The Bias-Variance Decomposition of Human Inequality

Suppose we have two groups of people experiencing different treatment–there is inequality of benefit between the two groups. We can write the difference between their benefits as:

(a-b)

Usually though, we can inspect the size of inequality using quadratic difference, or the quadratic inequality:

(a-b)^2

Take the expected value:

E[(a-b)^2]

And rearrange to produce:

(mean(a)-mean(b))^2+var(a)+var(b)-2cov(a,b)

This formula can be read off easily. The quadratic inequality between group having a experience and group having b experience is composed of

(mean(a)-mean(b))^2

the mean difference in benefit, i.e. the mean-induced inequality between the two groups is a primal component of inequality,

var(a)+var(b)

which is the internal inconsistency of group a and group b also increases the inequality, but it could be removed by

-2cov(a,b),

which says if treatment a and b we’re correlated, then that reduces the variance-induced inequality.

To think through an example: suppose I benefit boys by giving or takimg money from each by drawing a random number from \mathcal{N}(2,1). The benefit I give to girls is to drawn from a separate \mathcal{N}(1,100). Clearly boys on average receive $1 more than girls. But also there is an interesting difference in experience. Boys will more consistently receive money around $2, where as the the girls would have money taken or given to them with much more variability approaching $1 only when averaging a large number of experiences. It’s hard to say what the psychological effect of the larger variance is. For example is it unfair for boys or for girls if girls are drawn from \mathcal{N}(2,100)? In this analysis we can clearly detect that there is inequality between the two groups and that the inequality is induced by variance of the benefit to girls.

There is an intersting side note from this branch of social science that in such a circumstance, the number of boys we must sample to ascertain a highly likely interval of his mean reward is lower than the number of girls we must see in order to ascertain the same likely interval of her mean reward. So if the treatment of minority has high enough variability, then it becomes harder to be certain that there is mean-induced inequality. This decomposition enables us to identify such situations.

This is bit of Machine Learning math is really neat! We can now measure inequality in all aspects of life and understand, beyond an average, what is causing inequality and uncertainty.

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