

$X=6$ | 5.9254 | 6.0320 | 6.0694 | 6.1884 | 6.2304 | 6.2736 |

$X=7$ | 6.8504 | 6.9469 | 6.9848 | 7.1228 | 7.1790 | 7.2418 |

$X=8$ | 7.7792 | 7.8513 | 7.8833 | 8.0222 | 8.0917 | 8.1810 |

$X=9$ | 8.7114 | 8.7432 | 8.7593 | 8.8485 | 8.9103 | 9.0175 |

$X=10$ | 9.6468 | 9.6199 | 9.6042 | 9.4762 | 9.3056 | 8.1111 |

**Table 2.**Dependence of the value of the function ${f}_{m}\left(X\right)$ on the expert error $k$ and the grade $X$ in case 3, when $\mu =7$.

k | $\mathit{X}=1$ | $\mathit{X}=2$ | $\mathit{X}=3$ | $\mathit{X}=4$ | $\mathit{X}=5$ | $\mathit{X}=6$ | $\mathit{X}=7$ | $\mathit{X}=8$ | $\mathit{X}=9$ | $\mathit{X}=10$ |
---|---|---|---|---|---|---|---|---|---|---|

0.8 | 1.2729 | 2.2403 | 3.2035 | 4.1615 | 5.1131 | 6.0568 | 6.9905 | 7.9111 | 8.8145 | 9.6943 |

1 | 1.3431 | 2.3021 | 3.2556 | 4.2024 | 5.1410 | 6.0694 | 6.9848 | 7.8833 | 8.7593 | 9.6042 |

1.2 | 1.4143 | 2.3646 | 3.3081 | 4.2435 | 5.1688 | 6.0814 | 6.9777 | 7.8531 | 8.7 | 9.5077 |

**Table 3.**Dependence of the function ${f}_{m}\left(X\right)$ value on the mean of $f\left(\theta \right)$ and the grade $X$ in case 4.

${\mathit{f}}_{\mathit{m}}\left(\mathit{X}\right)$ | $\mathit{\mu}=5$ | $\mathit{\mu}=6.5$ | $\mathit{\mu}=7$ | $\mathit{\mu}=8.5$ | $\mathit{\mu}=9$ | $\mathit{\mu}=9.5$ |
---|---|---|---|---|---|---|

$X=1$ | 1.5024 | 1.6184 | 1.6447 | 1.7132 | - | - |

$X=2$ | 2.4033 | 2.5394 | 2.5854 | 2.6695 | 2.7105 | 2.7105 |

$X=3$ | 3.2837 | 3.4868 | 3.5024 | 3.6146 | 3.6711 | 3.3705 |

$X=4$ | 4.1469 | 4.3552 | 4.4033 | 4.5519 | 4.5568 | 4.6184 |

$X=5$ | 5 | 5.2171 | 5.2837 | 5.4554 | 5.5024 | 5.5443 |

$X=6$ | 5.8531 | 6.0741 | 6.1469 | 6.3459 | 6.4033 | 6.4554 |

$X=7$ | 6.7163 | 6.9259 | 7 | 7.2171 | 7.2837 | 7.3459 |

$X=8$ | 7.5967 | 7.7829 | 7.8531 | 8.0741 | 8.1469 | 8.2171 |

$X=9$ | 8.4976 | 8.6541 | 8.7163 | 8.9259 | 9 | 9.0741 |

$X=10$ | 9.4145 | 9.5446 | 9.5967 | 9.7829 | 9.8531 | 9.9259 |

**Table 4.**Dependence of the value of the function ${f}_{m}\left(X\right)$ on the expert error $k$ and the grade $X$ in case 4, when $\mathsf{\sigma}=1,$ $\mu =7$.

k | $\mathit{X}=1$ | $\mathit{X}=2$ | $\mathit{X}=3$ | $\mathit{X}=4$ | $\mathit{X}=5$ | $\mathit{X}=6$ | $\mathit{X}=7$ | $\mathit{X}=8$ | $\mathit{X}=9$ | $\mathit{X}=10$ |
---|---|---|---|---|---|---|---|---|---|---|

0.8 | 1.4632 | 2.2412 | 3.3498 | 4.2761 | 5.1916 | 6.6098 | 7 | 7.9018 | 8.8084 | 9.7239 |

1 | 1.6447 | 2.5854 | 3.5024 | 4.4033 | 5.2837 | 6.1469 | 7 | 7.8531 | 8.7163 | 9.5967 |

1.2 | 1.8281 | 2.7573 | 3.6633 | 4.5401 | 5.1842 | 6.2001 | 7 | 7.7992 | 8.6153 | 9.4599 |

**Table 5.**Dependence of the value of the function ${f}_{m}\left(X\right)$ on $\mathsf{\sigma},$ the $f\left(\theta \right)$ standard deviation parameter, and the grade $X$ in case 4, when $\mu =5$, $k=1$.

