Constrained abductive reasoning with fuzzy parameters in Bayesian networks

Abstract

This work proposes a novel approach for solving abductive reasoning problems in Bayesian networks involving fuzzy parameters and extra constraints. The proposed method formulates abduction problems using nonlinear programming. To maximize the sum of the fuzzy membership functions subjected to various constraints, such as boundary, dependency and disjunctive conditions, unknown node belief propagation is completed. The model developed here can be built on any exact propagation methods, including clustering, joint tree decomposition, etc.

Introduction

Bayesian networks are widely used knowledge representation and reasoning tools for various domains under uncertainty [1], [2], [3], [4], [5], [6], [7]. Since an expert system requires both predictive and diagnostic information, two types of reasoning commonly are employed, namely deduction and abduction. Deduction is a logical process from a hypothesis to deduce evidence where probabilistic relationships are involved, and abduction is a logical process that hypothetically explains experimental observations [7].

Several methods have been developed for solving abductive reasoning problems in Bayesian networks. Exact methods exploit the independence structure contained in the network to efficiently propagate uncertainty [1], [6], [7]. Meanwhile, stochastic simulation methods provide an alternative approach suitable for highly connected networks, in which exact algorithms can be inefficient [7]. Recently, search-based approximate algorithms, which search for high probability configurations through a space of possible values, have emerged as a new alternative [8]. On the other hand, two key approaches have been proposed for symbolic inference in Bayesian networks, namely: the symbolic probabilistic inference algorithm (SPI) [9] and symbolic calculations based on slight modifications of standard numerical propagation algorithms [1], [10].

The above methods have two limitations for abductive reasoning:

• All relevant parameters are assumed to be crisp.

• Extra constraints or knowledge regarding belief propagation in Bayesian networks are difficult to embed.

Those limitations restrict the usefulness of reasoning in Bayesian networks. First, the conditional probabilities between a node and its parents could be fuzzy parameters because of the difficulties of learning accurately the causal relationships among the nodes. Additionally, knowledge workers often acquire additional information regarding inferences in Bayesian networks, particularly when facing diverse diagnostic scenarios. This information can relate to boundary, dependency or disjunctive conditions. The above limitations are illustrated below using an example from Pearl [7].

Metastatic cancer is a possible cause of a brain tumor and is an explanation for increased total serum calcium. Either of these could explain a patient falling into a coma. Severe headache is also possibly associated with a brain tumor.

Fig. 1(a) shows a Bayesian network representing the above cause and effect relationships. Table 1 lists the causal influences in terms of conditional probability distributions. Each variable is characterized by the probability given the state of its parents. For instance: C∈{1,0} represents the dichotomy between having a brain tumor and not having one, +c denotes the assertion C=1 or “Brain tumor is present”, and −c is the negation of +c, namely, C=0. The root node, A, which has no parent, is characterized by its prior probability distribution. The above information can be used to solve the following reasoning problems.

Problem 1

Compute the posterior probability of every A, B, and C, given the conditional probabilities in Table 1, and a situation involving a patient who is suffering from a severe headache (E=1) but has not fallen into a coma (D=0); that is, compute P(a|−d,+e), P(b|−d,+e) and P(c|−d,+e).

Current abductive reasoning methods [1], [6], [7], [8], [9], [10] can solve Problem 1 successfully. However, if the parameters in Table 1 are fuzzy numbers, conventional methods may have difficulty in answering the queries. For instance, P(+b|+a) cannot be 0.8 but rather is a fuzzy number, say $\text{x}\text{̃}{}_1$, where $\text{x}\text{̃}{}_1\text{=P(+b|+a)}$, and is associated with a membership function $\text{μ}{}_1\text{(}\text{x}\text{̃}{}_1\text{)}$, represented as follows (see Fig. 2).

$$\text{μ}\text{̃}{}_1\text{(}\text{x}\text{̃}{}_1\text{)=5(}\text{x}\text{̃}{}_1\text{−0.6)−5(|}\text{x}\text{̃}{}_1\text{−0.8|+}\text{x}\text{̃}{}_1\text{−0.8),}\text{0.6⩽}\text{x}\text{̃}{}_1\text{⩽1,}$$

where “$\text{|∗|}$” denotes the absolute value of a term $\text{∗}$.

The above expression and Fig. 2 mean that the interval of $\text{x}\text{̃}{}_1$ is between 0.6 and 1.0. If $\text{x}\text{̃}{}_1\text{=0.8}$ then $\text{μ}{}_1\text{(}\text{x}\text{̃}{}_1\text{)=1}$, implying that $\text{x}\text{̃}{}_1\text{=0.8}$ is the most possible situation. If $\text{x}\text{̃}{}_1\text{⩽0.6}$ or $\text{x}\text{̃}{}_1\text{⩾1}$ then $\text{μ}{}_1\text{(}\text{x}\text{̃}{}_1\text{)=0}$, the least-possible manifestation of $\text{x}\text{̃}{}_1$. If $\text{x}\text{̃}{}_1\text{=0.7}$, then $\text{μ}{}_1\text{(}\text{x}\text{̃}{}_1\text{)=0.5}$.

