# What does it mean for a probability model to be "well-specified" or "misspecified"?

I read in several articles that a model $\{p_\theta , \theta \in \Theta\}$ is well-specified if there exists a $\theta^* \in \Theta$ such that $p_{\theta^*} = p^*$.

Similarly, a model $\{p_\theta , \theta \in \Theta\}$ is misspecified if there does not exists a $\theta^* \in \Theta$ such that $p_{\theta^*} = p^*$.

Here, the definition of $p^*$ is that it is the true "data-generating distribution".

I can understand the terminology but what is the underlying logic here? I feel that I am missing the deeper story here. What is an example of a misspecified model? thanks!

• If your model is misspecified, then you'll never learn the true data-generating distribution no matter how many examples you see. A simple example is if your model is that your data is being generated by a linear function with some parameters but in fact it is not linear at all. Mar 6, 2017 at 23:31

Well-specified means that the class of distribution $\mathcal{C}$ you are assuming for your modeling actually contains the unknown probability distribution $p$ from where the sample is drawn.
Misspecified means, on the other hand, that $\mathcal{C}$ does not contain $p$. You made a modeling assumption, and it is not perfect: for instance, you assume your sample is Gaussian, but (maybe due to noise, or just inherently) it is not actually originating from any Gaussian distribution. It may be close to some Gaussian, if you're lucky, but not exactly.