• Choose a category
• All
• Classic-Fiction

By Root 752 0

will be viewed from any angle outside the object, but it is insufficient for predicting how the object will be deformed if manipulated and squeezed by external forces. The additional information needed for making predictions such as the object’s resilience or elasticity is analogous to the information that causal assumptions provide. These considerations imply that the slogan “correlation does not imply causation” can be translated into a useful principle: One cannot substantiate causal claims from associations alone, even at the population level. Behind every causal conclusion there must lie some causal assumptions that are not testable in observational studies.

Any mathematical approach to causal analysis must acquire a new notation for expressing causal assumptions and causal claims. To illustrate, the syntax of probability calculus does not permit us to express the simple fact that “symptoms do not cause diseases,” let alone draw mathematical conclusions from such facts. All we can say is that two events are dependent—meaning that if we find one, we can expect to encounter the other, but we cannot distinguish statistical dependence, quantified by the conditional probability P(disease/symptom) from causal dependence, for which we have no expression in standard probability calculus. Symbolic representation for the relation “symptoms cause disease” is distinct from the symbolic representation of “symptoms are associated with disease.”

The need to adopt a new notation, foreign to the province of probability theory, has been traumatic to most persons trained in statistics partly because the adaptation of a new language is difficult in general, and partly because statisticians—this author included—have been accustomed to assuming that all phenomena, processes, thoughts, and modes of inference can be captured in the powerful language of probability theory. Causality formalization requires new mathematical machinery for cause–effect analysis and a formal foundation for counterfactual analysis including concepts such as “path diagrams,” “controlled distributions,” causal structures, and causal models.

12.5.1 Bayesian Networks

One of the powerful aspects of graphical models is that a specific graph can make probabilistic statements for a broad class of distributions. In order to motivate the use of directed graphs to describe probability distributions, consider first an arbitrary joint distribution p(a, b, c) over three variables a, b, and c. By application of the product rule of probability, we can write the joint distribution in the form

We now represent the right-hand side of the equation in terms of a simple graphical model as follows. First, we introduce a node for each of the random variables a, b, and c and associate each node with the corresponding conditional distribution on the right-hand side of the equation. Then, for each conditional distribution we add directed links, (arrows) to the graph from the nodes corresponding to the variables on which the distribution is conditioned. Thus, for the factor p(c|a, b), there will be links from nodes a and b to node c, whereas for the factor p(a) there will be no incoming links, as presented in Figure 12.33a. If there is a link going from a node a to a node b, then we say that node a is the parent of node b, and we say that node b is the child of node a.

Figure 12.33. A directed graphical model representing the joint probability distribution over a set of variables. (a) Fully connected; (b) partially connected.

For given K variables, we can again represent a joint probability distribution as a directed graph having K nodes, one for each conditional distribution, with each node having incoming links from all lower numbered nodes. We say that this graph is fully connected because there is a link between every pair of nodes. Consider now the graph shown in Figure 12.33b, which is not a fully connected graph because, for instance, there is no link from x1 to x2 or from x3 to x7. We may transform this graph to the corresponding representation of the joint probability distribution