L_p-norm (LP)

The L_p-norm (LP) measures the p-norm distance between the facet distributions of the observed labels in a training dataset. This metric is non-negative 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 p-norm distance between the points x and y is defined as follows:

        L_p(x, y) = (|x₁-y₁|^p + |x₂-y₂|^p + … +|x_n-y_n|^p)^1/p

The 2-norm is the Euclidean norm. Assume you have an outcome distribution with three categories, for example, y_i = {y₀, y₁, y₂} = {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:

        L₂(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.