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Danny C. Lee
first articulated the conditional probability relationship between a prior hypothesis, evidence, and a posterior hypothesis that is at the core of current belief
network methodology.
I do not cover the details of BBNs here nor discuss all the ramifications of their
use. Readers are referred to the references herein for such information. Rather, I
would hope to stimulate by way of example, sufficient interest in BBNs that
readers might pursue more advanced understanding on their own.
Pearl (1988) defines a BBN as
A directed acyclic graph in which each node represents a random variable that can take on
two or more possible values, and the arcs signify the existence of direct influences between
linked variables. The strengths of these influences are quantified by forward conditional
probabilities.
To illustrate, consider a simple example represented by a directed acyclic graph
containing three nodes (Fig. 9.2). Each node represents a system property that is
characterized by using a discrete random variable. The nodes represent (A) watershed history, (B) habitat condition, and (C) population trend. The directed arcs
signify that watershed history influences habitat condition directly, which, in turn,
affects the population trend.
Each random variable, or node, can take on a range of values. To simplify
matters, each variable is divided into three or four discrete values, each of which
may represent a range of conditions. At any given point in time, the state of the
system is reflected by the set of random variables, A i , B j , C k , a subset of all
possible combinations of A, B, C. Although there is only one true state of the
system, one may be uncertain of its nature. This uncertainty is expressed as a
belief vector, represented here by the histogram at each node. The belief (probability) attached to each level within each node represents the degree to which one
believes each level is the true state of the system. In this example, I am certain that
the watershed is undisturbed but less certain about habitat conditions and recent
population trend; the most likely combination is favorable habitat and a stable
population trend.
Simply stated, the purpose of a BBN is to track belief vectors and update them
systematically as information is added to the system. The terminology associated
with trees is often used to describe BBNs because of their branching structure.
Nodes that have no directed arcs entering them are referred to as root nodes,
whereas those with no directed arcs originating from them are called leaf nodes. In
this simple example, watershed history is a root node, and population trend is a
leaf node. Once the structure of a causal network has been defined, the bulk of the
remaining effort is dedicated to developing reasonable estimates for the link
matrices. Link matrices are conditional probability matrices that define the relationships between nodes (Table 9.1). The values within the matrices come from
combinations of empirical evidence and expert opinion or may be generated by
ancillary models as demonstrated below.
Updating the BBN involves entering findings (i.e., changing the belief vectors
of nodes in which you have information germane to the issue at hand). Findings
Danny C. Lee
first articulated the conditional probability relationship between a prior hypothesis, evidence, and a posterior hypothesis that is at the core of current belief
network methodology.
I do not cover the details of BBNs here nor discuss all the ramifications of their
use. Readers are referred to the references herein for such information. Rather, I
would hope to stimulate by way of example, sufficient interest in BBNs that
readers might pursue more advanced understanding on their own.
Pearl (1988) defines a BBN as
A directed acyclic graph in which each node represents a random variable that can take on
two or more possible values, and the arcs signify the existence of direct influences between
linked variables. The strengths of these influences are quantified by forward conditional
probabilities.
To illustrate, consider a simple example represented by a directed acyclic graph
containing three nodes (Fig. 9.2). Each node represents a system property that is
characterized by using a discrete random variable. The nodes represent (A) watershed history, (B) habitat condition, and (C) population trend. The directed arcs
signify that watershed history influences habitat condition directly, which, in turn,
affects the population trend.
Each random variable, or node, can take on a range of values. To simplify
matters, each variable is divided into three or four discrete values, each of which
may represent a range of conditions. At any given point in time, the state of the
system is reflected by the set of random variables, A i , B j , C k , a subset of all
possible combinations of A, B, C. Although there is only one true state of the
system, one may be uncertain of its nature. This uncertainty is expressed as a
belief vector, represented here by the histogram at each node. The belief (probability) attached to each level within each node represents the degree to which one
believes each level is the true state of the system. In this example, I am certain that
the watershed is undisturbed but less certain about habitat conditions and recent
population trend; the most likely combination is favorable habitat and a stable
population trend.
Simply stated, the purpose of a BBN is to track belief vectors and update them
systematically as information is added to the system. The terminology associated
with trees is often used to describe BBNs because of their branching structure.
Nodes that have no directed arcs entering them are referred to as root nodes,
whereas those with no directed arcs originating from them are called leaf nodes. In
this simple example, watershed history is a root node, and population trend is a
leaf node. Once the structure of a causal network has been defined, the bulk of the
remaining effort is dedicated to developing reasonable estimates for the link
matrices. Link matrices are conditional probability matrices that define the relationships between nodes (Table 9.1). The values within the matrices come from
combinations of empirical evidence and expert opinion or may be generated by
ancillary models as demonstrated below.
Updating the BBN involves entering findings (i.e., changing the belief vectors
of nodes in which you have information germane to the issue at hand). Findings