We have seen that directed graphical models specify a factorization of the joint distribution over a set of variables into a product of local conditional distributions. They also define a set of conditional independence properties that must be satisfied by any distribution that factorizes according to the graph. We turn now to the second major class of graphical models that are described by undirected graphs and that again specify both a factorization and a set of conditional independence relations.

A Markov random field, also known as a Markov network or an undirected graphical model, has a set of nodes each of which corresponds to a variable or group of variables, as well as a set of links each of which connects a pair of nodes. The links are undirected, that is they do not carry arrows. In the case of undirected graphs, it is convenient to begin with a discussion of conditional independence properties.

8.3.1 Conditional independence properties

In the case of directed graphs, we saw that it was possible to test whether a particular conditional independence property holds by applying a graphical test called d-separation. This involved testing whether or not the paths connecting two sets of nodes were ‘blocked‘. The definition of blocked, however, was somewhat subtle due to the presence of paths having head-to-head nodes. We might ask whether it is possible to define an alternative graphical semantics for probability distributions such that conditional independence is determined by simple graph separation. This is indeed the case and corresponds to undirected graphical models. By removing the directionally from the links of the graph, the asymmetry between parent and child nodes is removed, and so the subtleties associated with head-to-head nodes no longer arise.

Suppose that in an undirected graph we identify three sets of nodes, denoted A, B, and C, and what we consider the conditional independence property.

(1)

To test whether this property is satisfied by a probability distribution defined by a graph we consider all possible paths that connect nodes in set A to nodes in set B. If all such paths pass through one or more nodes in set C, then all such paths are ‘blocked‘ and so the conditional independence property holds. However, if there is at least one such path that is not blocked, then the property does not necessarily hold, or more precisely there will exist at least some distributions corresponding to the graph that do not satisfy this conditional independence relation. Note that this is exactly the same as the d-separation criterion except that there is no ‘explaining away‘ phenomenon. Testing for conditional independence in undirected graphs is therefore simpler than in directed graphs.

An alternative way t view the conditional independence test is to imagine removing all nodes in set C from the graph together with any node in A to any node in B. If there are no such paths, then the conditional independence property must hold.

The Markov blanket for an undirected graph takes a particularly simple form, because a node will be conditionally independent of all other nodes conditioned only on the neighbouring nodes.

8.3.2 Factorization properties

We now seek a factorization rule for undirected graphs that will correspond to the above conditional independence test. Again, this will involve expressing th joint distribution as a product of functions defined over sets of variables that are local to the graph. We therefore need to decide what is the appropriate notion of locality in this case.

If we consider two nodes and that are connected by a link, then these variables must be conditionally independent given all other nodes in the graph. This follows from the fact that there is no direct path between the two nodes, and all other paths pass through nodes that are observe, and hence those paths are blocked. This conditional independence property can be expressed as

(1)

where denotes the set of all variables with and removed. The factorization of the joint distribution must therefore be such that and do not appear in the same factor in order for the conditional independence property to hold for all possible distributions belonging to the graph.

This leads us to consider a graphical concept called a clique, which is defined as a subset of the nodes in a graph such that there exists a link between all pairs of nodes in the subset. In other words, the set of nodes in a clique is fully connected. Furthermore, a maximal clique is a clique such that it is not possible to include any are illustrated by the undirected graph over four variables. This graph has five cliques of two nodes given by , and , as well as two maximal cliques given by and . The set is not a clique because of the missing link from to .

We can therefore define the factors in the decomposition of the joint distribution to be functions of the variables in the cliques. In fact, we can consider functions of the maximal cliques, without loss of generality, because other cliques must be subsets of maximal cliques. Thus, if is a maximal clique and we define an arbitrary function over this clique, then including another factor defined over a subset of these variables would be redundant.

Let us denote a clique by and the set of variables in that clique by . Then the joint distribution is written as a product of potential functions over the maximal cliques of the graph

(2)

Here the quantity , sometimes called the partition function, is a normalization constant and is given by

(3)

which ensures that the distribution given by (2) is correctly normalized. By considering only potential functions which satisfy we ensure that . In (3) we have assumed that comprises discrete variables, but the framework is equally applicable t continuous variables, or a combination of the two, in which the summation is replaced by the appropriate combination of summation and integration.

Note that we do not restrict the choice of potential functions to those that have a specific probabilistic interpretation as marginal or conditional distributions. This is in contrast to directed graphs in which each factor represents the conditional distribution of the corresponding variable, conditioned on the state of its parents. However, in special cases, for instance where the undirected graph is constructed by starting with a directed graph, the potential functions may indeed have such an interpretation, as we shall see shortly.

One consequence of the generality of the potential functions is that their product will in general not be correctly normalized. We therefore have to introduce an explicit normalization factor given by (3). Recall that for directed graphs, the joint distribution was automatically normalized as a consequence of the normalization of each of the conditional distributions in the factorization.