Fuzzy membership functions can be expressed in various ways. For example, denote $\text{x}\text{̃}{}_7\text{=P(+d|+b,−c)}$ and express $\text{μ}{}_7\text{(}\text{x}\text{̃}{}_7\text{)}$ as the following function (Fig. 3).

$$\text{μ}{}_7\text{(}\text{x}\text{̃}{}_7\text{)=10(}\text{x}\text{̃}{}_7\text{−0.7)−5(|}\text{x}\text{̃}{}_7\text{−0.8|+}\text{x}\text{̃}{}_7\text{−0.8)−5(|}\text{x}\text{̃}{}_7\text{−0.85|+}\text{x}\text{̃}{}_7\text{−0.85),}\text{0.7⩽}\text{x}\text{̃}{}_7\text{⩽0.95.}$$

Here $\text{μ}{}_7\text{(}\text{x}\text{̃}{}_7\text{)}$ is a trapezoid membership function and comprises four line segments, where $\text{0.8⩽}\text{x}\text{̃}{}_7\text{⩽0.85}$ has the maximal membership. A fuzzy membership function is frequently a concave function.

This work defines the fuzzy parameters $\text{x}\text{̃}{}_{\mathrm{i}}$, i=1,2,…,8, where $\text{P(+b|+a)=}\text{x}\text{̃}{}_1$, $\text{P(+b|−a)=}\text{x}\text{̃}{}_2$, $\text{P(+c|+a)=}\text{x}\text{̃}{}_3$, $\text{P(+c|−a)=}\text{x}\text{̃}{}_4$, $\text{P(+d|+b,+c)=}\text{x}\text{̃}{}_5$, $\text{P(+d|−b,+c)=}\text{x}\text{̃}{}_6$, $\text{P(+d|+b,−c)=}\text{x}\text{̃}{}_7$, and $\text{P(+d|−b,−c)=}\text{x}\text{̃}{}_8$. Table 2 lists the membership functions of the fuzzy parameters, among which $\text{μ}{}_7\text{(}\text{x}\text{̃}{}_7\text{)}$ and $\text{μ}{}_8\text{(}\text{x}\text{̃}{}_8\text{)}$ are trapezoid membership functions while the remainder are triangular functions. Next, consider Problem 2.

Problem 2

Compute the belief distributions P(a|−d,+c), P(b|−d,+c), and P(c|−d,+c), given the fuzzy membership functions in Table 2 and some constraints related to belief propagation.

Current abductive reasoning methods have difficulties in solving Problem 2 since it involves fuzzy information and extra constraints.

Consider abductive reasoning with constraints. For a given Bayesian network, knowledge workers (such as clinicians) may have professional judgments regarding the features of certain nodes and the relationships among them in particular diagnostic backgrounds. These features and relationships can take the form of various constraints.

• Boundary constraints: From additional information or observations, clinicians can infer that the posterior probability of A given E=1 and D=0 should be higher than 0.1 but lower than 0.3, which is expressed as

  $$\text{0.1⩽P(+a|−d,+e)⩽0.3.}$$

• Functional dependency: The beliefs of certain nodes are functionally dependent. For example, clinicians can judge that the posterior probability of B is roughly a certain multiple of that of A given E=1 and D=0, which is expressed as

  $$\text{P(+a|−d,+e)⩽2P(+b|−d,+e).}$$

• Disjunctive constraints: Sometimes disjunction may occur between nodes. For example, a doctor may estimate that either P(+a|−d,+e) or P(+b|−d,+e) is equal to or below 0.2, which is expressed as

  $$\text{Either}\text{P(+a|−d,+e)⩽0.2}\text{or}\text{P(+b|−d,+e)⩽0.2.}$$

By introducing these constraints into the reasoning system, the following problems are formulated.

Problem 2.1

Compute the belief distributions P(a|−d,+c), P(b|−d,+c), and P(c|−d,+c), given the fuzzy membership functions in Table 2 and the following constraints.

$$\text{0.1⩽P(+a|−d,+e)⩽0.3,}$$$$\text{P(+b|−d,+e)⩽2P(+c|−d,+e)}$$$$\text{Either}\text{P(+a|−d,+e)⩽0.2}\text{or}\text{P(+b|−d,+e)⩽0.2.}$$

Problem 2.1 is more complicated and difficult than Problem 1 when solved using current propagation methods.

This study develops a new approach for solving Problem 2.1 based on optimization techniques. Section 2 first presents the mathematical expressions of reasoning with fuzzy parameters as well as the techniques for linearizing the nonlinear absolute terms. Section 3 then formulates Problem 2.1 as a nonlinear program and introduces some constraints relating to belief propagation. Next, Section 4 illustrates several numerical examples. The final section presents some concluding remarks.