Higher order beliefs and coherency in game theory
The way hierarchies of beliefs are specified, beliefs on the same events are encoded in different places. 

The basic idea is actually quite simple. You have two players, Ann and Bob, say. Ann's first order beliefs specify how likely she thinks each strategy choice of Bob. Ann's second order beliefs specify how likely she thinks each combination of a strategy choice of Bob and a belief of Bob on Ann's strategy choices. Second order beliefs are joint probabilities.

Now suppose Ann's first order beliefs specify that she believes that Bob plays the strategies C and D with probability $1/2$ each.

Suppose also that Ann believes with probability $1/4$ that "Bob plays C and believes with probability 1 that Ann plays D for sure" and believes with probability $3/4$ that "Bob plays D and believes Ann plays D for sure". 

If you ask someone who knows only Ann's first order beliefs how likely Ann thinks it is that Bob will play C, that person will answer with $1/2$. If you ask someone who knows only Ann's second order beliefs how likely Ann thinks it is that Bob will play C, that person will answer with $1/4$. Indeed, that is what the marginal of her second order belief on the space of strategies says.

This is absurd and makes no sense. Ann's beliefs on how likely it is that Bob plays C should be the same according to her first order beliefs as to her second order beliefs. Now coherency is exactly the condition that guarantees that. If some belief on an event E is specified in more than one level of the hierarchy of beliefs, each hierarchy should specify that the same belief on the event E.  

In terms of the formalism: The marginal of a probability measure is again a probability measure, so $marg(b_i^2;\Sigma_j)=b_i^1(E)$ is comparing apples and oranges; on the left is a probability measure, on the right a number. What you want is $marg(b_i^2;\Sigma_j)(E)=b_i^1(E)$, as prescribed by coherency- and indeed by common sense.