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Let's say I own a few hundred McDonalds locations. In a subset of those (say 100) I observe vegans eating there and I estimate the arrival time of vegans at these 100 restaurants using a Poisson distribution (so 100 different Poisson RVs with their own lambda values). Ie) at restaurant "A" I observe a lambda of 1.5 / hour and at "B" it is 3/hour etc.

Overall I model the distribution of these individual lambdas as a Gamma.

Now say I observe another characteristic - bearded customers for example - in another subset of restaurants (say again 100 for simplicity). And I follow the exact same process as above (model arrival times of beards with Poisson at each of the 100 locations and have 100 different lambdas modeled with a Gamma).

What can I say (if anything) about the intersection between these two gammas? Specifically, what does the distribution of the rate parameter for people that are vegan AND have a beard look like given I have the rate parameters for vegans and beards separately?

Additional knowns: Total # of locations observed with bearded vegans + total # of bearded vegans observed. Concretely say there were 50 total locations with bearded-vegans (out of the 100 with vegans and 100 with beards) and a total of 150 bearded-vegan customers across those 50 locations (so an average lambda of 3)

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    $\begingroup$ There is little you can deduce, because you have no information about people who are both vegans and bearded. What, for instance, if all the vegans are clean-shaven? What if they all have beards? Why not just keep more detailed data so you know of each individual what characteristics they have? $\endgroup$
    – whuber ♦
    Commented Feb 28, 2022 at 19:36
  • $\begingroup$ I do also provide the additional (small) bit of information of total number of locations with observed bearded vegans as well as the total number of customers that were observed to be bearded vegans. So we know for example, that not all vegans are clean shaven (in my toy sample I am able to say that we specifically observed 150 bearded vegan customers across 50 locations). So if I also wanted to use a gamma on the intersection - I know the average. I also should have bounds but I am wondering if anything else can be deduced about the shape (given known shape of the components) $\endgroup$
    – user2313135
    Commented Feb 28, 2022 at 19:49
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    $\begingroup$ That helps a tiny bit, but without making strong assumptions about how beards and being vegan might be associated, it only slightly narrows already very wide bounds on the answer. $\endgroup$
    – whuber ♦
    Commented Feb 28, 2022 at 19:52
  • $\begingroup$ OK well lets say we make some of those assumptions - say we assume independence between beardedness and vegan-ism (maybe ridiculous but for sake of the example). Or if we assume we are able to quantify the correlation between the two? How can this be used to solve? $\endgroup$
    – user2313135
    Commented Feb 28, 2022 at 19:56
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    $\begingroup$ Yes, independence gets you all the way there. The problem is most easily analyzed by thinking of arrival times in terms of homogeneous Poisson processes. stats.stackexchange.com/questions/288807 and stats.stackexchange.com/questions/180057 might help illustrate this way of thinking. $\endgroup$
    – whuber ♦
    Commented Feb 28, 2022 at 20:03

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