How can I get the average value of a class A∩B knowing the average values of classes A and B?

Question

I am trying to model the population density for my metropolitan region, but the published census data does not meet the precision requirements I need. The most accurate data I have avaiable are average household occupancy by district, average household occupancy by type, and the type and location of each household in my area of interest (from which I can obtain their absolute and relative frequencies).

Given:

• The average occupancy of dwellings by disctrict (acrosss all types) $a_i$: $a_1$ , $a_2$ ,$a_3$ ... $a_d$
• The average occupancy of dwellings by type (acrosss all disctricts) $b_j$: $b_1$ , $b_2$ ,$b_3$ ... $b_t$
• The relative frequency of each disctrict $w_i$: $w_1$ , $w_2$ ,$w_3$ ... $w_d$
• The relative frequency of each type $k_i$: $k_1$ , $k_2$ ,$k_3$ ... $k_t$
• The relative frequency of each combination of type and dictrict $wk_{ij}$: $w_{11}$ , $w_{12}$ ,$w_{13}$ ... $w_{dt}$

Is there a way to calculate all the average occupancies $ab_{ij}$ of each combination of type and disctrict? My first tought was that because of the apparent similarity with the combined probability none of the frequencies are needed. Also, my bayesian rudiments suggest that if there is no general formula, maybe reducing the number of disctricts and types could give me an approximate answer.

Thank you very much

Answer 1

After trying with a reduced know datased, I confirmed that, if there is some general way to calculate the average of any $ab_{ij}$ intersection subset, more information is needed.

Of course, you could just assign an average value for each subset according to the weights of their original subsets in order to keep the overall average. Nevertheless, this does not guarantee that the subsets' averages hold true, as I confirmed by brute force. So, if the know averages of the original subsets are not enough to calculate precisely their intersection's subset, it implies that multiple answers are possible (if not infinite), and it seems imposible that any other method could calculate the real intersection average.

Thank you anyways.