Summary: A new coding problem is introduced for a correlated source \((X^n,Y^n)_{n=1}^\infty\). The observer of \(X^n\) can transmit data depending on \(X^n\) at a prescribed rate \(R\). Based on these data the observer of \(Y^n\) tries to identify whether for some distortion measure \(\rho\) (like the Hamming distance) \(n^{-1}\rho (X^n,Y^n)\leq d\), a prescribed fidelity criterion. We investigate as functions of \(R\) and \(d\) the exponents of two error probabilities, the probabilities for misacceptance, and the probabilities for misrejection. In the case where \(X^n\) and \(Y^n\) are independent, we completely characterize the achievable region for the rate \(R\) and the exponents of two error probabilities; in the case where \(X^n\) and \(Y^n\) are correlated, we get some interesting partial results for the achievable region. During the process, we develop a new method for proving converses, which is called ``inherently typical subset lemma''. This new method goes considerably beyond the ``entropy characterization'', the ``image size characterization'', and its extensions. It is conceivable that this new method has a strong impact on multiuser information theory.