5 An Introduction to Imprecise Markov Chains
147
Fig. 5.1 A (partial) event tree for a binary state-space X = {a, b}. The vertices are situations, i.e.
elements of X ∗
, and the edges are induced by the prefix order ≺. Dashed lines represent branches
that are not shown in the figure. The tree has been augmented to a probability tree, by assigning to
each w ∈ X ∗ a local model p w . A time axis represents at which point in time the situations can
occur
Proceeding in this fashion, an infinite random walk along this tree generates a full
path ω : N 0 → X , where, for all t ∈ N 0 , the state ω(t) represents the (randomly
chosen) branch that we took along the tree at the (t + 1)-th step.
This ‘path construction’ view allows us also to connect back to the measuretheoretic definition that we encountered earlier. To obtain this correspondence in
one direction, fix a probability tree (X ∗
, ≺, p (·) ) and let (Ω, F ) be an appropriate
measurable space of discrete-time sample paths, on which we will aim to construct
the measure P quantifying, in the measure-theoretic sense, the uncertainty of the
corresponding stochastic process {X t } t∈N 0 on the resulting probability space.
We now reason intuitively by using the ‘random walk’ along the probability tree.
Starting from , we transition to a first situation x ∈ X with probability p (x).
From there, we could then perform the entire infinite random walk to generate the
remainder of the path. So, a different way of saying this is that, of all the random
paths ω ∈ Ω that could be generated, a fraction of p (x) of them will start with
ω(0) = x. Using also the interpretation given by Corollary 5.1, it therefore makes
sense to define the first-step marginal measure P ∗ (X 0 = x) := p (x) for all x ∈
X .
Let us now consider the next step, and assume the first step down the tree resulted
in a situation x ∈ X . Then, with probability p x (y), y ∈ X , the next situation will
be xy. In terms of paths that could be generated, a fraction of p x (y) of the paths that
147
Fig. 5.1 A (partial) event tree for a binary state-space X = {a, b}. The vertices are situations, i.e.
elements of X ∗
, and the edges are induced by the prefix order ≺. Dashed lines represent branches
that are not shown in the figure. The tree has been augmented to a probability tree, by assigning to
each w ∈ X ∗ a local model p w . A time axis represents at which point in time the situations can
occur
Proceeding in this fashion, an infinite random walk along this tree generates a full
path ω : N 0 → X , where, for all t ∈ N 0 , the state ω(t) represents the (randomly
chosen) branch that we took along the tree at the (t + 1)-th step.
This ‘path construction’ view allows us also to connect back to the measuretheoretic definition that we encountered earlier. To obtain this correspondence in
one direction, fix a probability tree (X ∗
, ≺, p (·) ) and let (Ω, F ) be an appropriate
measurable space of discrete-time sample paths, on which we will aim to construct
the measure P quantifying, in the measure-theoretic sense, the uncertainty of the
corresponding stochastic process {X t } t∈N 0 on the resulting probability space.
We now reason intuitively by using the ‘random walk’ along the probability tree.
Starting from , we transition to a first situation x ∈ X with probability p (x).
From there, we could then perform the entire infinite random walk to generate the
remainder of the path. So, a different way of saying this is that, of all the random
paths ω ∈ Ω that could be generated, a fraction of p (x) of them will start with
ω(0) = x. Using also the interpretation given by Corollary 5.1, it therefore makes
sense to define the first-step marginal measure P ∗ (X 0 = x) := p (x) for all x ∈
X .
Let us now consider the next step, and assume the first step down the tree resulted
in a situation x ∈ X . Then, with probability p x (y), y ∈ X , the next situation will
be xy. In terms of paths that could be generated, a fraction of p x (y) of the paths that