Expectations and Entropy
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1 Expectations and Entropy Data Science: Jordan Boyd-Graber University of Maryland SLIDES ADAPTED FROM DAVE BLEI AND LAUREN HANNAH Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 1 / 9
2 Expectation An expectation of a random variable is a weighted average: E[f(X)] = f(x) p(x) (discrete) = x f(x) p(x) dx (continuous) Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 2 / 9
3 Expectation Expectations of constants or known values: E[a] = a Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 3 / 9
4 Expectation Intuition E[x] is most common expectation Average outcome (might not be an event: 2.4 children) Center of mass Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 4 / 9
5 Expectation of die / dice What is the expectation of the roll of die? Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 5 / 9
6 Expectation of die / dice What is the expectation of the roll of die? One die = Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 5 / 9
7 Expectation of die / dice What is the expectation of the roll of die? One die = 3.5 Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 5 / 9
8 Expectation of die / dice What is the expectation of the roll of die? One die = 3.5 What is the expectation of the sum of two dice? Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 5 / 9
9 Expectation of die / dice What is the expectation of the roll of die? One die = 3.5 What is the expectation of the sum of two dice? Two die = Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 5 / 9
10 Expectation of die / dice What is the expectation of the roll of die? One die = 3.5 What is the expectation of the sum of two dice? Two die = 7 Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 5 / 9
11 Entropy Measure of disorder in a system In the real world, entroy in a system tends to increase Can also be applied to probabilities: Is one (or a few) outcomes certain (low entropy) Are things equiprobable (high entropy) In data science We look for features that allow us to reduce entropy (decision trees) All else being equal, we seek models that have maximum entropy (Occam s razor) Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 6 / 9
12 Aside: Logarithms lg(1)=0 lg(2)=1 lg(x) = b 2b = x Makes big numbers small Way to think about them: cutting a carrot lg(4)=2 lg(8)=3 Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 7 / 9
13 Aside: Logarithms lg(1)=0 lg(2)=1 lg(x) = b 2b = x Makes big numbers small Way to think about them: cutting a carrot Negative numbers? lg(4)=2 lg(8)=3 Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 7 / 9
14 Aside: Logarithms lg(1)=0 lg(x) = b 2b = x Makes big numbers small Way to think about them: cutting a carrot Negative numbers? Non-integers? lg(2)=1 lg(4)=2 lg(8)=3 Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 7 / 9
15 Entropy Entropy is a measure of uncertainty that is associated with the distribution of a random variable: H(X) = E[lg(p(X))] = p(x) lg(p(x)) = x p(x) lg(p(x)) dx (discrete) (continuous) Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 8 / 9
16 Entropy Entropy is a measure of uncertainty that is associated with the distribution of a random variable: H(X) = E[lg(p(X))] = p(x) lg(p(x)) = x p(x) lg(p(x)) dx (discrete) (continuous) Does not account for the values of the random variable, only the spread of the distribution. H(X) 0 uniform distribution = highest entropy, point mass = lowest suppose P(X = 1) = p, P(X = 0) = 1 p and P(Y = 100) = p, P(Y = 0) = 1 p: X and Y have the same entropy Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 8 / 9
17 Wrap up Probabilities are the language of data science You ll need to manipulate probabilities and understand marginalization and independence In Class: Working through probability examples Next: Conditional probabilities Data Science: Jordan Boyd-Graber UMD Expectations and Entropy 9 / 9
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