Identifiability condition

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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.

[edit] 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(2\pi) = \sin(2\pi) = \cdots

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

[edit] Example 2

Let the function be y = f(x) = x3. This function clearly satisfies the condition as it is a one-to-one function.

[edit] 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