Jensen’s inequality, put simply, dictates that the expected value of a convex function applied to a random variable is either equal to or larger than the convex function of the expected value of the variable. A profound mathematical result, known as Jensen’s inequality, plays a crucial role in the realms of probability theory and convex analysis.
It unveils a connection between the anticipated outcome of a convex function when subjected to a stochastic variable and the function of the average value of said variable. In simple words, if you have y as a random variable and g(y) is the convex function in the range of y.
Significance of Jensen’s Inequality in Python
Jensen’s inequality, a tool in probability theory, can be used to establish connections between moments of random variables. As an illustration, one application is proving that variance, being the second moment of a distribution, is always non-negative due to the convexity of the function.
Convex functions often employ Jensen’s inequality in optimization problems, as it simplifies finding the minimum by considering the relationship between the function and its expected value. Entropy for certain distributions is proven to have concavity using Jensen’s inequality in information theory, which is used in entropy-related concepts. Particularly in the realm of machine learning, Jensen’s inequality finds its practical application in deriving various optimization algorithms, specifically for convex loss functions.
Mathematical Representation of Jensen’s Inequality
Jensen’s inequality is represented in the form of a relation between two functions.
In this equation, E[g(X)] is the expected value applied to the function, and g(E[X]) is a convex function applied to the random variable X.
Implementation of Jensen’s Inequality in Python
To execute Jensen’s inequality in Python, we need to create some functions that will calculate some required values. These two values are the expected value and the convex function. We are computing both values and creating one more function to calculate Jensen’s inequality. The expected value of the random variable and convex function is calculated here.
Then, we will review Jensen’s inequality If the condition is true, we will return true; otherwise, we will return false. Let’s try to implement it according to our approach.
def calculate_expected_value(values):
return sum(values) / len(values)
def convex_function(x):
return x ** 2
def jensens_inequality(values):
ex = calculate_expected_value(values)
egx = sum(convex_function(x) for x in values) / len(values)
if egx >= convex_function(ex):
return True
else:
return False
random_values = [1, 2, 3, 4, 5]
result = jensens_inequality(random_values)
if result:
print("Jensen's inequality holds for the convex function g(x) = x^2 and the given random variable X.")
else:
print("Jensen's inequality does not hold for the convex function g(x) = x^2 and the given random variable X.")
In this code, three different functions help to run and verify Jensen’s condition in Python. The result should be valid for the allotted random variable. Let’s check the results for validation.
Implementation of Jensen’s Inequality Using Numpy
The numpy package is used to confirm Jensen’s inequality in Python. Let’s attempt to execute this.
import numpy as np
import matplotlib.pyplot as plt
def convex_function(x):
return np.exp(x)
def calculate_expected_value(values):
return np.mean(values)
def jensens_inequality(values):
ex = calculate_expected_value(values)
egx = np.mean(convex_function(values))
if egx >= convex_function(ex):
return True
else:
return False
random_values = [1, 2, 3, 4, 5]
result = jensens_inequality(random_values)
if result:
print("Jensen's inequality holds for the convex function g(x) = e^x and the given random variable X.")
else:
print("Jensen's inequality does not hold for the convex function g(x) = e^x and the given random variable X.")
x_values = np.linspace(0, 5, 100)
plt.plot(x_values, convex_function(x_values), label='g(x) = e^x')
plt.axhline(y=calculate_expected_value(random_values), color='r', linestyle='--', label='E[X]')
plt.legend()
plt.xlabel('x')
plt.ylabel('y')
plt.title("Jensen's Inequality Example")
plt.grid(True)
plt.show()
In this code, we are using the same function to calculate the expected value, convex functions, and the other function to confirm the inequality. In this code, we are also importing Matplotlib, which will help us plot the graph. Let’s see the outcome.
Real-World Application of Jensen’s Inequality
In studying entropy, mutual information, and Kullback-Leibler divergence in information theory, Jensen’s inequality is a vital principle. It assists our comprehension of the effectiveness of data compression algorithms and the quantification of information transmission in communication systems. Employed in multiple statistical analyses, Jensen’s inequality provides proof for the variance inequality of a random variable, along with limits on the moments and variance of random variables. Performance characteristics for control systems featuring uncertain parameters are commonly established using Jensen’s inequality in control theory. It is applied in signal processing to derive bounds on signal estimation and detection performance, as well as analyze signal-to-noise ratios.
Investment managers in finance utilize Jensen’s inequality to evaluate and analyze the performance of investment portfolios. This inequality is valuable for gauging whether an investment strategy outperforms a benchmark, like a market index. When the expected return of the portfolio surpasses the convex function of the benchmark’s expected return, it is considered to possess “positive alpha,” indicating it has generated extra returns. During convex optimization in the field of machine learning, Jensen’s inequality proves to be a useful tool. Updating model parameters intelligently helps optimization algorithms minimize various convex loss functions. Notably, it also helps establish bounds on generalization errors in statistical learning.
Summary
Jensen’s inequality is about discovering the relationship between two mathematical functions and terms. One of them is the function of the expected value, and the second is the convex function of the expected values in terms of random variables. We attempted to confirm the condition using two additional methods. One method is based on simple function implementations, and the second method performs with the numpy and matplotlib libraries. The overall real-world applications, significance of Jensen’s inequality, and the basics of Jensen’s inequality in Python are explained in particular. I hope you will enjoy this writing.
References
Read more about the NumPy library here.