A hierarchical model is one which has an ordered structure, such that a sequence of
parameters of interest are conditionally independent of each other. For example, if
one has a population of similar specimens, then one might expect that they have
similar average crack growth rates. Let the average rate in specimen i be
, the observed rates at j times be 
and the overall average be
L. Then are assumed to be distributed with mean , and
distributed with mean L. Then 
that is that the are conditionally independent of L
given the . The parameter L, refers to the distribution of the parameters
, and is sometimes called a hyperparameter. Such models are used in many
different circumstances to model population effects [56].
In order to visualise the relationships between different parameters and hyperparameters in a model, a useful tool is the directed acyclic graph representation. Such graphs may be referred to as DAGs, and it is noted that the important properties of these graphs are that the arrows have direction, and that no cycles exist in the graph. Spiegelhalter has been an active proponent of directed acyclic graphs for some time [50], [26].
The DAG in Figure 
demonstrates such a graphical
structure. Following [50], v represents a node
from the set of nodes, V. A parent of v is any node which has
an arrow pointing from it to v. A descendant of v is any node
for which a sequence of arrows exists, starting at v and
finishing at the descendant.
A box around a node indicate that it is a constant (or fixed, known) quantity, whereas a circle around a node indicates that the quantity is a random variable. Solid arrows represent a probabilistic dependency, whereas a dotted arrow represents a deterministic relationship. The stacked boxes represent a sequence of plates, which may be thought of as a collection of exchangeable random variables. The graph is structured, so that all dependencies are visible. The model suggests that, conditional on knowing the parents of a node, the random variable is independent of all others in the graph, apart from its own descendants.
A DAG assists one with writing down the distribution of quantities of interest, since the independence structure is given by the graph. Define
and
Since by definition v is
conditionally independent of 
given 
, in order to factorise the joint
distribution of all the parameters one need only consider, for each node, 
.
This is useful for Gibbs sampling. The computer package WinBUGS
[51], allows one to specify certain types of models directly using
DAGs.