# Re-calibrate interaction matrix in population sample

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?