Uncertain Reasoning

1
Uncertain Reasoning

• Dempster-Shafer theory (cont.)

2
Reasoning on Markov trees

• Let us consider, instead of simply a domain O, a
  domain encoded as a product space OM.
• We will abuse notation a little, and use M both
  as a natural number and the set 1, 2, ..., M.
• Let N c M, and vN be a function that takes
  subsets of ON to non-negative real numbers. We
  call vN a valuation.
• Let V be the set of all valuations. Later we will
  constrain valuations to be mass distributions.

3
Reasoning on Markov trees

• Mass distributions will constitute a special
  class of valuations. We call any such special
  class admissible valuations.
• Let S, T c M, vS a valuation of subsets of OS and
  vT a valuation of subsets of OT. We define vSxvT
  as a valuation of subsets of OS U T as follows
• If vS is not admissible or vT is not admissible,
  then vSxvT is not admissible.
• If both vS and vT are admissible, then not
  necessarily vSxvT is admissible. If vSxvT is
  admissible, then vS and vT are combinable.

4
Reasoning on Markov trees

• Let S c T c M. A projection ?(TS) is a mapping
  between valuations of subsets of OT and of OS,
  that preserves admissibility if vT is
  admissible, then so is its projection if vT is
  not admissible, then so is not its projection.
• If the valuations are mass distributions,
  projection corresponds exactly to the notion of
  projection defined in previous lectures, and the
  product corresponds to the combination rule for
  information fusion.

5
Reasoning on Markov trees

• Our goal is, given the valuations for product
  spaces of the form OTi, obtain the product
  valuation (through information fusion the so
  called Dempster's rule of combination) and then
  project the resulting valuation to determine the
  valuation of an event of interest.

6
Reasoning on Markov trees

• These calculations are in general computationally
  intractable.
• As proved by Prakash Shenoy and Glenn Shafer, if
  the dependencies among events are structured as a
  Markov tree, the calculations can be done
  relatively efficiently, through an interleaving
  of projections and fusions.

7
Reasoning on Markov trees

• Let T be a twig, vT be its valuation, and S be
  its neighbour in the Markov tree. S can be the
  neighbour of more twigs. T sends as a message
  to S the value ?(T TnS)(vT), i.e. its valuation
  projected to the variables that T and S have in
  common.

8
Reasoning on Markov trees

• S, in turn, updates its valuation given the
  messages it receives, calculatingvS x ?(Ti
  TinS)(vTi), for all twigs Ti to which S is
  neighbour.

9
Reasoning on Markov trees

• Example we can use this framework for uncertain
  reasoning to permanently update the belief state
  of an agent in the face of new evidence. Previous
  belief state is taken as an independent
  observation, and the incoming observations are
  combined with the previous ones to gradually
  update them.

10
Reasoning on Markov trees

• Example let us consider the following simple
  Markov tree, where the twigs are fed directly by
  the sensors of a robot. To simplify our exemple,
  each variable admits only two values (true and
  false).

11
Reasoning on Markov trees

• Example

T L F
A T
L F B
L S B
12
Reasoning on Markov trees

• Example
• We have a mass distribution for each hyperedge.
  Let us say that our interest is to know the
  belief distribution for the root hyperedge, i.e.
  the distribution for the possible values
  respectively for the variables T, L and F 000,
  001, 010, 011, etc.

13
Reasoning on Markov trees

• Example
• We assume that we already have mass distributions
  for each hyperedge. We now assume, however, that
  new observations update the values in the twigs.

14
Reasoning on Markov trees

• Example
• In this case, the twigs generate the
  corresponding projections. For example, to go up
  from the twig LSB to the node LFB, we use the
  projection ?(LSB LB). This gives partial
  information about LFB, to be combined using
  Dempster's rule with the previous beliefs of LFB.
  The process can be repeated up to the root of the
  tree.