11.4 Jensen's inequality and its applications

Jensen's inequality is a cornerstone of convex analysis, linking expected values and convex functions. It states that for a convex function, the function of the average is less than or equal to the average of the function.

This powerful tool has wide-ranging applications in probability, statistics, and optimization. It underpins many important inequalities and helps solve problems in fields from finance to information theory, making it a crucial concept to grasp.

Understanding Jensen's Inequality

Jensen's inequality for convex functions

Top images from around the web for Jensen's inequality for convex functions

• 

  probability or statistics - Reproducing a graphic on Jensen's inequality for the probabilistic ... View original

  Is this image relevant?

• 

  Jensen's inequality - Wikipedia View original

  Is this image relevant?

• 

  Jensen's inequality - Wikipedia View original

  Is this image relevant?

• 

  probability or statistics - Reproducing a graphic on Jensen's inequality for the probabilistic ... View original

  Is this image relevant?

• 

  Jensen's inequality - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Jensen's inequality for convex functions

• 

  probability or statistics - Reproducing a graphic on Jensen's inequality for the probabilistic ... View original

  Is this image relevant?

• 

  Jensen's inequality - Wikipedia View original

  Is this image relevant?

• 

  Jensen's inequality - Wikipedia View original

  Is this image relevant?

• 

  probability or statistics - Reproducing a graphic on Jensen's inequality for the probabilistic ... View original

  Is this image relevant?

• 

  Jensen's inequality - Wikipedia View original

  Is this image relevant?

1 of 3

• Jensen's inequality relates expected value of convex function to function of expected value
• For convex function $f$ and random variable $X$: $f(E[X]) \leq E[f(X)]$
• Discrete case with probabilities $p_i$ and values $x_i$: $f(\sum_{i=1}^n p_i x_i) \leq \sum_{i=1}^n p_i f(x_i)$
• Key components include convex function, expected value, and inequality direction
• Graphically interpreted as chord lying above function curve (parabola)

Proof of Jensen's inequality

• Convexity definition: $f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$ for $\lambda \in [0,1]$
• Proof steps:
  1. Start with two points, apply definition
  2. Extend to finite number of points using induction
  3. Use weighted average concept
  4. Generalize to continuous case with integrals
• Leverages linearity of expectation and properties of convex functions
• Demonstrates inequality holds for both discrete and continuous random variables

Applications of Jensen's Inequality

Applications in probability and statistics

• Probability inequalities derived:
  • Markov's inequality bounds probability of exceeding expected value
  • Chebyshev's inequality estimates deviation from mean
• Statistical applications:
  • Lower bound for variance $Var(X) \geq 0$
  • Information inequality in estimation theory
• Machine learning uses:
  • Cross-entropy loss function optimization
  • Kullback-Leibler divergence in probabilistic models
• Financial mathematics: option pricing, risk assessment

Derivation of mathematical inequalities

• Arithmetic-Geometric mean inequality: $\sqrt{ab} \leq \frac{a+b}{2}$
• Hölder's inequality generalizes Cauchy-Schwarz inequality
• Minkowski's inequality extends triangle inequality to $L^p$ spaces
• Log-sum inequality useful in information theory
• Power mean inequality compares different types of averages
• Applications span analysis, geometry, and information theory
• Provides powerful tools for bounding complex expressions