A realistic example of a non identifiable model? Given the definition of statistical model identifiability : identifiable iff $P_\theta = P_{\theta'} \implies \theta = \theta' ~~ \forall \theta, \theta' \in \Theta$ we can see that, for example, $X \sim N(\frac{\theta_1}{\theta_2},1)$ would not be identifiable.
But can you make a simple, intuitive real-world example of a case when one would meet with an unidentifiable model ?

Another classic example of non-identification is a dose response model like the Emax model, which assumes that some (continuous) outcome for patients $i=1,\ldots,N$ for a drug that you can give in different doses obeys the following:
$$Y_i \sim N( E_0 + E_\text{max} \frac{\text{dose}_i}{\text{dose}_i + \text{ED}_{50}}, \sigma^2),$$
where $\text{dose}_i$ is the dose given to patient $i$.
Let's assume we have observed the following data:




i (patient)
$\text{dose}_i$
$Y_i$




1
0 (placebo)
1.2


2
0 (placebo)
0.2


3
0 (placebo)
-0.1


4
0 (placebo)
0.7


5
0 (placebo)
-0.3


6
0 (placebo)
-0.8


7
100 mg
11.7


8
100 mg
12.5


9
100 mg
13.1


10
100 mg
11.4


11
100 mg
10.9


12
100 mg
11.8




The parameters $E_0$ and $\sigma$ are identified by the model/data, but $E_\text{max}$ and $\text{ED}_{50}$ are not. No matter how much data you collect for dose 0 and 100, you will have a whole curve of combinations of those latter two parameters that will fit the data equally well. The way out of this situation is of course to try additional doses of the drug.