$\mathit{\sigma}$ | $\mathit{X}=1$ | $\mathit{X}=2$ | $\mathit{X}=3$ | $\mathit{X}=4$ | $\mathit{X}=5$ | $\mathit{X}=6$ | $\mathit{X}=7$ | $\mathit{X}=8$ | $\mathit{X}=9$ | $\mathit{X}=10$ |
---|---|---|---|---|---|---|---|---|---|---|

0.5 | 1.8451 | 2.7838 | 3.6618 | 4.4050 | 5 | 5.5949 | 6.3382 | 7.2162 | 8.1549 | 9.1197 |

1 | 1.5024 | 2.4033 | 3.2837 | 4.1469 | 5 | 5.8531 | 6.7163 | 7.5967 | 8.4976 | 9.4145 |

1.5 | 1.2693 | 2.2058 | 3.1391 | 4.0702 | 5 | 5.9298 | 6.8609 | 7.7942 | 8.7307 | 9.6711 |

**Table 6.**Dependence of the function ${f}_{m}\left(X\right)$ value on the mean of $f\left(\theta \right)$ and the grade $X$ in case 5.

${\mathit{f}}_{\mathit{m}}\left(\mathit{X}\right)$ | $\mathit{\mu}=5$ | $\mathit{\mu}=6.5$ | $\mathit{\mu}=7$ | $\mathit{\mu}=8.5$ | $\mathit{\mu}=9$ | $\mathit{\mu}=9.5$ |
---|---|---|---|---|---|---|

$X=1$ | 1.6808 | 1.770 | 1.7909 | 1.8367 | 1.8481 | 1.8581 |

$X=2$ | 2.5851 | 2.7162 | 2.7456 | 2.8085 | 2.8237 | 2.8367 |

$X=3$ | 3.4435 | 3.6377 | 3.6808 | 3.7702 | 3.7909 | 3.8085 |

$X=4$ | 4.2444 | 4.5211 | 4.5851 | 4.7162 | 4.7456 | 4.7702 |

$X=5$ | 5 | 5.3513 | 5.4435 | 5.6377 | 5.6808 | 5.7162 |

$X=6$ | 5.7555 | 6.1256 | 6.2444 | 6.5211 | 6.5851 | 6.6377 |

$X=7$ | 6.5565 | 6.8744 | 7 | 7.3513 | 7.4435 | 7.5211 |

$X=8$ | 7.4149 | 7.6487 | 7.7555 | 8.1256 | 8.2444 | 8.3513 |

$X=9$ | 8.3192 | 8.4789 | 8.5565 | 8.8744 | 9 | 9.1256 |

$X=10$ | 9.2544 | 9.3623 | 9.4149 | 9.6487 | 9.7555 | 9.8744 |

**Table 7.**Dependence of the function ${f}_{m}\left(X\right)$ value on the expert error $k$ and the grade $X$ in case 5, when $\mu =8.5$, $\mathsf{\sigma}=1$.

k | $\mathit{X}=1$ | $\mathit{X}=2$ | $\mathit{X}=3$ | $\mathit{X}=4$ | $\mathit{X}=5$ | $\mathit{X}=6$ | $\mathit{X}=7$ | $\mathit{X}=8$ | $\mathit{X}=9$ | $\mathit{X}=10$ |
---|---|---|---|---|---|---|---|---|---|---|

0.8 | 1.6395 | 2.6129 | 3.5777 | 4.5293 | 5.4619 | 6.3680 | 7.2414 | 8.0847 | 8.9153 | 9.7586 |

1 | 1.8367 | 2.8085 | 3.7702 | 4.7162 | 5.6377 | 6.5211 | 7.3513 | 8.1256 | 8.8744 | 9.6487 |

1.2 | 2.0336 | 3.0038 | 3.9628 | 4.9038 | 5.8154 | 6.6785 | 7.4672 | 8.1699 | 8.8301 | 9.5328 |

**Table 8.**Dependence of the value of the function ${f}_{m}\left(X\right)$ on the standard deviation parameter $\sigma $ of the function $f\left(\theta \right)$ and the grade $X$ in case 5, when $\mu =8.5$ , $k=1$ .

$\mathit{\sigma}$ | $\mathit{X}=1$ | $\mathit{X}=2$ | $\mathit{X}=3$ | $\mathit{X}=4$ | $\mathit{X}=5$ | $\mathit{X}=6$ | $\mathit{X}=7$ | $\mathit{X}=8$ | $\mathit{X}=9$ | $\mathit{X}=10$ |
---|---|---|---|---|---|---|---|---|---|---|

0.5 | 1.9606 | 2.9534 | 3.9431 | 4.9271 | 5.8996 | 6.8444 | 7.7069 | 8.3217 | 8.6783 | 9.2931 |

1 | 1.8367 | 2.8085 | 3.7702 | 4.7162 | 5.6377 | 6.5211 | 7.3513 | 8.1256 | 8.8744 | 9.6487 |

1.5 | 1.6449 | 2.5971 | 3.5391 | 4.6912 | 5.3857 | 6.2886 | 7.1790 | 8.0607 | 8.9393 | 9.8209 |

**Table 9.**Dependence of the value of the function ${f}_{m}\left(X\right)$ on the standard deviation parameter $\mathsf{\sigma}$ of the function $f\left(\theta \right)$ and the grade X in case 5, when $\mu =5$, $k=1$.

