Bayesian Network Inference Algorithms - Bayesian Networks in R: With Applications in Systems Biology - page 89

4.1.3 Causal Inference

When a Bayesian network is given a causal interpretation, the interpretation of
queries and evidence changes as well. Just as the arcs in the network describe causal
relationships instead of probabilistic dependencies, queries evaluate the probability
of known causes given their effects or vice versa.
In this setting, posterior probabilities are not interpreted in terms of beliefs chang-
ing according to some observed evidence but rather as measures of the effects of
interventions on the causal structure. To distinguish the latter from the former, we
will denote interventions with I while keeping the same general notation for both.
Interventions play the same role that evidence had in Sect. 4.1.1 , and like evidence,
they can be classified either as ideal ( perfect ) interventions or stochastic ( imperfect )
interventions ( Korb et al. , 2004 ).
Ideal interventions represent the causal analogous of hard evidence; they describe
an action whose only effect is to fix the values of the variables in I to particular set
of values

= X i 1 = x k .

I x 1 , X i 2 = x 2 ,..., X i k = (4.9)

Conditional probability queries of the form

P ( Q | I , G , Θ )= P ( X j 1 ,..., X j l | I , G , Θ ) , (4.10)

involving ideal interventions are called intervention queries . They evaluate the con-
sequences of the intervention I on Q through its posterior distribution. If some hard
evidence on a third set of variables is included in the query as well, so that

P ( Q | I , E , G , Θ )= P ( X j 1 ,..., X j l | I , E , G , Θ ) , (4.11)

the query is called a counterfactual query , and it evaluates the consequences of
intervention I in a particular scenario defined by the hard evidence E .Inotherwords,
it evaluates the consequences of I in an alternate world in which E happened instead
of the values actually observed for the sample; hence the name.
Stochastic interventions are very difficult to handle in their most general form.
Unlike soft evidence, not only the variables in I are not fixed, but the set of variables
that are included in I is a random variable. For this reason, they are rarely used
in practice even under the simplifying assumption that the set I is not random. In
most cases, assuming that interventions are ideal results in significant computational
savings without noticeably degrading the quality of the query. This is the case, for
example, in the protein-signaling data from Sachs et al. ( 2005 ) studied in Sect. 2.5 .
Even though the stimulatory cues and the inhibitory interventions applied to the
various parts of the data set are hardly ideal, they are assumed to be so to include
their effects in the structure learning process. The conclusions of the original paper
indicate how this assumption did not invalidate the results of structure learning, but
improved its ability to correctly identify causal relationships instead.