In probabilistic reasoning, random variables (RVs) are used to represent events and objects. By making various assignments to these RVs, we can model the current state of the world and weight the states according to the joint probabilities.
A Bayesian network is a directed acyclic graph. Directed arcs between RVs represent conditional dependencies. When all the parents of a given RV are instantiated, that RV is said to be conditionally independent of the remaining RVs given it's parents.
Allen's interval algebra is governed by 13 relations on the intervals.
Basically, there is a time interval in which each event occurs denoted
by [a, b] where a is the starting time point and b is the
termination point. Temporal relationships between events are expressed
as relations between their intervals. The relations between intervals,
denoted 
, are 
(figure 1).
For example, event A = [a,b] preceding event B = [c,d] is denoted
A < B indicating that a < b < c < d. The set of 13 relations
is mutually exclusive and exhaustive.
Figure 1: Allen's thirteen possible relations.
Uncertainty in the exact relationship between intervals is
expressed as disjunctions. For example, ``interval A precedes or
meets interval B'' is written as 
. Some commonly
used disjunctions are disjoint, written 
, and
contains, written 
[1].
A temporal random variable is a set of states, e.g. 
,
, or 
, and a set of
temporal intervals each having an associated random variable (RV).
Each RV has defined a density function giving the probability for each
state.
There is no requirement that the (i,r) pairs have any semantic
relationship to each other. The TRV, however, represents the state of
an event or process in time and should have a clear semantic meaning.
This is especially important in more restricted models where only one
(i,r) pair can be 
and the state of that one pair provides the
visible state of the entire TRV.
An assignment to a TRV consists of an assignment to each RV in the TRV:
Sometimes the state of all of the RVs in a TRV is not available. A partial temporal assignment is a subset of a temporal assignment.
A TBN is a directed acyclic graph in which the nodes are TRVs and the edges indicate that the source exerts direct causal influence on the destination. Furthermore, the causal influence is tempered with the thirteen temporal relations described above.
While, a TBN can only hold TRVs, a special class of TRV is used to represent random variables. These TRVs have only one interval which spans the entire model. For convenience, these TRVs are referred to as RVs.
What does it mean for one TRV, 
, to exert temporal causal influence on
another TRV, 
? The probability of to take on some particular state
is dependent on taking on some state on some interval fitting the
temporal relation, e.g.
``no interval in can have state unless that interval is before an
interval in having state .''
This is written
with every 
having 
. The MAP is an
function mapping from a set of RVs in to a single state in
of . The OR function is defined below.
In order to preserve Bayesian syntax/semantics, the value of a TRV can only
appear as a singleton to other variables in the TBN. This is the role of the
map function. A mapping must be done
from the set of values held by the RVs in the TRV to a single element of
. Just as care must be taken to avoid cycles in Bayesian networks, care
must be taken to ensure that the mapping functions are total. For many models,
these maps will be extremely simple, e.g.
which performs an exclusive or on the set.
The map 
maps from a singleton RV set to an element of
. The following example is for 
Thus the relationship 
, read
`` exerts direct causal influence on under all temporal relationships.''
is equivalent to the causal relation in Bayesian networks. This relationship
does not need to be explicitly stated for relations between RVs and from RVs to
TRVs.
If each random variable and temporal random variable is assigned, then the TBN is said to be completely assigned. The set of all of these assignments and there associated random variables forms a complete assignment to the TBN.
A partial assignment is a partial specification of the state of the TBN consisting of a subset of the variables of the TBN and the associated temporal assignments. More formally:
A partial assignment, 
, is said to be a subset of another
partial assignment, 
, (denoted 
)
if every 
in
(except those having 
)
has a corresponding 
in such that
and 
.
