This is the expression of the posterior risk but I don't understand the step of the line 2. How can we developped the first line in this way ?

$T^*$ is defined as $E[\theta | X]$

enter image description here


Add and subtract $T^*$ inside the square: $$\mathbf E[(T -T^* +T^* - \theta)^2|X)] = \mathbf E[(T - T^*)^2 + (T^* - \theta)^2 + 2(T - T^*)(T^* - \theta)].$$

Now, we use the linearity of expectations and the fact that $T$ and $T^*$ are $X$-measurable (they act as constants in expectations conditional on $X$). Hence, $$\ldots = (T - T^*)^2 +\mathbf E[ (T^* - \theta)^2 + 2(T - T^*)\mathbf E[T^* - \theta].$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.