# Is there a word or phrase to describe a model that is basically unidentifiable in practice?

So suppose we have a model $$f(x|\theta)$$ that is theoretically identifiable, so that $$\theta_1 \neq \theta_2$$ implies $$f(x|\theta_1) \neq f(x|\theta_2)$$.

However, suppose that data collection is very limited and that, for all samples $$X$$ taken in practice, for every $$\theta$$ there exists a $$\hat{\theta}$$ such that $$f(X|\theta) \approx f(X|\hat{\theta})$$.

And an example of this would be if $$X=(x_1,\dots,x_n)$$ is a time series model in equilibrium, i.e., it is basically constant, over the time period considered, such that $$X_i \approx x_c$$ for all $$i=1,2,3,\dots,N.$$

If this was the case, and if for example we had $$f(x|\theta) \propto \exp \left(-\prod_i \left(\Delta x_i-\alpha x_i^\beta \right)^2 \right)$$, then the model would, for practical considerations, be unidentifiable since we would have

$$\log f(x|\theta) \propto - \prod_i \left(\Delta x_i-\alpha x_i^{\beta_1}\right)^2 \approx - \prod_i \left(\Delta x_i-\alpha x_c^{\beta_1}\right)^2 = -\prod_i \left(\Delta x_i-\tilde{\alpha} x_c^{\beta_2} \right)^2\\ \approx -\prod_i \left(\Delta x_i-\tilde{\alpha} x_i^{\beta_2}\right)^2$$

for any $$\beta_1,\beta_2 > 0$$, where $$\hat{\alpha} = \alpha x_c^{\beta_1-\beta_2}$$. If the above was the case, then it would be practically impossible to determine between $$(\alpha,\beta_1)$$ and $$(\hat{\alpha},\beta_2)$$, even though the model is theoretically identifiable. I am curious if there is a word or phrase to describe this situation? And whether my concerns make sense.

• Your example doesn't really illustrate what I think you are asking about. The "parameter" $\alpha$ is just the scaling constant for the density, so it is a function of $\beta$. Moreover, although you have not specified the bounds, the example density is monotone in $\beta$. Can you rethink your example and use another that better illustrates your query? (A more realistic example would be when you have explanatory variables in a regression that are almost perfectly correlated.)
– Ben
Feb 18, 2021 at 6:02
• Thanks @Ben for pointing that out, I chose the first density that came to mind without thinking that through. I updated it to be of Gaussian form, as is typically assumed in time-series models. Is the problem more sensical now?
– user281754
Feb 18, 2021 at 6:35

Surely this is the case for almost every statistical model with a continuous valued parameter $$\theta$$ and a likelihood that is reasonably smooth. Perhaps this is an issue of computational precision rather than a statistical issue?
For instance, take the likelihood to be that of a Normal with mean $$\theta$$ and known variance $$1$$. This is an identifiable model.
Thus, for a sample size of $$1$$, $$\log f(x \mid \theta) = C -(x - \theta)^2$$ where $$C$$ is a scaling constant which ensures the likelihood is valid.
Now $$\log f(x \mid \theta + \delta) = C -(x - \theta - \delta)^2$$.
Now if $$\delta$$ is sufficiently small relative to $$\theta$$, (say $$\theta = 5$$, $$\delta = 10^{-20}$$) a computer could never tell $$\log f(x \mid \theta)$$ apart from $$\log f(x \mid \theta + \delta)$$. There is a mathematical distinction, but if a computer can't tell the difference then, to a modern statistician, they are effectively the same.