Identifiability

Remarks

Example: the normal model $P_{(\mu, \sigma^2)} = \mathcal{N}(\mu, \sigma^2)$ with $\mu \in \R$ and $\sigma^2 > 0$ is identifiable, because the mean and the variance of the law recover $\mu$ and $\sigma^2$. Non-example: let $P_{(a, b)} = \mathcal{N}(a + b, 1)$ with $(a, b) \in \R^2$. Then $P_{(1, 0)} = P_{(0, 1)}$ while $(1, 0) \ne (0, 1)$, so the model is not identifiable. A function $\tau(\theta)$ is identifiable when $P_{\theta_1} = P_{\theta_2}$ implies $\tau(\theta_1) = \tau(\theta_2)$; in the non-example, $\tau(a, b) = a + b$ is identifiable.