Uncertain Reasoning

1
Uncertain Reasoning

• Dempster-Shafer theory (cont.)

2
Reasoning on Markov trees

• An undirected graph looks like this

3
Reasoning on Markov trees

• Undirected graphs are powerful mathematical
  modelling tools. Among many other uses, they can
  be used for knowledge representation, as may have
  been hinted at in previous lectures of this
  course.
• Undirected graphs admit several special cases of
  interest. One special case which is particularly
  interesting are trees.

4
Reasoning on Markov trees

• A tree looks like this

5
Reasoning on Markov trees

• Tree
• There is no path leaving a node and reaching the
  same node.
• The path between two different nodes is always
  unique.

6
Reasoning on Markov trees

• A different representation for graphs

a
c
b
d
g
h
f
e
i
7
Reasoning on Markov trees

• A different representation for graphs

a
c
b
d
g
h
f
e
i
8
Reasoning on Markov trees

• A different representation for graphs

a
c
b
d
g
h
f
e
i
9
Reasoning on Markov trees

• A different representation for graphs
• Edges are two-element sets.
• Nodes are (singleton) intersections of edges.

10
Reasoning on Markov trees

• A different representation for graphs
• ... so what?

11
Reasoning on Markov trees

• A different representation for graphs
• Set-based representation of graphs is easier to
  generalise, so that we can define hypergraphs.

12
Reasoning on Markov trees

• Hypergraph
• Set of non-empty subsets of a finite set V.
• Non-empty sets (of any size) are hyperedges.
• Intersections of hyperedges are hypernodes. A
  hypernode can have more than one element.

13
Reasoning on Markov trees

• Skeletal hypergraph
• No hyperedge is a subset of another hyperedge.

14
Reasoning on Markov trees

• Hypertrees
• In a hypergraph, a path between elements v and w
  is a sequence of hyperedges E1, ..., Eq such
  that
• v ? E1, w ? Eq.
• For all k 1, ..., (q-1), Ekn Ek1 ? .

15
Reasoning on Markov trees

• Hypertrees
• A twig is a hyperedge T such that there is
  another hyperedge B with the following
  properties
• B ? T.
• B n T ? .
• For all hyperedge E such that E n T ? and E ?
  T, v ? E n T implies v ? B.

16
Reasoning on Markov trees

• Hypertrees
• A hypertree is a hypergraph such that the
  hyperedges can be ordered as E1, ..., En, in such
  way that hyperedge Ei is a twig of hypergraph E1,
  ..., Ei for all i 1, ..., n.

17
Reasoning on Markov trees

• Markov trees
• A Markov tree is a pair (G, M) such that
• G is a set of hyperedges forming a hypergraph.
• M is a set of edges, i.e. two-element sets of
  elements of G.
• If E, F ? M, then E n F ? .
• If E, F ? G, E ? F, and there is a v such that v
  ? E and v ? F, then v ? all elements of M along
  the path from E to F.

18
Reasoning on Markov trees

• Markov trees
• For every hypertree, one can construct a
  corresponding Markov tree such that the
  hyperedges of the hypertree are nodes of the
  Markov tree.
• For every Markov tree, one can construct a
  corresponding hypertree such that the nodes of
  the Markov tree are hyperedges of the hypertree.

19
Reasoning on Markov trees

• Markov trees
• ... so what?

20
Reasoning on Markov trees

• Markov trees
• Markov trees can be used for knowledge
  representation, to characterise independent
  evidences gathered about certain events. We can
  relate mass distributions to nodes of a Markov
  tree, and then propagate beliefs along paths.

21
Reasoning on Markov trees

• How to interpret an interval? A possible
  intuitive guideline
• Bel, Pl Interpretation
• 0,1 no knowledge at all
• 0, 0 event is false
• 1, 1 event is true
• 0.3, 1 partial support for event
• 0, 0.7 partial support for not(event)
• 0.3, 0.7 partial support for both event and
  not(event)
• 0.3, 0.3 exact probability of event is known