Bayesian network terminology and topology

According to Jensen (2001, p28), “… in principle there are two kinds of decisions, namely test decisions and action decisions. A test decision is a decision to look for more evidence to be entered into the model, and an action decision is a decision to change the state of the world.” As Jensen points out, “… in real life, this distinction is not very sharp; tests may have side-effects, and by performing a treatment against a disease, evidence on the diagnosis may be acquired. In order to be precise, we should say that decisions have two aspects, namely a test aspect and an action aspect.”

The two aspects are handled differently in connection with Bayesian networks, and accordingly are treated separately. Actions should be divided into two types, namely

)

intervening actions, which force a change of state for some variables in the model, and non-intervening actions, where the impact is not part of the model. Although both observations and intervening actions change the probability distributions in the model, they are fundamentally different. To illustrate this, consider the examples in Figure 2.4 and Figure 2.5. From the point of view of entering evidence and propagating probabilities, the two Bayesian networks shown are equivalent.

However, the difference becomes apparent when taking an aspirin. If the same utility and decision nodes are added in Figure 2.4, taking an aspirin will cure flu but will have no effect on sleepiness. This would not be a correct (causal) representation because aspirin does not actually cure the Flu (although some may think so) (Jensen, 2001).

Figure 2.4. A Bayesian network of diagnostic reasoning equivalent to the one in Figure 2.5

Figure 2.5. A simple flu decision model with an action (aspirin=A) and a test temperature=T) attached. The action has no impact on P (Flu)

In practice, diagnostic (reverse) links (Figure 2.5) are only used for test decisions (i.e.

when carrying out tests to estimate the value of collecting evidence on different indicator variables). For modelling the impact of intervening, or action decisions - in this case taking an asprin - the links should follow the cause-effect relationship, so as to represent the impact of the action.

A Bayesian network (Bn) consists of a set of variables and a set of directed links between variables. When describing the relations in a Bn the wording of family relations is generally used (e.g. Jensen, 2001; Castelletti and Soncini-Sessa, 2007):

Flu Fever Sleepy

T

A

Flu Fever Sleepy

if there is a link from A to B, then B is referred to as a child of A and A is a parent of B. A node which does not have any parents is called a root node and represents an input variable. A node without children is a leaf node and constitutes an output variable. Each node in the network is assigned with a set of discrete values or states, which represent all the possible conditions that that variable, represented by the node, can take. In Bns node states can be either quantitative or qualitative. For each node (except the root nodes) a conditional probability table (CPTs) is specified. The probabilities entered in the cpt describe the strength and weights of causal relationships (parameters) between nodes when other nodes are in a particular state.

Once the prior probabilities of a number of variables (usually the input variables) have been specified in chance nodes, it is possible to calculate the posterior probabilities for all the nodes in the network (belief propagation). This is done by employing basic probability calculus and Bayes’ theorem, described in Section 2.3.1.

As new knowledge about the system is obtained in the form of observations (evidence) about one or more variables, the prior probabilities for node states are updated. The procedure of adding evidence, also referred to as instantiation, results in the beliefs about states of other connected variables in the network to be updated through belief propagation, described below. Belief propagation in Bns is essentially a computational tool for communicating probabilistic inference between nodes within a Bn model. In practice, the combination of belief propagation and Bayes’ theorem in Bns produces a powerful modelling tool that allows both bottom-up (or backward-looking) probabilistic inference to address diagnostic tasks or top-down (forward-looking) probabilistic inference for predictive/explanatory purposes (Castelletti and Soncini-Sessa, 2007). In the first case, the evidence of an effect is given and the most likely cause is inferred. In the second, the probability of an effect is computed once the evidence for one or more of its causes is provided.

For Bayesian networks the graphical convention shown in Figure 2.6, below, is used, where a rectangle denotes a decision node, a diamond denotes a utility node and an oval denotes a chance node. Each state in the chance node connected to the utility node is assigned a corresponding value in the utility node.

DETERMINISTIC

LAYER

PROBABILISTIC

LAYER

DECISION NODE UTILITY NODE

CHANCE NODE Child CHANCE NODE

(Parent 1)

CHANCE NODE (Parent 2)

CHANCE NODE (Parent 3)

◊

□

○

Figure 2.6. Influence Diagram structure showing three node types

Chance nodes in a Bayesian network represent the probabilistic layer of the problem domain. As such they are an objective representation of the world. Decision and utility nodes represent human intervention in the model

They represent the utility (or the value) that results from a given decision will have, given the updated probabilities in the chance nodes. Utility nodes use subjective values (utility functions) to quantify the value of different states in the connected chance nodes. The maximum expected utility (MEU) is calculated by factorising the different utility functions using the probabilities in the connected chance node.

The model in Figure 2.7, below, is a simplified Bayesian network of a water demand management decision for a city.

Figure 2.7. A simplified Influence Diagram of a WDM decision

The model considers whether, given the reservoir level forecast based on the evidence added in the two parent chance nodes, it is necessary to reduce raw water abstraction by implementing a WDM programme. The directed links show that raw water abstraction and meteorological conditions impact on future reservoir levels.

Figure 2.8 (overleaf) shows the model in Figure 2.7 in use. Four intervention options are assigned in the decision node (no programme, or minimum, moderate, maximum programmes). Node states are shown in the monitor windows overlapping each node. The different model instantiations show (A) the model in its resting state, (B) propagated conditional probabilities give evidence on population of city and meteorological conditions and (C) propagated conditional probabilities and utilities for the decision ‘maximum programme’. The utility functions and conditional probabilities are shown in the box in the bottom-left.

2.3.2.1 Decision trees and utility theory

Decisions trees are an alternative way to structure Bns and the decision tree below, Figure 2.9, which is a decision tree for a section of the ID in Figure 2.8, demonstrates how the posterior probability distribution given a set of evidence permits calculation of the maximum expected utility (MEU) for a decision.

A Bayesian network software package called Hugin was selected for use in the case study field work based on a number of technical criteria. The full review of different platforms and technical criteria used is presented in Appendix C.

A

B

C

Propagated conditional probabilities, given evidence on ‘population of city

and ‘meteorological conditions’.

Propagated conditional probabilities, given decision

‘maximum program’

Utility functions Maximum expected

utilities (MEU)

Figure 2.8. Model instantiations and populated conditional probability tables for the simplified ID of the WDM decision for the Iskar Dam near Sofia

Raw water

Figure 2.9. Decision tree showing maximum expected utilities derived from the posterior probabilities

The following section discusses Bayesian modelling approaches for decision analysis and managing uncertainty.