Definition: A probability model \(p(x|\bt)\) is said to be identifiable if for any \(\bt_1, \bt_2 \in \bT\), the set
\[\{x: p(x|\bt_1) \ne p(x|\bt_2) \}\]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.