Identifiability condition

Identifiability condition

In mathematics, the identifiability condition is defined as

:f(x) = f(y) Leftrightarrow x = y quad forall x,y

which says that if a function evaluates the same, then the arguments must be the same. I.e., a function is "identifiable" iff it is one-to-one. The article on injective functions deals with this same topic more abstractly.

Example 1

Let the function be the sine function.This function does not satisfy the condition when the parameter is allowed to take any real number.

:sin(0) = 0 = sin(2pi) = sin(4pi) = cdots

However, if the parameter is restricted to [-pi/2, pi/2] then it satisfies the condition.

Example 2

Let the function be y = f(x) = x^3.This function clearly satisfies the condition as it is a one-to-one function.

Example 3

Let the function be the normal distribution with zero mean.For a fixed random value and nonfixed variance, this function satisfies the condition

:f(x; sigma_1^2) = f(x; sigma_2^2) Leftrightarrow sigma_1^2 = sigma_2^2

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