Definition: A probability model $p(x \mid \bt)$ is said to be identifiable if for any $\bt_1, \bt_2 \in \bT$, the set
\[\{x: p(x \mid \bt_1) \ne p(x \mid \bt_2) \}\]has nonzero probability.
If a model does not meet this criterion, then it is impossible to choose between $\bt_1$ and $\bt_2$ objectively; even with an infinite amount of data, we would never know which parameter values were true.