# Prior and Posterior Linear Pooling for Combining Expert Opinions: Uses and Impact on Bayesian Networks—The Case of the Wayfinding Model

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## Abstract

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## 1. Introduction

## 2. Bayesian Networks

## 3. Linear Pooling Methods for Combining Opinions

#### Advantages and Disadvantages of Linear Pooling Methods for Combining Opinions

## 4. The Wayfinding Bayesian Network Model

## 5. Case Study: Linear Pooling Methods and the Wayfinding Bayesian Network Model

#### 5.1. Prior Linear Pooling (PrLP)

#### 5.2. Posterior Linear Pooling (PoLP)

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The Wayfinding Bayesian Network [12].

**Figure 3.**Mean and Standard Deviation bar plots for the subgroups Gender, Travel Purpose, and Travel Experience, grouped for each node of interest.

**Figure 4.**Histograms of the responses from the subgroups of interest for the node Human Factors, in the state Good.

**Figure 5.**Histograms of the responses from the subgroups of interest for the node Wayfinding, in the state Effective.

**Figure 6.**Boxplot showing the spread of responses for the six nodes of interest. For simplicity, only one state per node is shown since the states are binary.

**Figure 8.**Boxplots showing the spread of responses for each node of interest for Female, Inexperienced Travelers on Personal Travel, and for Experienced Male Travelers traveling for Business. For simplicity, only one state per node is shown since the states are binary.

**Table 1.**Advantages and disadvantages of using Prior Linear Pooling and Posterior Linear Pooling for combining the opinions of multiple experts.

Method | Advantages | Disadvantages |
---|---|---|

Prior Linear Pooling (PrLP) | · Smaller number of steps are required to obtain marginal probabilities of interest. · Having only one BN makes updating information easier and more timely. · Diagnostic, predictive, and intercausal reasoning are easier to undertake. | · Pooling, when used with BNs do not follow a coherent probability model [25]. · Since each of the probabilities given by the experts are pooled within each entry of the CPT, the resulting averages are not a reflection of what was originally given by the expert for that entry, and so the conditional independence structure is lost. |

Posterior Linear Pooling (PoLP) | · The conditional independence structure of the BN is maintained. | · More steps are required in order to obtain the marginal probabilities of interest. · Updating information can be time consuming if there are a large number of experts, and hence BNs. · Diagnostic, predictive, and intercausal reasoning is also time consuming if there are a large number of experts. This is because each individual BN must be modified and then pooling once again done to obtain the marginal probabilities of interest. |

**Table 2.**The six nodes of interest, their definitions and respective states, as found in Farr et al. [36].

Node | Description | States |
---|---|---|

Communication | The effectiveness of communication in the airport terminal | Effective, Ineffective |

Environmental Factors | The level of the environmental factors such as terminal design and navigation pathway complexity that contribute to effective wayfinding in airport terminals | Good, Bad |

Human Factors | The level of the human factors such as spatial anxiety and cognitive and spatial skills that contribute to effective wayfinding in airport terminals | Good, Bad |

Navigation Pathway | The complexity of the navigation pathway that a passenger must traverse in order to reach a desired destination in the airport terminal | Simple, Complex |

Visual Elements of Communication | The quality of the visual elements of communication in the airport terminal | Good, Bad |

Wayfinding | The effectiveness of wayfinding in the airport terminal | Effective, Ineffective |

**Table 3.**The marginal probabilities for the Human Factors and Wayfinding nodes, using Prior Linear Pooling, for the full network and the three subgroups Gender (Female, Male), Travel Purpose (Business, Personal), and Travel Experience (Experienced, Inexperienced). Since the nodes have binary states, the probability for one state per node is shown.

Group | Human Factors Good | Wayfinding Effective |
---|---|---|

All | 0.8033 | 0.8057 |

Female | 0.7790 | 0.7876 |

Male | 0.8369 | 0.8305 |

Business | 0.8033 | 0.8057 |

Personal | 0.8033 | 0.8057 |

Experienced | 0.8135 | 0.8132 |

Inexperienced | 0.7458 | 0.7683 |

**Table 4.**The marginal probabilities for Human Factors and Wayfinding for the full BN, as well as for combinations of the subgroups Gender and Travel Experience. The absolute value of the differences between the full model and each subgroup is shown in the square brackets.

