Probability Applications

368 Chapter 15. Signal Validation Using Bayesian Belief Networks and Fuzzy Logic
influence of the parent node on the child node. Associated with each child node is a probability
distribution conditional on its parents. The nodes with no parents are called "root nodes"
and each has an a priori probability distribution.
At any instant, a certain body of external information is made available to the network.
This "evidence" is incorporated by instantiating the corresponding nodes. The directed
structure of the network and the conditional probabilities associated with each link store
a body of prior knowledge. This allows for the calculation of probability distributions of
the uninstantiated nodes in the network, given the evidence. This represents the conclusion
inferred by the network in light of the evidence presented and is called "inference." The
process of message passing that results in the inference is called "propagation" of evidence.
The inference scheme uses the Bayesian method to calculate updated "beliefs," i.e.,
the probabilities. A typical inference process consists of one initialization step, a series of
updating probabilities, and exchanging n and A messages among nodes. A ^.-message, which
is sent by a child node to its parent, represents the diagnostic information. A TT-message from
a parent node to a child node represents the causal influence. Once the network stabilizes,
the updated probabilities associated with each node represent the belief about the state of the
node in light of the evidence.
Consider a simple system that consists of a temperature sensor/indicator (Figure 15.1).
If the sensor is operational, then there is a correlation between its value and that of the
temperature. The lack of correlation in a faulty sensor is represented as a uniform probability
distribution in the conditional probability table of the indicator node; i.e., both high and low
are equally probable values for sensor reading, irrespective of the high or low value of the
temperature. When the temperature is above a certain threshold (high), the indicator is red;
otherwise it is blue. The network representation of this system uses three discrete-valued
nodes: one for the indicator reading (node /), one for the status of the sensor (node S),
and another (node T) for temperature, whose actual value is unknown at any given time.
The causal relationships and dependencies between the nodes are shown in Figure 15.1.
Node / has two possible values of "red" and "blue." Node S can be either "functional" or
"inoperative" representing the corresponding status of the sensor. Node T represents the
actual temperature that can be "high" if it is above the threshold or "low" otherwise. The
actual states of nodes S and T are not known at any given time. The network helps to form
a probabilistic inference about these states based on the indicator reading.
Based on the manufacturer's specifications, let the a priori probability of the functional
state of node 5 be 0.65. Since each node is binary-valued, a total of four conditional
probability statements are needed to uniquely define the dependence of / on T and S. These
values are shown in boxes attached to each node in Figure 15.1. At this point, the a priori
distribution for the node T needs to be provided for the network to be completely defined.
As a starting point, assume a noninformative prior, assigning 0.5 for the two possible states.
An observation about the indicator reading is presented to the network via instantiation of
the node /. Given this evidence, the network updates its beliefs about the other two nodes, S
and T. For example, if the indicator is observed to be red, the probability of the temperature
being high increases from 50% to 82.5%. Beliefs about node S remain unchanged, due to
the noninformative prior. The calculations showing the propagation of probabilities in the
network and the resultant inference about the status of the sensor are shown in detail below.
Initialization. The objective of this stage is to initialize the BBN such that the network
becomes ready to accept evidence. Probabilities for all nodes and all n and A. messages are
initialized for this purpose.