Let's say I have a population {$(G, p)$} where $G$ is a group within the population, and $p$ is their proportion distributed as such:

(A, 0.2)
(B, 0.3)
(C, 0.2)
(D, 0.3)

Now, let's say I have an interaction matrix $A$, where $A_{ij}$ represents the distribution of relationships in the population that are of the form $(i,j)$. This matrix is symmetric, and all non-diagonal values are thus double-counted.

So, let's say that the interaction matrix looks as follows: $\begin{bmatrix} 0.04&0.06&0.04&0.06\\0.06&0.10&0.07&0.07\\0.04&0.07&0.05&0.04\\0.06&0.07&0.04&0.13 \end{bmatrix}$

Now, let's say there's a certain sample of the same population whose {$(G, p)$} distribution looks like this:

(A, 0.4)
(B, 0.3)
(C, 0.2)
(D, 0.1)

How would I go about estimating the interaction matrix for this subgroup given the above information? Is that even possible?


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