Expert Knowledge Makes Towards an . . . Towards an . . . - PowerPoint PPT Presentation

Empirical Observation . . . Towards an Explanation Expert Knowledge Makes Towards an . . . Towards an . . . Predictions More Accurate: Reference Theoretical Explanation of Home Page an Empirical Observation Title Page ◭◭ ◮◮ Julio Urenda 1 , 2 , Marco Cardiel 2 , Laura Hinojos 2 , ◭ ◮ Oliver Martinez 2 , and Vladik Kreinovich 2 1 Department of Mathematical Sciences Page 1 of 6 2 Department of Computer Science Go Back University of Texas at El Paso, El Paso, TX 79968, USA jcurenda@utep.edu, macardiel@miners.utep.edu, Full Screen ljhinojos@miners.utep.edu, omartinez14@miners.utep.edu, vladik@utep.edu Close Quit

1. Empirical Observation That Needs Explaining Empirical Observation . . . Towards an Explanation • It is known that the use of expert knowledge makes Towards an . . . predictions more accurate. Towards an . . . • For example, computer-based meteorological forecasts Reference are regularly corrected by experts. Home Page • A typical improvement is that the accuracy consis- tently improves by 10%. Title Page ◭◭ ◮◮ • How can we explain this? ◭ ◮ Page 2 of 6 Go Back Full Screen Close Quit

2. Towards an Explanation Empirical Observation . . . Towards an Explanation • Use of expert knowledge means, in effect, that we com- Towards an . . . bine: Towards an . . . – an estimate produced by a computer model and Reference – an expert estimate. Home Page • Let σ m and σ e denote the standard deviations, corre- Title Page spondingly, of the model and of the expert estimate. ◭◭ ◮◮ • In effect, the only information that we have about com- paring the two accuracies is that ◭ ◮ – expert estimates are usually less accurate Page 3 of 6 – than model results: Go Back Full Screen σ m < σ e . Close • So, if we fix σ e , then the only thing we know about σ m is that σ m is somewhere between 0 and σ e . Quit

3. Towards an Explanation (cont-d) Empirical Observation . . . Towards an Explanation • We have no reason to assume that some values from Towards an . . . the interval [0 , σ e ] are more probable than others. Towards an . . . • Thus, it makes sense to assume that all these values Reference are equally probable. Home Page • So, we have a uniform distribution on this interval. Title Page • For this uniform distribution, the average value of σ m is equal to 0 . 5 · σ e . ◭◭ ◮◮ • Thus, we have σ e = 2 · σ m . ◭ ◮ • In general: Page 4 of 6 – if we combine two estimates x m and x e with accu- Go Back racies σ m and σ e , Full Screen – then the combined estimate x c is obtained by min- Close imizing the sum ( x m − x c ) 2 + ( x e − x c ) 2 . Quit σ 2 σ 2 m e

4. Towards an Explanation (cont-d) Empirical Observation . . . Towards an Explanation • The resulting estimate is x c = x m · σ − 2 m + x e · σ − 2 e , with Towards an . . . σ − 2 m + σ − 2 e 1 Towards an . . . accuracy σ 2 c = . Reference σ − 2 m + σ − 2 e • For σ e = 2 σ m , we have σ − 2 = 0 . 25 · σ − 2 m , thus σ 2 c = Home Page e 1 1 1 + 0 . 25 = σ 2 1 . 25 = 0 . 8 · σ 2 σ 2 m , thus σ c ≈ 0 . 9 · σ m . m · m · Title Page • So we indeed get a 10% increase in the resulting pre- ◭◭ ◮◮ diction. ◭ ◮ Page 5 of 6 Go Back Full Screen Close Quit

5. Reference Empirical Observation . . . • N. Silver, The Signal and the Noise: Why So Many Towards an Explanation Towards an . . . Decisions Fail – but Some Don’t , Penguin Press, New Towards an . . . York, 2012. Reference Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 6 Go Back Full Screen Close Quit