CPTs. Note, however, that the network in Figure 11a is consistent with common perceptions of causal influences, yet the one in Figure 11b violates these perceptions due to edges A → E and A → B. Is there any significance to this discrepancy? In other words, is there some additional information that can be extracted from one of these networks, which cannot be extracted from the other? The answer is yes according to a body of work on causal Bayesian networks, which is concerned with a key question: 16, 30 how can one characterize the additional information captured by a causal Bayesian network and, hence, what queries can be answered only by Bayesian networks that have a causal interpretation?

According to this body of work, only causal networks are capable of updating probabilities based on interventions, as opposed to observations. To give an example of this difference, consider Figure 11 again and suppose that we want to compute the probabilities of various events given that someone has tampered with the alarm, causing it to go off. This is an intervention, to be contrasted with an observation, where we know the alarm went off but without knowing the reason. In a causal network, interventions are handled as shown in Figure 12a: by simply adding a new direct cause for the alarm variable. This local fix, however, cannot be applied to the non-causal network in Figure 11b. If we do, we obtain the network in Figure 12b, which asserts the following (using d-separation): if we observe the alarm did go off, then knowing it was not tampered with is irrelevant to whether a burglary or an earthquake took place. This independence, which is counterintuitive, does not hold in the causal structure and represents one example of what may go wrong when using a non-causal structure to answer questions about interventions.

Causal structures can also be used to answer more sophisticated queries, such as counterfactuals. For example, the probability of “the patient would have been alive had he not taken the drug” requires reasoning about interventions (and sometimes might even require functional information, beyond standard causal Bayesian networks30). Other types of queries include ones for distinguishing between

one of the most
intriguing aspects
of Bayesian
networks is
the role they play
in formalizing
causality.

direct and indirect causes and for determining the sufficiency and necessity of causation. 30 Learning causal Bayesian networks has also been treated, 16, 30 although not as extensively as the learning of general Bayesian networks.

 

Beyond Bayesian networks

Viewed as graphical representations of probability distributions, Bayesian networks are only one of several other models for this purpose. In fact, in areas such as statistics (and now also in AI), Bayesian networks are studied under the broader class of probabilistic graphical models, which include other instances such as Markov networks and chain graphs (for example, Edwards11 and Koller and Friedman22). Markov networks correspond to undirected graphs, and chain graphs have both directed and undirected edges. Both of these models can be interpreted as compact specifications of probability distributions, yet their semantics tend to be less transparent than Bayesian networks. For example, both of these models include numeric annotations, yet one cannot interpret these numbers directly as probabilities even though the whole model can be interpreted as a probability distribution. Figure 9b depicts a special case of a Markov network, known as a Markov random field (MRF), which is typically used in vision applications. Comparing this model to the Bayesian network in Figure 9a, one finds that smoothness constraints between two adjacent pixels Pi and Pj can now be represented by a single undirected edge Pi – Pj instead of two directed edges and an additional node, Pi → Cij ← Pj. In this model, each edge is associated with a function f (Pi, Pj) over the states of adjacent pixels. The values of this function can be used to capture the smoothness constraint for these pixels, yet do not admit a direct probabilistic interpretation.

Bayesian networks are meant to model probabilistic beliefs, yet the interest in such beliefs is typically motivated by the need to make rational decisions. Since such decisions are often contemplated in the presence of uncertainty, one needs to know the likelihood and utilities associated with various decision outcomes. A