# Identifiability condition

Identifiability condition

In mathematics, the identifiability condition is defined as

:$f\left(x\right) = f\left(y\right) Leftrightarrow x = y quad forall x,y$

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.

:$sin\left(0\right) = 0 = sin\left(2pi\right) = sin\left(4pi\right) = cdots$

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

Example 2

Let the function be $y = f\left(x\right) = x^3$.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

:$f\left(x; sigma_1^2\right) = f\left(x; sigma_2^2\right) Leftrightarrow sigma_1^2 = sigma_2^2$

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