# What is a "weakly identified" parameterization?

I understand that a parameterization is identified if it's true that

$$\theta_1 \neq \theta_2 \Rightarrow p(y|\theta_1) \neq p(y|\theta_2)$$

Intuitively, it means that two different parameter values must result in two different probability of the observed data.

However, I don't understand what people means by "weakly identified" parameterization? Is that when different values of $\theta$ leads to close, by not exactly the same, $p(y|\theta$)?

You asked if weak identification referred to the case where different parameter values lead to similar likelihoods. This is partly correct. If these "nearly equivalent" parameters are all in one region of space (the "ridge" referred to by Gelman), then we have weak identifiability. If, however, the likelihood function has several distinct regions of near-maximum likelihood, then it is more common to call the model "locally identifiable." For example, when fitting a Gaussian distribution to observations, the standard deviation $$\sigma$$ is technically only locally identifiable, because both $$\hat\sigma$$ and $$-\hat\sigma$$ are maximum likelihood solutions. However, when we impose the restriction that $$\sigma>0$$, the model becomes identifiable.