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A precise definition of identifiability of a parameter is given in terms of consistency in probability for the parameter estimate. Under some mild Uniformity assumptions on the conditional density parameterized by the unknown parameter, necessary and sufficient conditions for the unknown parameter to be identifiable are established. The assumptions and identifiability criteria are expressed in terms of the density of individual observations, conditioned upon all past observations. The results are applied to linear system identification problems.
Tse et al. (Sun,) studied this question.