$\mathit{\sigma}$ | $\mathit{X}=1$ | $\mathit{X}=2$ | $\mathit{X}=3$ | $\mathit{X}=4$ | $\mathit{X}=5$ | $\mathit{X}=6$ | $\mathit{X}=7$ | $\mathit{X}=8$ | $\mathit{X}=9$ | $\mathit{X}=10$ |
---|---|---|---|---|---|---|---|---|---|---|

0.5 | 1.9154 | 2.8774 | 3.7927 | 4.56 | 5 | 5.4399 | 6.2073 | 7.1226 | 8.0846 | 9.0639 |

1 | 1.6808 | 2.5851 | 3.4435 | 4.2444 | 5 | 5.7555 | 6.5565 | 7.4149 | 8.3192 | 9.2544 |

1.5 | 1.4292 | 2.3388 | 3.2352 | 4.1206 | 5 | 5.8794 | 6.6748 | 7.6612 | 8.5708 | 9.4943 |

**Table 10.**Machine learning and neural networks’ course evaluations; $\mu =8.638$ , $\mathsf{\sigma}=1.463$.

Expert Error | Course Evaluation X | $\mathbf{Updated}\text{}\mathbf{Evaluation}\text{}{\mathit{f}}_{\mathit{m}}\left(\mathit{X}\right)$ | ||
---|---|---|---|---|

Case 3 | Case 4 | Case 5 | ||

k = 0.8 | 9 | 8.9047 | 8.9826 | 8.9698 |

k = 0.8 | 8 | 8.0385 | 8.0307 | 8.0532 |

k = 1 | 9 | 8.8628 | 8.9733 | 8.9539 |

k = 1 | 7 | 7.1377 | 7.1198 | 7.2036 |

k = 1 | 8 | 8.0398 | 8.047 | 8.0811 |

k = 1.2 | 10 | 9.2597 | 9.8598 | 9.7626 |

Mean: | 8.5 | 8.3739 | 8.5022 | 8.5040 |

**Table 11.**

**‘**Structural engineering’ course evaluations; $\mu =8.864\text{}$ , $\mathsf{\sigma}=0.995.$

Expert Error | Course Evaluation X | $\mathbf{Updated}\text{}\mathbf{Evaluation}\text{}{\mathit{f}}_{\mathit{m}}\left(\mathit{X}\right)$ | ||
---|---|---|---|---|

Case 3 | Case 4 | Case 5 | ||

k = 0.8 | 5 | 5.2063 | 5.3429 | 5.4916 |

k = 0.8 | 4 | 4.2273 | 4.4068 | 4.5509 |

k = 1 | 6 | 6.2189 | 6.3911 | 6.5719 |

k = 1 | 5 | 5.2562 | 5.4933 | 5.6727 |

k = 1 | 9 | 8.8903 | 8.9796 | 8.9652 |

k = 1.2 | 9 | 8.8424 | 8.9721 | 8.9529 |

Mean: | 6.3333 | 6.4402 | 6.5976 | 6.7009 |

**Table 12.**True and upgraded ‘Building management’ course grades; $\mu =8.864\text{}$ , $\mathsf{\sigma}=0.995.$

Expert Error | Course Evaluation X | $\mathbf{Updated}\text{}\mathbf{Evaluation}\text{}{\mathit{f}}_{\mathit{m}}\left(\mathit{X}\right)$ | ||
---|---|---|---|---|

Case 3 | Case 4 | Case 5 | ||

k = 0.8 | 10 | 9.5598 | 9.8877 | 9.8111 |

k = 0.8 | 9 | 8.9283 | 8.9864 | 8.9766 |

k = 1 | 9 | 8.8903 | 8.9796 | 8.9652 |

k = 1 | 10 | 9.3730 | 9.8323 | 9.7230 |

k = 1 | 8 | 8.0711 | 8.1284 | 8.2149 |

k = 1.2 | 10 | 9.1388 | 9.7710 | 9.6292 |

Mean: | 9.3333 | 8.9936 | 9.2642 | 9.22 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vinogradova-Zinkevič, I.
Application of Bayesian Approach to Reduce the Uncertainty in Expert Judgments by Using a Posteriori Mean Function. *Mathematics* **2021**, *9*, 2455.
https://doi.org/10.3390/math9192455

**AMA Style**

Vinogradova-Zinkevič I.
Application of Bayesian Approach to Reduce the Uncertainty in Expert Judgments by Using a Posteriori Mean Function. *Mathematics*. 2021; 9(19):2455.
https://doi.org/10.3390/math9192455

**Chicago/Turabian Style**

Vinogradova-Zinkevič, Irina.
2021. "Application of Bayesian Approach to Reduce the Uncertainty in Expert Judgments by Using a Posteriori Mean Function" *Mathematics* 9, no. 19: 2455.
https://doi.org/10.3390/math9192455