Is there a way to prove that, if we achieve equal proportions of a certain outcome among 4 intersectional groups formed from Sex and Ethnicity, we won't create inequalities among the single groups?
sex = {male, female}, ethnicity = {white, black}
single groups: [math]P(male|y=1)[/math] [math]P(female|y=1)[/math] [math]P(white|y=1)[/math] [math]P(Black|y=1)[/math]
intersectional groups: [math]P(white,male|y=1)[/math] [math]P(black,male|y=1)[/math] [math]P(white,female|y=1)[/math] [math]P(black,female|y=1)[/math]
My question is how do I show that if: [imath]P(white,male|y=1)[/imath] == [imath]P(black,male|y=1)[/imath] == [imath]P(white,female|y=1)[/imath] == [imath]P(black,female|y=1)[/imath], it is (not)possible to achieve [imath]P(male|y=1)[/imath] == [imath]P(female|y=1[/imath] and [imath]P(white|y=1)[/imath] == [imath]P(black|y=1)[/imath]
sex = {male, female}, ethnicity = {white, black}
single groups: [math]P(male|y=1)[/math] [math]P(female|y=1)[/math] [math]P(white|y=1)[/math] [math]P(Black|y=1)[/math]
intersectional groups: [math]P(white,male|y=1)[/math] [math]P(black,male|y=1)[/math] [math]P(white,female|y=1)[/math] [math]P(black,female|y=1)[/math]
My question is how do I show that if: [imath]P(white,male|y=1)[/imath] == [imath]P(black,male|y=1)[/imath] == [imath]P(white,female|y=1)[/imath] == [imath]P(black,female|y=1)[/imath], it is (not)possible to achieve [imath]P(male|y=1)[/imath] == [imath]P(female|y=1[/imath] and [imath]P(white|y=1)[/imath] == [imath]P(black|y=1)[/imath]