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Slide 1 Tutorial: Optimal Learning in the Laboratory Sciences The value of information December 10, 2014 Warren B. Powell Kris Reyes Si Chen Princeton.

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Presentation on theme: "Slide 1 Tutorial: Optimal Learning in the Laboratory Sciences The value of information December 10, 2014 Warren B. Powell Kris Reyes Si Chen Princeton."— Presentation transcript:

1 Slide 1 Tutorial: Optimal Learning in the Laboratory Sciences The value of information December 10, 2014 Warren B. Powell Kris Reyes Si Chen Princeton University http://www.castlelab.princeton.edu Slide 1

2 Lecture outline 2  The value of information

3 Value of Information Best estimate: the maximum value of the utility function, e.g. the length of Al 2 O 3 +Fe After one experiment, the belief changes 3 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length Best estimate before the experiment Best estimate after the experiment Marginal value of information

4 Value of Information Truth and measurement noise are random  Truth is unknown: we have many possible truths  Measurements are noisy around the unknown truth Fe Possible truths about the performance of Fe Nanotube Length

5 Average Marginal Value of Information Truth and measurement noise are random  Truth is unknown: we have many possible truths  Measurements are noisy around the unknown truth To calculate the average marginal value of information: average over all possible outcomes (truths + noise) 5 Fe Ni PHN Al 2 O 3 +Fe Al 2 O 3 +Ni FeNi PHN Al 2 O 3 +Fe Al 2 O 3 +Ni Fe Ni PHN Al 2 O 3 +Fe Al 2 O 3 +Ni FeNi PHN Al 2 O 3 +Fe Al 2 O 3 +Ni

6 A Numerical Example We want to calculate the average marginal information of growing nanotubes using Ni  Assumes we have 4 possible truths  These truths can be captured by our belief 6 Fe Ni PHN Al 2 O 3 +Fe Al 2 O 3 +Ni FeNi PHN Al 2 O 3 +Fe Al 2 O 3 +Ni Fe Ni PHN Al 2 O 3 +Fe Al 2 O 3 +Ni FeNi PHN Al 2 O 3 +Fe Al 2 O 3 +Ni 5.2 7.1 8.5 9.8 9.0 3.7 8.6 10.01 9.6 7.1 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length 5.8 7.4 8.7 10.0 9.1

7 9.7 A Numerical Example We want to calculate the average marginal information of growing nanotubes using Ni  For each possible truth, we generate a random measurement around this truth and update beliefs  Calculate Marginal Value of Information 7 0.9 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length 5.2 7.1 8.5 9.8 9.0 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length 5.0 7.5 7.7 8.8 7.0 MVI

8 7.8 A Numerical Example We want to calculate the average marginal information of growing nanotubes using Ni  For different measurements, we get different MVIs 8 0.0 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length 5.2 7.1 8.5 9.8 9.0 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length 5.0 7.5 7.7 8.8 7.0

9 9.1 A Numerical Example We want to calculate the average marginal information of growing nanotubes using Ni  For different measurements, we get different MVIs 9 0.3 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length 5.0 7.5 7.7 8.8 7.0 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length 5.2 7.1 8.5 9.8 9.0

10 A Numerical Example We want to calculate the average marginal information of growing nanotubes using Ni  For each possible truth, we generate a measurement  For different measurements, we get different MVIs  Average over 3 different observations: (0.9+0+0.3)/3=0.4 10 5.2 7.1 8.5 9.8 9.0

11 A Numerical Example We want to calculate the average marginal information of growing nanotubes using Ni  For different truths, we repeat the process and get different averages over noisy observations, e.g 0.4, 1.3, 0.7, 1.6 11 10.4 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length 5.0 7.5 7.7 8.8 7.0 FeNi PHN Al 2 O 3 +FeAl 2 O 3 +Ni Nanotube Length 1.6 3.7 8.6 10.01 9.6 7.1

12 A Numerical Example We want to calculate the average marginal information of growing nanotubes using Ni  For different possible truths, we repeat the process and get different averages over observations, e.g 0.4, 1.3, 0.7, 1.6  We then average the results of the experiments, which combines variations in truths and noise, giving us: MVI = (0.4+1.3+0.7+1.6)/4=1.0  Average marginal value of information of Ni is 1.0 12

13 Knowledge Gradient scores Low confidence in prior DMS data This plot tells us how much information can be gained from targeting each region. Highest scoring regions = most information to be gained

14 Knowledge Gradient scores Use KG scores as a guideline to picking the next experiment. The primer for this highest scoring probe need to be ordered But we have this primer in stock, and it has a reasonably large KG value.

15 Policy function approximations Lookup table policies arise in many settings in everyday life © 2013 W.B. Powell Slide 15

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