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Bayesian network (BN) is a well-accepted framework for representing and inferring uncertain knowledge. As the qualitative abstraction of BN, qualitative probabilistic network (QPN) is introduced for probabilistic inferences in a qualitative way. With much higher efficiency of inferences, QPNs are more suitable for real-time applications than BNs. However, the high abstraction level brings some inference conflicts and tends to pose a major obstacle to their applications. In order to eliminate the inference conflicts of QPN, in this paper, we begin by extending the QPN by adding a mutual-information-based weight (MI weight) to each qualitative influence in the QPN. The extended QPN is called MI-QPN. After obtaining the MI weights from the corresponding BN, we discuss the symmetry, transitivity and composition properties of the qualitative influences. Then we extend the general inference algorithm to implement the conflict-free inferences of MI-QPN. The feasibility of our method is verified by the results of the experiment.

Bayesian network (BN) is a well-accepted model to represent a set of random variables and their probabilistic relationships via a directed acyclic graph (DAG) [

However, the high abstraction level of QPN results in the problem that there is no information left to compare two different qualitative influences in a QPN. Thus, when a node receives two inconsistent signs from its two different neighbor nodes during inferences, it is hard to know which sign is more suitable for the node, so that the “? (i.e., unknown)” sign is obtained as the result sign on the node. This means that there are inference conflicts generated on the node, which lead to less powerful expressiveness and inference capabilities than expected for real-world prediction and decision-making of uncertain knowledge in economics, health care, traffics, etc [

To provide a conflict-free inference mechanism of QPN is paid much attention. Various approaches have been proposed from various perspectives. However, in some representative methods [

Therefore, in this paper we are to consider eliminating the inference conflicts of general QPNs and developing a conflict-free inference method. We derive the quantitative QPN by adding a weight to each qualitative influence from the corresponding BN, while sample data and threshold values are not required. The weight is adopted as the information to compare two different qualitative signs. Therefore, when a node in a general QPN faces an inference conflict, a trade-off will be incorporated to avoid the conflict based on the weights.

Mutual information (MI) is a quantity that measures the mutual dependence of the two random variables [

Generally speaking, the main contributions of this paper can be summarized as follows:

We define the MI-based weight of a qualitative influence, called MI weight, and extend the traditional QPN by adding a MI weight to each qualitative influence. We call the extended QPN as MI-QPN.

We propose an efficient algorithm to derive MI weights from the conditional probability tables and prior probability distributions in the corresponding BN instead of the sample data.

We discuss the symmetry, transitivity and composition properties of qualitative influences in the MI-QPN. Then, we extend the general QPN’s inference algorithm to achieve conflict-free inferences with the MI- QPN.

We give preliminary experiments to verify the feasibility and correctness of our method.

BN has been successfully established as a framework to describe, manage uncertainty using the probabilistic graphical approach [

Lv et al. [

Renooij et al. [

We extended QPN to solve inference conflicts by adding the weights to qualitative influences based on the rough set theory and interval probability theory respectively in [

A QPN has the same graphical structure as the corresponding BN, also represented by a DAG. In a QPN, each node accords with a random variable, and the influence between each pair of the nodes can only be one of the signs including “+”, “−”, “?” and “0”, where “+” (“−”) means the probability of a higher (lower) value for the corresponding variable increases, sign “?” denotes the unknown influence by giving an evidence and “0” represents initial state of the variable without observations. Each edge with a qualitative sign means the qualitative influence between two corresponding variables. The definition of the qualitative influence [

Definition 3.1 We say that A positively influences C, written

This definition means the probability of a higher value of C is increased when given a higher value of A, regardless of any other direct influences on C. A negative qualitative influence S^{−} and a zero qualitative influence S^{0} are defined analogously, by replacing ³ in the above formula by £ and = respectively. If the qualitative influence between A and C does not belong to the above three kinds, written

Example 3.1 Based on Definition 3.1 and from the BN shown in

It is known that the qualitative influence of a QPN exhibits various useful properties [

erty expresses if there is

Building on these three properties and operators, Druzdzel et al. proposed an efficient inference algorithm based on sign propagation [

The inference conflicts take place when a node receives two different kinds of qualitative signs (“+” and “−”). In fact, the weights of qualitative influences in a QPN are not always equivalent. Thus, in this paper, we will add a weight to each qualitative influence so that a node facing a conflict will take a sign (“+” or “−”) instead of “?” by comparing the corresponding weights.

It is well known that information entropy quantifies the information contained in a message and is a measure of the uncertainty associated with a random variable. Now we introduce relevant definition [

Definition 4.1 The information entropy

Definition 4.2 The conditional information entropy of

where

We know

Definition 4.3 The mutual information of two discrete random variables

It is known that the dependency quantity

tween

Now, we define the weight of a quantitative influence based on a normalized variant of the mutual information. The weight is called MI weight that satisfies the above properties.

Definition 4.4 The MI weight of a qualitative influence

From the above definition, we know that the MI weight further satisfies the third property: 3)

Then, we consider the concerned computation in each of the three properties of the MI weight as follows:

・ For the first property, if

・ For the second property, if

since if

If

If

Therefore, the second property can be derived from Definition 4.4.

・ For the third property, if

If

Thus, we have

By Formulae (9), (10) and (11), we can obtain

Definition 4.5 If we have

By the symmetry property of the qualitative influence, we know

sent bidirectional qualitative influences with the MI weights between the corresponding two nodes.

