5 An Introduction to Imprecise Markov Chains
161
imprecise probability theory—which is weaker than strong independence a different
type of independence, and what would hold of all P ∈ P were Markov chains.
In a similar vein, the notion of homogeneity is here only enforced on the lower
envelope. So, for an IDTMC P that is homogeneous, there may be processes P ∈ P
that are neither Markov nor homogeneous.
The reason why we stress this so strongly is twofold. First of all, it implies that
the structural assumptions of an imprecise Markov chain are in fact much weaker
than those of a precise Markov chain—we no longer assume that future events are
fully independent of the history, given the current state, or that their distribution
is independent of the point in time. They might be, of course—there are elements
P ∈ P that satisfy those properties—but it’s not enforced as strictly. In other words,
this model also represents ‘higher-order’ uncertainty about the structural properties
of the system that we are trying to model.
The second reason is that this property is central to all the efficient computational
methods that have been developed for working with imprecise Markov chains. We
will next illustrate this point by moving the discussion to the representation of
IDTMCs as imprecise probability trees.
5.3.1 Imprecise Probability Trees
Recall that for precise probability trees, we associate with each situation w ∈ X ∗
a local model p w , which is a probability mass function on X . In contrast, in order
to define imprecise probability trees, we will consider imprecise local models. Such
an imprecise local model P w is simply a set of probability mass functions on X .
This leads to the following definition:
Definition 5.11 (Imprecise probability tree) An imprecise probability tree is a
tuple (X ∗
, ≺, P (·) ), where (X ∗
, ≺) is an event tree and P (·) is a set-valued
function such that, for all w ∈ X ∗
, P w is a non-empty set of probability mass
functions on X .
An obvious question is how one should interpret such imprecise probability trees.
As a first step, we consider the (precise) probability trees that are compatible with a
given imprecise probability tree:
Definition 5.12 Let (X ∗
, ≺, P (·) ) be an imprecise probability tree. Then a (precise) probability tree (X ∗
, ≺, p (·) ) is called compatible with this imprecise probability tree, if p w ∈ P w for all w ∈ X ∗
.
This immediately lets us connect back to the sets-of-measures that we discussed
before. Specifically, consider an imprecise probability tree (X ∗
, ≺, P (·) ), and
suppose the tree (X ∗
, ≺, p (·) ) is compatible with it. Then, using the method
outlined in Sect. 5.2.1, we can associate a (precise) measure P to this precise
tree. Collecting in the set P all the associated measures of all precise trees that are
compatible with the imprecise tree, we obtain a set representation as in Sect. 5.3.
161
imprecise probability theory—which is weaker than strong independence a different
type of independence, and what would hold of all P ∈ P were Markov chains.
In a similar vein, the notion of homogeneity is here only enforced on the lower
envelope. So, for an IDTMC P that is homogeneous, there may be processes P ∈ P
that are neither Markov nor homogeneous.
The reason why we stress this so strongly is twofold. First of all, it implies that
the structural assumptions of an imprecise Markov chain are in fact much weaker
than those of a precise Markov chain—we no longer assume that future events are
fully independent of the history, given the current state, or that their distribution
is independent of the point in time. They might be, of course—there are elements
P ∈ P that satisfy those properties—but it’s not enforced as strictly. In other words,
this model also represents ‘higher-order’ uncertainty about the structural properties
of the system that we are trying to model.
The second reason is that this property is central to all the efficient computational
methods that have been developed for working with imprecise Markov chains. We
will next illustrate this point by moving the discussion to the representation of
IDTMCs as imprecise probability trees.
5.3.1 Imprecise Probability Trees
Recall that for precise probability trees, we associate with each situation w ∈ X ∗
a local model p w , which is a probability mass function on X . In contrast, in order
to define imprecise probability trees, we will consider imprecise local models. Such
an imprecise local model P w is simply a set of probability mass functions on X .
This leads to the following definition:
Definition 5.11 (Imprecise probability tree) An imprecise probability tree is a
tuple (X ∗
, ≺, P (·) ), where (X ∗
, ≺) is an event tree and P (·) is a set-valued
function such that, for all w ∈ X ∗
, P w is a non-empty set of probability mass
functions on X .
An obvious question is how one should interpret such imprecise probability trees.
As a first step, we consider the (precise) probability trees that are compatible with a
given imprecise probability tree:
Definition 5.12 Let (X ∗
, ≺, P (·) ) be an imprecise probability tree. Then a (precise) probability tree (X ∗
, ≺, p (·) ) is called compatible with this imprecise probability tree, if p w ∈ P w for all w ∈ X ∗
.
This immediately lets us connect back to the sets-of-measures that we discussed
before. Specifically, consider an imprecise probability tree (X ∗
, ≺, P (·) ), and
suppose the tree (X ∗
, ≺, p (·) ) is compatible with it. Then, using the method
outlined in Sect. 5.2.1, we can associate a (precise) measure P to this precise
tree. Collecting in the set P all the associated measures of all precise trees that are
compatible with the imprecise tree, we obtain a set representation as in Sect. 5.3.