Finding the relevant
past and future of a
decision problem
• Optimal policy for a decision D:
•
A rule how to act, in order to maximise expected utility
•
for every possible configuration of the past
• Optimal policy for a decision D:
•
A rule how to act, in order to maximise expected utility
•
for every possible configuration of the past
• Optimal policy for a decision D:
•
A rule how to act, in order to maximise expected utility
•
for every possible configuration of the past
..
optimise
..
x
x
x
x
required variable
• D, a decision in an influence diagram I
• variable X past(D) is required for D, if
there exists
• a realisation R of I
• a configuration y’ over dom(δ )\{X}
• states x1 and x2 of X
D
• So that the optimal strategies δ
D
differ, with different states of X
δD(x1, y’) != δD(x2, y’)
for D
relevant utility nodes
• the utility function U is relevant for decision D
• if there exists two realisations R and R of
1
I that differ only on U
• such that the optimal policies for D are
different in R1 and R2.
2
Proposition I
• Let D
be the last decision variable in the
influence diagram I.
n
• Let U be a utility node in I.
• Then U is relevant for D if and only if
n
there is a directed path from Dn to U.
Proposition 2
• Let D be the last decision variable in the
influence diagram I.
• Let X be a variable in past(D).
• Then X is required for D if and only if X is
d-connected to a utility node U relevant
for D, given past(D)\{X}.
algorithm
identify required variables
•
Let I be an influence diagram and let D1,D2,..,Dn be the decision variables
in I, indexed by order. to determine variables required for Di, req(Di):
•
•
set i:=n
for each decision variable Di not considered (i>0)
•
•
let Vi be the set of utility nodes to which there exists a path from Di
•
replace Di with a chance-variable representation of the policy for Di,
and let I be the resulting model
•
set i:=i-1
let req(Di) be the set of nodes X such that X
past(Di) and X is
d-connected to a utility node Vi, given past(Di)\{X}
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
change to chance-variable
representation
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
change to chance-variable
representation
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
cha
nge
to
rep chan
res
ceent
v
atio ariabl
e
n
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
Prop 11.1: Let Dn be the last decision variable in the influence diagram I, and let U be a utility node in I. Then U is relevant for
Dn if and only if there is a directed path from Dn to U.
Prop 11.2: Let D be the last decision variable in the influence diagram I, and let X be a variable in past(D). Then X is required
for D if and only if X is d-connected to a utility node relevant for D, given past(D)\{X}.
identifying relevant
future
identifying relevant
future
identifying relevant
future