Definition 4.6 MI-QPN is a DAG

We can obtain a prior probability distribution from each orphans node and a CPT from each non-orphans one taking as input the BN directly. If

denotes the total number of possible states of

In order to derive the MI weights between _{ }, it is necessary to obtain the conditional probability sets associated with

It is also necessary to compute the prior probability distribution of _{ }and _{ }, respectively,

where _{i}-th state for each

It can be seen that we need traverse the CPT once to derive the conditional probability set associated with one node and one of its parents. With the CPTs and the prior probability distributions of a BN, we propose Algorithm 1 to compute the MI weights for each qualitative influence in the MI-QPN.

As the basic operation in Algorithm 1, the multiplication operation for computing relevant probabilities in get conditional probability is the most time-consuming operation. Let m and n be the number of non-orphans nodes and the maximal in-degree of these nodes respectively. For convenience, we suppose the number of each node’s possible values, denoted as

Example 4.3 We consider the MI weights for the QPN in

It is known that

Algorithm 1. Deriving MI weights from a BN.

In order to address the transitivity property, we consider the MI-QPN fragment in

network, we have

are to derive the qualitative influence of

First, based on QPN’s transitivity property, we can obtain

In order to discuss the composition property, we consider the MI-QPN fragment in

the predecessors of

where

Intuitively, the composition operator “Ú” ought to satisfy the following properties: 1) The composition weight belongs to [0, 1]; 2) The composition operation is commutative; 3) The composition operation is associative; 4) Combining two influences with the same qualitative signs (e.g., two “+” signs or two “−” signs) will result in an influence with a greater MI weight; 5) Combining two influences with different qualitative signs will result in an influence dependent on but less than the larger one. Inspired by the evidence theory and the basic idea of evidence combination [

Definition 5.1

・ if

・ if

・ if

・ if _{ }and

Based the above properties, we give Algorithm 2 for conflict-free inferences with the MI-QPN.

Now, we discuss the time complexity of Algorithm 2 for MI-QPN inferences. First, whether a node will be visited is determined by its node sign whose changes are specified in QPN’s general inference algorithm [

Algorithm 2. Conflict-free inference with an MI-QPN.

To test the performance of our method in this paper, we implemented our algorithms for constructing and inferring MI-QPN. We take BN’s inference results obtained from Netica [

It is well known that Wet-Grass network is a classic BN for whether the Our Grass is wet, related to Our Wall, Rain, Our Sprinkler and Neighbor’s Grass status, containing 5 binary variables and 6 edges [

First, we derive the corresponding QPN and MI-QPN shown in

Then, we compare the inference results on the modified Wet-Grass BN and those on the derived MI-QPN. We take each node of the BN as evidence, and then record the inference results of other nodes shown in

Evidence | QPN | MI-QPN | |||||||
---|---|---|---|---|---|---|---|---|---|

Node | Sign | Rain | Sprinkler | Grass | Wall | Rain | Sprinkler | Grass | Wall |

Rain | + | + | − | ? | ? | + | − | + | − |

Sprinkler | + | − | + | ? | ? | − | + | + | + |

Grass | + | ? | ? | + | ? | + | + | + | + |

Wall | + | ? | ? | ? | + | − | + | + | + |

E-Node | E-Sign | Rain | Sprinkler | Grass | Wall | |
---|---|---|---|---|---|---|

BN | Rain | 0 → 1 | 0 → 1 | 0.5 → 0.10 | 0.475 → 0.905 | 0.355 → 0.165 |

Sprinkler | 0 → 1 | 0.31 → 0.0476 | 0 → 1 | 0.314 → 0.902 | 0.0379 → 0.702 | |

Grass | 0 → 1 | 0.0433 → 0.323 | 0.0934 → 0.676 | 0 → 1 | 0.0782 → 0.504 | |

Wall | 0 → 1 | 0.245 → 0.104 | 0.183 → 0.931 | 0.408 → 0.892 | 0 → 1 | |

MI-QPN | Rain | + | + | - | + | - |

Sprinkler | + | - | + | + | + | |

Grass | + | + | + | + | + | |

Wall | + | - | + | + | + |

In this paper, we introduced mutual-information based weights (MI weights) to qualitative influences in QPNs to resolve conflicts during the inferences. We first defined the MI weights based on mutual information and MI-QPN by extending the traditional QPN. Then, we proposed the method to derive the MI weights for the MI-QPN from the corresponding BN without sample data or threshold values. By theoretic analysis, we know the method for deriving MI weights is effective. Furthermore, we discussed the symmetry, transitivity and composition properties in the MI-QPN, and extended the general influence algorithm to implement the conflict-free inferences of MI-QPN. The feasibility of our method was verified by the results of the preliminary experiment.

Our work in this paper also leaves open some other interesting research issues. We are to further consider adding the MI weights to the qualitative synergies and discussing the method to resolve the inference conflicts caused by two inconsistent signs with the same MI weight. As well, we will further resolve the conflicts that take place during the fusion or integration of multiple QPNs by adding the MI weights to the qualitative influences in the QPNs. These are exactly our future work.

This work was supported by the National Natural Science Foundation of China (61472345), the Natural Science Foundation of Yunnan Province (2014FA023, 2013FA013), the Yunnan Provincial Foundation for Leaders of Disciplines in Science and Technology (2012HB004), and the Program for Innovative Research Team in Yunnan University (XT412011).