MSE part 1 - Estimation X X R Estimate R Estimate ^ X ^ X N...

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Estimation X: a student’s height, R:weight. X: daily crude oil price, R: KOSPI ˆ Observe and come up with an estimate of R XX . Estimation Error: ˆ eXX = Cost Function: () Ce Average Cost: ,| (,) ( | ) () XR R C C e f x r dxdr C e f x r f r dxdr = = ∫∫ The optimal estimate is the estimate that minimizes the average cost. R X ζ X N R Estimate X ˆ Estimate X ˆ
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Cost Functions and Optimal Estimates Square cost 2 () Ce e = leads to the minimum mean square error estimate (MSE). The best estimate is the conditional mean . [ ] | ˆ | (|) XR X E X R r xf x r dx = = = Absolute cost Ce e = . The best estimate is the median of the conditional probability. ˆ || ˆ X X f x r dx f x r dx −∞ = ∫∫
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Delta cost 0 () 1 for e Ce for e ≤∆ = >∆ leads to the MAP (maximum a posteriori probability ) estimate, where is an arbitrarily small positive number. The best estimate is the maximum a posteriori probability. || ˆ ( |) m a x (|) XR x f Xr f xr = Note. When the a posteriori probability is qaussian, the above three estimates result in the same value.
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MSE part 1 - Estimation X X R Estimate R Estimate ^ X ^ X N...

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