L_{p}norm (LP)
The L_{p}norm (LP) measures the pnorm distance between the facet distributions of the observed labels in a training dataset. This metric is nonnegative and so cannot detect reverse bias.
The formula for the L_{p}norm is as follows:
L_{p}(P_{a}, P_{d}) = ( ∑_{y}P_{a}  P_{d}^{p})^{1/p}
Where the pnorm distance between the points x and y is defined as follows:
L_{p}(x, y) = (x_{1}y_{1}^{p} + x_{2}y_{2}^{p} + … +x_{n}y_{n}^{p})^{1/p}
The 2norm is the Euclidean norm. Assume you have an outcome distribution with three categories, for example, y_{i} = {y_{0}, y_{1}, y_{2}} = {accepted, waitlisted, rejected} in a college admissions multicategory scenario. You take the sum of the squares of the differences between the outcome counts for facets a and d. The resulting Euclidean distance is calculated as follows:
L_{2}(P_{a}, P_{d}) = [(n_{a}^{(0)}  n_{d}^{(0)})^{2} + (n_{a}^{(1)}  n_{d}^{(1)})^{2} + (n_{a}^{(2)}  n_{d}^{(2)})^{2}]^{1/2}
Where:

n_{a}^{(i)} is number of the ith category outcomes in facet a: for example n_{a}^{(0)} is number of facet a acceptances.

n_{d}^{(i)} is number of the ith category outcomes in facet d: for example n_{d}^{(2)} is number of facet d rejections.
The range of LP values for binary, multicategory, and continuous outcomes is [0, √2), where:

Values near zero mean the labels are similarly distributed.

Positive values mean the label distributions diverge, the more positive the larger the divergence.
