Decision Network

Description:

• Bayes’ Net but there is action and utility
  • 
• Square denotes action
• Eclipse denotes the chance nodes
• Kite denotes utility
• Arc denotes influence

Expected utility (EU):

• $EU(a)=P(s).U(s,a)$: Expected utility for taking action a

Maximum Expected Utility (MEU):

• $MEU(E=e)={\max }_{a}EU(a|E=e)={\max }_a{\sum }_{s}P(s|e)U(s,a)$
• action maximizes the expected utility given the evidence

Outcome Tree:

• Almost like an Expectimax Search but for outcome trees we annotate our nodes with what we know at any given moment (inside the curly braces)
• 

Value of Perfect Information (VPI):

• Think of the forecast as a separate variable, and we perfectly know the outcome that that variable
  • Forecast variable has their own distribution of being good/bad
  • different from probability of var given that forecast var
• With knowing more evidence, we can have different probablities
  • $MEU(e,E^{\prime })={\max }_a{\sum }_{s}P(s|e,e^{\prime })U(s,a)$
    • where $P(s|e,e^{\prime })$ is probability of actual variable given other evidence and forecast
• $VPI(E^{\prime },e)=MEU(e,E^{\prime })-MEU(e)$
• Properties of VPI:
  • $\forall E^{\prime },e:VPI(E^{\prime }|e)\ge 0$: non negative: 0. Observing new information always allows you to make a more informed decision, and so your max-imum expected utility can only increase (or stay the same)
  • $VPI(E_j,E_k|e)\ne VPI(E_j|e)+VPI(E_k|e)$: Nonadditivity, observed twice doesnt increase VPI by sum of them
  • $VPI(E_j,E_k|e)=VPI(E_j|e)+VPI(E_k|e,E_j)+VPI(E_k|e)+VPI(E_j|e,E_k)$: Order independent, observe 2 variables in any order is the same