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Application example of maximum entropy principle
Example 3. 1 is a random variable, which is estimated by the maximum entropy principle.

Solution: Entropy of the system

The constraint condition is

Constructing Lagrangian function

Solve the 6 yuan equation (as a variable)

The unconstrained maximum entropy distribution is

The entropy at this time is. Because constraints provide more information, the uncertainty of the system is reduced.

Example 3.2

Solution: Based on Theorem 2. 1, Euler equation is

Solution:

By substituting this result into two constraints, we can get the probability density that makes the objective functional reach the extreme value.

This is the probability density of normal distribution.

The probability density of functional extremum should satisfy

Auxiliary functional corresponding to this formula

soluble

Constraints can be solved recursively.

The maximum value of continuous entropy is complex, and there are various constraints, such as plastic constraints, differential constraints, isoperimetric constraints and so on. Some problems may be accompanied by some boundary conditions, and the above examples are just some basic examples. For complex problems, numerical calculation within the allowable error range is also a way to solve the problem.