What does this notation in minimal square error mean?

Question

My book defines minimal square error as:

$$MSE(\theta, T) = E_\theta\Vert T - \theta \Vert^2$$

What does the $E_\theta$ mean? Is it an expectation? If yes, what does the theta supscript do there?

Answer 1

The notation $E_\theta$ usually means expectation taken over the values of $\theta$, i.e. $$ E_\theta \|T-\theta\|^2 = \int_{-\infty}^{+\infty} \|T - \theta\|^2 \, p(\theta)\, d\theta. $$

But here I think the writer is being sloppy and means $$ E_{T|\theta} \|T-\theta\|^2 = \int \|t - \theta\|^2 \, p(t|\theta)\, dt. $$