Modification of belief in evidential causal networks

doi:10.1016/S0950-5849(99)00023-3

Information and Software Technology

Volume 41, Issue 9, 25 June 1999, Pages 597-603

https://doi.org/10.1016/S0950-5849(99)00023-3 Get rights and content

Abstract

This paper introduces a new evidential approach for the updating of causal networks which is to be added to an existing general data mining system prototype—the Mining Kernel System (MKS). We present a data mining tool which addresses both the discovery and update of causal networks hidden in database systems. It contributes to the discovery of knowledge which links rules—knowledge which would normally be considered domain knowledge (to be elicited from domain experts). We used different methods for generating networks such as our heuristic algorithm (HNG), which is briefly discussed in this paper. Evaluation of such knowledge presents difficulties but some anecdotal appraisal is presented here in the form of a simple case study.

Applications of this prototype with its new causal updating supplement are under way. Our approach is based on Evidence Theory and offers important advantages over conventional Bayesian methods for the applications envisaged. These approaches allow certainty levels of rules in causal networks to be kept up to date. When a causal network has been discovered, any subsequent new evidence may be fed into the model. After updating the belief function for any node the complete network is updated through communication between neighbouring nodes.

Introduction

Causal networks, which are directed acyclic graphs, provide a computational model for many purposes in the real world, one of which is reasoning under uncertainty. Fig. 1 below is an example of a 5-node network which represents causal relationship between pairs of propositions. Large networks, of several hundred nodes, often exist in business, engineering and scientific systems. Their complexity is sometimes such that conventional mathematical methods and their inter-linkage is beyond the current knowledge of the domain experts. Cruder methods based on “causal linkage” rather than detailed equations are the only feasible ways at present. For example, we are working on a system for fault diagnosis in telecommunications systems where the number of causal linkage is very large and the engineering mathematics, which is very sophisticated for limited sections of the graph of inter-relationships, cannot cope with the full network. Similar diagnostic systems can be found in many other engineering applications, and in medicine, and large complex causal networks can also be found in scientific and business systems. Often these networks are only partially known to domain experts, and our overall motivation for the present study is to investigate the extent to which they can be derived/mined from databases, and to provide tools to derive and use them where appropriate.

In evidential causal networks the nodes represent Boolean algebras and the edges represent dependencies between related nodes. The strengths of dependencies are determined by conditional probabilities which are stored in rule-strength matrices. Acquisition of new evidence and beliefs make it necessary to update rule strengths and various belief measures. This process is performed by using the links in the network to direct and activate the data flow in various computations which bring the network into a new equilibrium.

Probabilistic reasoning is the approach normally taken to do this job. Dempster–Shafer theory of evidence, the basic concepts of which are discussed in Section 2, is an alternative method which may be cheaper computationally, and simpler too. It offers a number of advantages over probabilistic reasoning, including the explicit handling of ignorance—particularly important in mining from databases [1]. In the latter approach mass functions are used to represent both the strengths of rules and belief functions for nodes. This is the core of our novel updating method, which is the primary subject of this paper, for use in causal networks.

Section 3 gives an overview of our mining tool, called Causal Network Management System (CANEMAS) [2]. This system searches for a structure of a particular (network) kind which governs the inter-working of many rules. Although CANEMAS is a stand-alone prototype it can be seen as an addition to MKS [3], [4] which generates individual rules using various numerical techniques such as machine-learning and clustering. At the present moment we are linking the causal updating method described in this paper to the other techniques for use in building a data mining solution for a particular application in the telecommunications industry [5].

The basic outline of our evidential approach for stabilising a causal network is described in Section 4. The next two sections (5 The, 6 An example of applying the) show the structure of our algorithm (PropBel) and an example of how it operates. The telecommunications application has used the network generation aspects of CANEMAS [5], and the plan is to use the updating method presented here subsequently. Evaluation of the networks and their updating is a long and difficult task which is still in its early stages. The purpose of the present paper is to show how the coherence and consistency of Evidence Theory provide a real alternative to Bayesian methods of network discovery and updating, which promises practical advantages for applications.

Section snippets

Basics of Dempster–Shafer theory

The Dempster–Shafer theory is based on the idea of using a number between zero and one to indicate the degree of belief of evidence for a proposition [6]. The theory also includes reasoning based on the rule of combination of degrees of belief based on different pieces of evidence.

The frame of discernment, denoted by Θ, is a finite non-empty set of propositions which are mutually exclusive and exhaustive.

A mass function (m) is a basic probability assignment function where $m : 2^{Θ} →[0,1]$ such that

Generating causal trees

Our Causal Network Management System (CANEMAS) [2] is a prototype which we recently designed to support a particular kind of knowledge discovery in databases (KDD). The ultimate purpose of this prototype is to provide a computational model supporting reasoning under uncertainty which captures the concept of causation, and can be used to extract causal relationships from evidence (held at multiple sources), and support decision-making based on these relationships.

One of the main components of

Reasoning a causal tree with activated data

In the causal network with n nodes each X has r children (Y₁,…,Y_r) and a single parent E. A reasoning process for a causal network begins when any node receives activated data. Belief updating and belief propagating are the two key steps of a reasoning process. Three parameters are made available to accomplish this process. These parameters are:

•
Causal support (m_E↓X) contributed by the parent of X, $m_{E↓X} =[m_{E↓X} (E_{1}),…,m_{E↓X} (E_{l})], on E={E_{1},…E_{l}}.$
•
Diagnostic support (m_{Y_i↑X}) contributed by the ith child Y_i(∀

The PropBel algorithm

The following relation (R) describes some features of the weather's effects on the heating control systems of a house. A frequency (F) attribute indicates the number of tuples in the original database characterised by each combination of events (Table 1).

HNG is a heuristic network generation algorithm we recently developed to create a causal network. Relation R is used as the form of input for HNG. An informal specification of this algorithm is presented below.

begin
After inputting a “cleaned”

An example of applying the PropBel algorithm

In relation to the causal network shown in Fig. 3 created by the HNG algorithm for the weather's effects on the heating control systems of a house new evidence is supplied for the warm room node (x₃). After updating the belief function for this node new diagnostic support is passed on to its parent (x₄). The mass functions which represent the strengths of supports at this node for the next stage are assumed to be $m_{3↑4} =[0.5,0.2,0.3] and m_{5↑4} =[0.5,0.5,0.0] for x_{4} ={0},{1} and {0,1}$ $m_{1↓4} =[0.8,0.1,0.1] for x_{1}$

Summary

We have introduced evidential causal networks in which nodes represent Boolean algebras and the strength of these dependencies are given by rule-strength matrices. Matrices and their products are used to express the reasoning algorithms. The network generation aspects of this method have been implemented for a general-purpose knowledge discovery tool MKS, which provides linkage to databases and to libraries of discovery routines. The resulting software system therefore provides an early

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¹: Sadly, Francis McErlean died before publication of this paper. His contribution was invaluable and he will be greatly missed by his friends and colleagues.

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