The presence of this normalization constant is one of the major limitations of undirected graphs. If we have a model with discrete nodes each having states, then the evaluation of the normalization term involves summing over states and so (in the worst case) is exponential in the size of the model. The partition function is needed for parameter learning because it will be a function of any parameters that govern the potential functions . However, for evaluation of local conditional distributions, the partition function is not needed because a conditional is the ratio of two marginal, and the partition function cancels between numerator and denominator when evaluating this ratio. Similarly, for evaluating local marginal probabilities we can work with the unnormalized joint distribution and then normalize the marginals explicitly at the end. Provided the marginals only involves a small number of variables, the evaluation of their normalization coefficient will be feasible.

So far, we have discussed the notion of conditional independence based on simple graph separation and we have proposed a factorization of the joint distribution that is intended to correspond to this conditional independence structure. However, we have not made any formal connection independence structure. However, we have not made any formal connection between conditional independence and factorization for undirected graphs. To do so we need to restrict attention to potential functions that are strictly positive (i.e., never zero or negative for any choice of ). Given this restriction, we can make a precise relationship between factorization and conditional independence.

To do this we again return to the concept of a graphical model as a filter. Consider he set of all possible distributions defined over a fixed set of variables corresponding to the nodes of a particular undirected graph. We can define to be the set of such distributions that can be expressed as a factorization of the form with respect to the maximal cliques of the graph. The Hammersley-Clifford theorem states that the sets and are identical.

Because we are restricted to potential functions which are strictly positive it is convenient to express them as exponentials, so that

(4)

where is called an energy function, and the exponential representation is called the Boltzmann distribution. The joint distribution is defined as the product of potentials, and so the total energy is obtained by adding the energies of each of the maximal cliques.

In contrast to the factors in the joint distribution for a directed graph, the potentials in an undirected graph do not have a specific probabilities interpretation. Although this gives greater flexibility in choosing the potential functions, because there is no normalization constraint, it does raise the question of how to motivate a choice of potential function for a particular application. This can be done by viewing the potential function as expressing which configurations of the local variables are preferred to others. Global configurations that have a relatively high probability are those that find a good balance in satisfying the (possibly conflicting) influences of the clique potentials.

8.3.4 Relation to directed graphs

We have introduced two graphical frameworks for representing probability distributions, corresponding to directed and undirected graphs, and it is instructive to discuss the relation between these. Consider first the problem of taking a model that is specified using a directed graph and trying to convert it to an undirected graph. In some cases this is straightforward. Here the joint distribution for the directed graph is given as a product of conditionals in the form

(1)

Now let us convert this to an undirected graph representation. In the undirected graph, the maximal cliques are simply the pairs of neighbouring nodes, and so from (2) we wish to write the joint distribution in the form

(2)

Let us consider how to generalize this construction, so that we can convert any distribution specified by a factorization over a directed graph into one specified by a factorization over an undirected graph. This can be achieved if the clique potentials of the undirected graph are given by the conditional distributions of the directed graph. In order for this to be valid, we must ensure that the set of variables that appears in each of the conditional distributions is a member of at least on clique of the undirected graph. For nodes on the directed graph having just one parent, this is achieved simply by replacing the directed link with an undirected link. However, for nodes in the directed graph having more than one parent, this is not sufficient. These are nodes that have ‘head-to-head‘ paths encountered in our discussion of conditional independence. The joint distribution for the directed graph takes the form

(3)

We see that the factor involves the four variables , and , and so these must all belong to a single clique if this conditional distribution is to be absorbed into a clique potential. To ensure this, we add extra links between all pairs of parents of the node . Anachronistically, this process of ‘marrying the parents‘ has become known as moralization, and the resulting undirected graph, after dropping the arrows, is called the moral graph. It is important to observe that the moral graph in this example is fully connected and so exhibits no conditional independence properties, in contrast to the original directed graph.

Thus in general to convert a directed graph into an undirected graph, we first add additional undirected links between all pairs of parents for each node in the graph and then drop the arrows on the original links to give the moral graph. Then we initialize all of the clique potentials of the moral graph to 1. We then take each conditional distribution factor in the original directed graph and multiply it into one of the clique potentials. There will always exist at least one maximal clique that contains all of the variables in the factor as a result of the moralization step. Note that in all cases the partition function is given by .

The process of converting a directed graph into an undirected graph plays an important role in exact inference techniques such as the junction tree algorithm. Converting from an undirected to a directed representation is much less common and in general presents problems due to the normalization constraints.

We saw that in going from a directed to an undirected representation we had to discard some conditional independence properties from the graph. Of course, we could always trivially convert any distribution over a directed graph into one over an undirected graph by simply using a fully connected undirected graph. This would, however, discard all conditional independence properties and so would be vacuous. The process of moralization adds the fewest extra links and so retains the maximum number of independence properties.

We have seen that the procedure for determining the conditional independence properties is different between directed and undirected graphs. It turns out that the two types of graph can express different conditional independence properties, and it is worth exploring this issue in more detail. To do so, we return to the view of a specific (directed or undirected) graph as a filter, so that the set of all possible distributions over the given variables could be reduced to a subset that respects the conditional independence implied by the graph. A graph is said to be a D map (for ‘dependency map‘) of a distribution if every conditional independence statement satisfied by the distribution is reflected in the graph. Thus a completely disconnected graph (no links) will be a trivial D amp for any distribution.