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Questions and Answers
What is the marginal probability of X being rainy?
What is the value of H(X | Y) computed in hartlys?
What is the method used to compute conditional entropy H(X | Y)?
What is H(Y | X) computed in hartlys?
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Which values are part of the joint probability mass function for X and Y?
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Study Notes
Conditional Entropy
- Conditional Entropy H(X|Y) quantifies the uncertainty of random variable X given that another variable Y is known.
- Mathematically defined as:
- H(X|Y) = Σ p(y) H(X|Y=yi) for all y in Y
- Equivalently expressed as H(X|Y) = -Σ Σ p(x, y) log p(x|y) for all x in X and y in Y.
- Specific case for conditional entropy of X given a particular outcome of Y (yi):
- H(X|Y=yi) = -Σ p(xj|yi) log p(xj|yi).
Chain Rule
- The chain rule for joint entropy illustrates the relationship between individual entropies and conditional uncertainties:
- H(X, Y) = H(X) + H(Y|X).
- A symmetrical expression also exists:
- H(X, Y) = H(Y) + H(X|Y).
- This rule clarifies that joint uncertainty can be decomposed into the uncertainty of individual variables and their dependencies.
Example Calculations
- Two variables X (weather: sunny or rainy) and Y (temperature: above or below 70 degrees) are utilized to compute conditional entropies.
- Marginal probabilities derived from the example are:
- P(X): {3/4 rainy, 1/4 sunny}
- P(Y): {3/4 above 70, 1/4 below 70}
- Conditional entropy H(X|Y) calculated at 0.207 hartlys.
Joint Probability Mass Function
- A second example presents a joint probability mass function for variables X and Y with several discrete outcomes.
- The joint distribution is defined with probabilities summing to 1 over all possible X and Y combinations.
- Calculated entropies:
- H(X|Y) yields a result of approximately 1.284 hartlys.
- H(Y|X) results in approximately 0.324 hartlys.
Assignments
- Tasks include calculations of H(X), H(Y), and H(X,Y) in hartlys based on joint distributions and marginal probabilities provided from previous sections.
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Description
This quiz covers the concept of conditional entropy, H(X|Y), and its mathematical definitions. It explores how uncertainty in one random variable can be understood given knowledge of another. Test your understanding of the chain rule in joint entropy as well.