Good Human Factors | Effective Wayfinding | |
---|---|---|

All | 0.8033 | 0.8057 |

Female, Experienced | 0.7880 [0.0153] | 0.7943 [0.0114] |

Female, Inexperienced | 0.7282 [0.0751] | 0.7500 [0.0557] |

Male, Experienced | 0.8487 [0.0454] | 0.8393 [0.0336] |

Male, Inexperienced | 0.7701 [0.0332] | 0.7810 [0.0247] |

**Table 5.**Marginal probabilities for diagnostic reasoning for the WBNM. Changes in the nodes of interest are shown when the Wayfinding node is set to be 100% effective. The absolute value of the differences between the full model and each subgroup is shown in the square brackets. The comparison shown is made between the full model with the Wayfinding node at 100% Good, and then each node chosen at 100% for each subsequent subgroup.

Group | Communication Effective | Environmental Factors Good | Human Factors Good | Navigation Pathway Simple | Visual Elements of Communication Good |
---|---|---|---|---|---|

Full network | 0.8115 | 0.7697 | 0.8033 | 0.6893 | 0.7087 |

Female | 0.8183 [0.0068] | 0.7931 [0.0234] | 0.9410 [0.1377] | 0.6941 [0.0048] | 0.7114 [0.0027] |

Male | 0.8239 [0.0012] | 0.7901 [0.0020] | 0.9587 [0.1554] | 0.6935 [0.0042] | 0.7137 [0.0050] |

Business | 0.8207 [0.0092] | 0.7918 [0.0021] | 0.9487 [0.1454] | 0.6939 [0.0046] | 0.7124 [0.0037] |

Personal | 0.8207 [0.0092] | 0.7918 [0.0021] | 0.9487 [0.1454] | 0.6939 [0.0046] | 0.7124 [0.0037] |

Experienced | 0.8201 [0.0086] | 0.7947 [0.0022] | 0.9518 [0.1485] | 0.6937 [0.0044] | 0.7121 [0.0034] |

Inexperienced | 0.8246 [0.0013] | 0.7947 [0.0025] | 0.9300 [0.1267] | 0.6945 [0.0052] | 0.7139 [0.0052] |

**Table 6.**The marginal probabilities for the six nodes of interest, using Posterior Linear Pooling, for the full network and the three subgroups Gender (Female, Male), Travel Purpose (Business, Personal), and Travel Experience (Experienced, Inexperienced). The absolute value of the differences between the full model and each subgroup is shown in the square brackets. Since the nodes have binary states, the probabilities for one state per node is shown.

Group | Communication Effective | Environmental Factors Good | Human Factors Good | Navigation Pathway Simple | Visual Elements of Communication Good | Wayfinding Effective |
---|---|---|---|---|---|---|

All | 0.7415 | 0.7672 | 0.7082 | 0.6509 | 0.8188 | 0.7517 |

Female | 0.7430 [0.0014] | 0.7680 [0.0007] | 0.7400 [0.0318] | 0.6524 [0.0015] | 0.8194 [0.0006] | 0.7546 [0.0029] |

Male | 0.7403 [0.0012] | 0.7666 [0.0006] | 0.6811 [0.0270] | 0.6495 [0.0013] | 0.8182 [0.0005] | 0.7492 [0.0024] |

Business | 0.7413 [0.0002] | 0.7674 [0.0001] | 0.6698 [0.0383] | 0.6517 [0.0008] | 0.8206 [0.0018] | 0.7585 [0.0067] |

Personal | 0.7416 [0.0001] | 0.7672 [0.00006] | 0.7247 [0.0164] | 0.6505 [0.0003] | 0.8180 [0.0007] | 0.7488 [0.0029] |

Experienced | 0.7363 [0.0052] | 0.7666 [0.0006] | 0.7006 [0.0076] | 0.6503 [0.0005] | 0.8180 [0.0007] | 0.7485 [0.0031] |

Inexperienced | 0.7690 [0.0274] | 0.7705 [0.0032] | 0.7482 [0.0399] | 0.6537 [0.0028] | 0.8230 [0.0041] | 0.7683 [0.0165] |

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**MDPI and ACS Style**

Farr, C.; Ruggeri, F.; Mengersen, K.
Prior and Posterior Linear Pooling for Combining Expert Opinions: Uses and Impact on Bayesian Networks—The Case of the Wayfinding Model. *Entropy* **2018**, *20*, 209.
https://doi.org/10.3390/e20030209

**AMA Style**

Farr C, Ruggeri F, Mengersen K.
Prior and Posterior Linear Pooling for Combining Expert Opinions: Uses and Impact on Bayesian Networks—The Case of the Wayfinding Model. *Entropy*. 2018; 20(3):209.
https://doi.org/10.3390/e20030209

**Chicago/Turabian Style**

Farr, Charisse, Fabrizio Ruggeri, and Kerrie Mengersen.
2018. "Prior and Posterior Linear Pooling for Combining Expert Opinions: Uses and Impact on Bayesian Networks—The Case of the Wayfinding Model" *Entropy* 20, no. 3: 209.
https://doi.org/10.3390/e20030209