Observational and Intervention BNs

One of the advantages of using BNs is the possibility of modelling causal effects between variables. With different causal representation, the reasoning process may differ. For example, inference on a model with a variable that is observed with a state (“seeing”) and the same state is generated by an intervention (“doing”) is different [160, 137]. These two types of modelling are named as an observation model and an intervention model respectively.

When performing the inference, observing a variable results in updated probabilities in both its causes and effects. For example, assume the overloading of a bridge can cause cracks, and cracks may lead to the bridge to fail. This can be modelled by a BN with a causal chain structure. With the inference, if we observed that there are severe cracks in a bridge deck, we can diagnose this bridge has a higher probability of being overloaded for over 10 years (cause), and predict this bridge has a higher probability of failing (effect). These inferences can be adjusted based on the causal structure of the model. Three basic causal structures are given in Figure 3.4 (top level) in the form of observational models developed by Hagmayer et al. [64], where their joint probabilities are modelled as:

• Causal chain: P(X ,Y, Z) = P(X )P(Y |X )P(Z|Y )

• Common effect: P(X ,Y, Z) = P(X )P(Y |X , Z)P(Z)

• Common cause: P(X ,Y, Z) = P(X )P(Y |X )P(Z|X )

By setting the variable to the observed value (represented by shaded ovals in the figure), we can calculate other events conditional on the observed variable in the usual way. Following the bridge cracks example, with the causal chain structure model, where X represents whether the bridge is overloaded for over 10 years, Y represents whether cracks exist, and Z represents whether the bridge is going to fail. Each variable is modelled as a Boolean variable where the state of true is represented as 1, and false is 0. The probability of the bridge is overloaded for over 10 years P(X = 1) given an observation that cracks exists P(Y = 1), we have P(X = 1|Y = 1) that is computed according to Bayes’s theorem from Equation 3.1:

P(X = 1|Y = 1) = P(Y = 1|X = 1)P(X = 1)

P(Y = 1|X = 1)P(X = 1) + P(Y = 1|X = 0)P(X = 0) (3.5)

Assume the probability of a bridge that is overloaded over the last 10 years is 0.2, P(X = 1) = 0.2. Since in this example the variable states are all binary, thus P(X = 0) = 1 − 0.2 = 0.8. Assume the probability of an overloaded bridge will cause cracks in 10 years is 0.7, P(Y = 1|X = 1) = 0.7, and the probability of having cracks even the bridge

Figure 3.4 Examples of three basic causal models represented in observational and interven- tion modes from Hagmayer et al. [64].

is not overloaded is 0.4, P(Y = 1|X = 0) = 0.4. From Equation 3.5, P(X = 1|Y = 1) is therefore equals to 0.3. Given an observation that there are cracks, the probability of bridge is overloaded for over 10 years increases from 0.2 to 0.3. Similar procedure can be performed to infer the probability of bridge failure given that there are cracks.

In this example the variables are linked with is a causal chain structure, but the conditional probabilities will be reasoned differently given different causal structures. Hagmayer et al. [64] and Meder et al. [111] provided a detail explanation of the use of other structures. These causal structures can also be extended to model a number of other structures, such as causal chain confounder and common cause confounder structures [112, 155]. Of these, the common cause confounder structure is later modelled in this thesis (see Section 4.4).

In an intervention BN model, the intervened variable is assigned a value in the same way as an observation. However, we must remove all the links from its causes to prevent backward reasoning about the causes given the value assigned by the intervention. The corresponding intervention models (we use a square node to represent an intervention) with the three basic causal structures are shown in Figure 3.4 (bottom level) altered from their observational models (showed by red crosses).

Follows the same example of bridge failure, in the observational model, the evidence of having cracks indicates a higher probability of bridge is overloaded for 10 years, and a higher probability of bridge will fail. But if the cracks are caused by environmental issues, it does not provide evidence about a bridge is overloaded either historically or in the future.

Manipulating the cracks existence (e.g. by repair) independently of the causes of cracks disconnects the cracks from their overloading cause, that is, mended the cracks does not imply the bridge was or will be overloaded. Hence, in the intervention model, the probability of bridge failure only depends on the existence of cracks, not on its loading.

To distinguish interventions from observations, Pearl introduced the use of do-operator [137]. For example, instead of P(X |Y = 1) that denotes the probability of X given Y is observed with a true state, P(X |do(Y = 1)) refers to the probability of X given that the state of Y is fixed to be true by an intervention. This operator renders a variable independent of all its causes when an intervention is performed. This process is also known as graph surgery.

For example, in Figure 3.4, assume in all observational causal structures, event Y is observed as Y = 1, while in the intervention models, event Y is manipulated as do(Y = 1), we have their joint distributions:

• Observations:

– Causal chain: P(X ,Y = 1, Z) = P(X )P(Y = 1|X )P(Z|Y = 1) – Common effect: P(X ,Y = 1, Z) = P(X )P(Y = 1|X , Z)P(Z) – Common cause: P(X ,Y = 1, Z) = P(X )P(Y = 1|X )P(Z|X )

• Interventions:

– Causal chain: P(X , do(Y = 1), Z) = P(X )P(Z|Y = 1) – Common effect: P(X , do(Y = 1), Z) = P(X )P(Z) – Common cause: P(X , do(Y = 1), Z) = P(X )P(Z|X )

With the observational BN model, we can use observation of evidence to update the probability of other variables. For example, with the backward reasoning, given a safety or reliability criterion as an observation, the observational model can tell you what repair action was more likely taken in history while the intervention model can give us an estimation of the effectiveness of different repair actions. By comparing the effectiveness of different maintenance actions, we can prioritise the maintenance actions. These models are developed in this thesis and presented in Section 4.4.