Talk about the use of entropy from information theoretic sense as a viable measure visualizing uncertainty

1. Visualization of Uncertainty without a Mean
Authors:
 Kristin Potter
 Samuel Gerber
 Erik W. Anderson
Presented by:
Subhashis Hazarika,
Ohio State University

2. Motivation
• Mean and Standard Deviation are the most common ways of quantifying
and visualizing uncertainty when we have a probability distribution
function for the dataset.
• For Categorical data which is inherently discrete, we cannot define mean
at a voxel by using data-points from the different categories.
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3. Categorical Data
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4. Entropy as a Measure of Uncertainty
• Let 𝑋 be a random variable with probability density 𝑝 on discrete sample
space 𝑝: {x1, …, xn}  R+
• Mean: 𝐸 𝑋 = 𝑥𝑖 𝑝(𝑥𝑖)𝑛
𝑖=1
• Variance: Var 𝑋 = 𝐸 𝑋 − 𝑥𝑖
2𝑛
𝑖=1
• Entropy: 𝐻 𝑋 = − 𝑝 𝑥𝑖 log(𝑝 𝑥𝑖 )𝑛
𝑖=1
• Discrete Distributions’ Entropy Range : [0, - log(1/n)]
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5. Visualizing Entropy
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6. Limitations
• Information about the value of the random variable is lost.
• Provides only a measure of randomness.
• Whereas mean and variance indicates a range of most likely values.
• Only the probability density influences the entropy not the actual value of
the random variables
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7. Information Theory Perspective
• Entropy can be thought of as the minimal expected number of binary
questions (question with yes/no answer) we can ask to determine the value of
an observation of X.  #Q
• H[X] <= E[#Q] < H[X] + 1 (in log2)
• p1 = {1, 0, 0, 0} H[X] = 0.00
p2 = {0.85, 0.15, 0, 0} H[X] = 0.61
p3 = {0.85, 0.05, 0.05, 0.05} H[X] = 0.85
p4 = {0.25, 0.25, 0.25, 0.25} H[X] = 2.00
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8. Visualization
• Find the maximal probability tissue type and take its color tag.
– arg maxx = X(i,j)p(i,j)(x)
• Blend this color through white on the basis of that locations entropy value.
– H[X(i,j)]
• Maximal entropy voxel will appear white and minimal entropy voxel will
have the characteristic color tag.
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9. Visualization
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10. In 3D Spatial Domain
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11. Thank You
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