Bayesian Network - Example

Example

Suppose that there are two events which could cause grass to be wet: either the sprinkler is on or it's raining. Also, suppose that the rain has a direct effect on the use of the sprinkler (namely that when it rains, the sprinkler is usually not turned on). Then the situation can be modeled with a Bayesian network (shown). All three variables have two possible values, T (for true) and F (for false).

The joint probability function is:

where the names of the variables have been abbreviated to G = Grass wet, S = Sprinkler turned on, and R = Raining.

The model can answer questions like "What is the probability that it is raining, given the grass is wet?" by using the conditional probability formula and summing over all nuisance variables:

As is pointed out explicitly in the example numerator, the joint probability function is used to calculate each iteration of the summation function, marginalizing over in the numerator, and marginalizing over and in the denominator.

If, on the other hand, we wish to answer an interventional question: "What is the likelihood that it would rain, given that we wet the grass?" the answer would be governed by the post-intervention joint distribution function obtained by removing the factor from the pre-intervention distribution. As expected, the likelihood of rain is unaffected by the action: .

If, moreover, we wish to predict the impact of turning the sprinkler on, we have with the term removed, showing that the action has an effect on the grass but not on the rain.

These predictions may not be feasible when some of the variables are unobserved, as in most policy evaluation problems. The effect of the action can still be predicted, however, whenever a criterion called "back-door" is satisfied. It states that, if a set Z of nodes can be observed that d-separates (or blocks) all back-door paths from X to Y then . A back-door path is one that ends with an arrow into X. Sets that satisfy the back-door criterion are called "sufficient" or "admissible." For example, the set Z=R is admissible for predicting the effect of S=T on G, because R d-separate the (only) back-door path S←R→G. However, if S is not observed, there is no other set that d-separates this path and the effect of turning the sprinkler on (S=T) on the grass (G) cannot be predicted from passive observations. We then say that P(G|do(S=T)) is not "identified." This reflects the fact that, lacking interventional data, we cannot determine if the observed dependence between S and G is due to a causal connection or due to spurious created by a common cause, R. (see Simpson's paradox)

To determine whether a causal relation is identified from an arbitrary Bayesian network with unobserved variables, one can use the three rules of "do-calculus" and test whether all do terms can be removed from the expression of that relation, thus confirming that the desired quantity is estimable from frequency data.

Using a Bayesian network can save considerable amounts of memory, if the dependencies in the joint distribution are sparse. For example, a naive way of storing the conditional probabilities of 10 two-valued variables as a table requires storage space for values. If the local distributions of no variable depends on more than 3 parent variables, the Bayesian network representation only needs to store at most values.

One advantage of Bayesian networks is that it is intuitively easier for a human to understand (a sparse set of) direct dependencies and local distributions than complete joint distribution.