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14 Jan, 2016

Decision Theory


Decision Theory

Notation:

  1. aA: a is an “action” and A is the set of all possible actions
  2. L(θ,a): loss function for θΘ and aA, L(θ,a)K>
    • useful loss functions:
    • Quadratic loss: (θa)2
    • Absolute loss: |θa|
    • 0-1 loss: I(θa)
  3. δ(x): decision rule, function from X to A, X is the sample space

Example:

  • Drug company needs to market a new pain killer:
    • θ: proportion of the market that the drug will capture
    • θΘ=[0,1], at A=[0,1]
    • L(θ,a)=θa if θa
    • L(θ,a)=2(aθ) if θ<a
  • Survey of x peolpe out of n responded “yes”
    • possible model x|nBin(n,θ)
    • possible decision rule: δ(x)=x/n
  • Classical Decision Theory
    • Def: The risk function of a decision rule δ(x)
    • R(θ,δ)=xL(θ,δ(x))p(x|θ)dx
    • we say that δ1(x) is “better” than δ2(x) if R(θ,δ1)R(θ,δ2)θ.
    • How do we pick an estimator ˆθ for θ?
    • Choose ˆθ(δ(x)) that minimizes R.
      • Typically, we need to constrain θ space to get an optimum. For example, unbiased, linear, etc.
      • Otherwise, you have many local minimums.
  • Bayesian Decision Theory
    • θ unknown random variable
    • x are observed data
    • Def: Let π(θ) a pdf at the time of decision making. The Bayesian expected loss for an action a is ρ(π\*,a)=ΘL(θ,a)π(θ)dθ=Eπ(θ)[L(θ,a)]
    • Ex|θ[L(θ,a)]=R(θ,a)
    • Eθ[L(θ,a)]=ρ(π,a), π is prior
    • Eθ|x[L(θ,a)]=ρ(p,a), ρ is posterior
    • Bayesian decision principle: choose aA that minimizes ρ(π,a). This action is called a Bayes action.
    • Example (Drug Company):
    • Assume:
      • π(θ)=1/10 if .1<θ<.2, 0 o.w.
      • if no data: ρ(π,a)=a0L(θ,a)π(θ)dθ
        • =a0 2(aθ)π(θ) dθ+1a (θa)π(θ) dθ
      • case 1: a.1ρ(π,a)=.15a, minimum at a=.1
        • ρπ,.1)=.05
      • case 2: .1<a<.2a=2/15 is optimal and ρ(π,2/15)=.03
      • case 3: a.2ρ(π,a)=2a.3, optimal at a=.2. So ρ(π,.2)=.1
  • The posterior expected loss of an action aA is ρ(p(x|θ),a)=ΘL(θ,a)p(θ|x)dθ we could have a situation like this: picture1

Minimizing ρ(p(x|θ),a)=ΘL(θ,a)p(θ|x)dθ

  • Say you obtained a Bayes action by minimizing ρ(p(θ|x),a), say δp(θ|x)
  • We can compute the “Bayes Risk” = Ex[ρ(p(θ|x),ˆδ)]=Xp(x)ΘL(θ,ˆδ)p(x|θ)π(θ)dθdx
  • =ΘR(θ,ˆδ)π(θ)dθ=Eθ[R(θ,δ)]

Recap:

FrequentistBayesianEstimatorδ(x)δπ(x)likelihoodp(x|θ)p(x|θ)PriorNAπ(θ)RiskR(θ,δ)=Ex|θ[L(θ,δ)]ρ(p(θ|x),δπ(x))=Eθ|x[L(θ,δπ(x)]Bayes RiskEθ[R(θ,δ)]Ex[ρ(π,δ)] The Bayes Risk are equivalent and are equal to r(π,δ)=Eθ,x[L(θ,δ)] = XΘ L(θ,δ(x)) f(x|θ) π(θ) dθ dX.