I understand that the Maximum Likelihood estimate of variance is given by:
And I understand mathematically how to show that the expectation of the maximum likelihood variance is:
However, I cannot seem to figure out intuitively, why they are different? What is the difference between the maximum likelihood of variance and the expectation of the maximum likelihood of variance? What does the expectation of the ML variance even mean?
What is different in the expectation that leads to the term:
Again I am not looking for the mathematical proof, I am just looking to understand what mechanism is leading to the creation of the bias term.