Entropy Function Overview

Questions and Answers

Match the following terms related to entropy with their definitions:

Entropy = Measure of uncertainty of a random variable H(X) = Entropy of a discrete random variable Distribution = How probabilities are assigned to random variable outcomes Logarithmic function = Mathematical function used to measure information content

Match the following entropy characteristics with their descriptions:

H(X) ≥ 0 = Entropy is always non-negative Maximized entropy = Occurs when outcomes are uniformly distributed Log K = Maximum entropy for K values Zero entropy = Indicates no uncertainty in random variable outcomes

Match the following entropy components with their formulas:

H(X) = $-rac{1}{n} imes ext{sum}(P(x) imes ext{log}(P(x)))$ P(x) = $ ext{Pr}ig{X=x\big ext{ for specific x}$ K values = Number of distinct outcomes in random variable p = Probability of a specific outcome in the random variable

Match the types of measurements of entropy with their units:

<p>Bits = Used when logarithm base is 2 Hartleys = Used when logarithm base is 10 Nats = Used when logarithm base is e Entropy function = Quantitative measure of uncertainty</p> Signup and view all the answers

Match the concepts of entropy with their effects or outcomes:

<p>Entropy maximization = Increases information content Entropy minimization = Reduces uncertainty Uniform distribution = Maximizes entropy value Probability extremes = No uncertainty about outcomes</p> Signup and view all the answers

Study Notes

Entropy Function Overview

Entropy measures the uncertainty of a random variable.
For a discrete random variable X with alphabet x and probability mass function P(x) = Pr{X=x}, the probabilities describe the likelihood of outcomes.

Definition of Entropy

The entropy H(X) is expressed mathematically:
H(X) = 1 * ∑ P(x) log_b(P(x)) for x ∈ X
Here, b denotes the logarithm base, which impacts the units of measurement.

Measurement Units

Entropy reflects the average information contained in a random variable.
Units of measurement include:
- Bits (base 2)
- Hartleys (base 10)
- Nats (base e)

Properties of Entropy

Entropy is dependent solely on the distribution of X, not the specific values.
The value of H(X) is always ≥ 0, indicating non-negative uncertainty.

Example of Entropy Calculation

For a binary random variable X that takes values 0 with probability (1-p) and 1 with probability p, the entropy is given by: H(X) = -p log(p) - (1-p) log(1-p)
Commonly represented as H(p, 1-p).

Maximum and Minimum Entropy

Entropy reaches its maximum when p = 0.5 (equal uncertainty).
H(X) is zero when p = 0 or p = 1, indicating no uncertainty about the outcome.

General Case with K Values

When X can assume K values, the entropy is maximized when X is uniformly distributed across these values.
In this uniform distribution case, entropy simplifies to:
H(X) = log(K)

Jensen's Inequality Application

Using Jensen's inequality, it is established that:
H(X) ≤ log(Σ P(x)) for normalized probabilities.
This confirms that maximum entropy occurs with a uniform distribution, supporting H(X) ≤ log(K).

Description

This quiz covers the concept of entropy in information theory, including its definition, mathematical formulation, and measurement units. Understand how entropy quantifies uncertainty and learn about its properties and calculation methods with specific examples.