Difference in Proportions of Labels (DPL)
The difference in proportions of labels (DPL) compares the proportion of observed outcomes with positive labels for facet d with the proportion of observed outcomes with positive labels of facet a in a training dataset. For example, you could use it to compare the proportion of middleaged individuals (facet a) and other age groups (facet d) approved for financial loans. Machine learning models try to mimic the training data decisions as closely as possible. So a machine learning model trained on a dataset with a high DPL is likely to reflect the same imbalance in its future predictions.
The formula for the difference in proportions of labels is as follows:
DPL = (q_{a}  q_{d})
Where:

q_{a} = n_{a}^{(1)}/n_{a} is the proportion of facet a who have an observed label value of 1. For example, the proportion of a middleaged demographic who get approved for loans. Here n_{a}^{(1)} represents the number of members of facet a who get a positive outcome and n_{a} the is number of members of facet a.

q_{d} = n_{d}^{(1)}/n_{d} is the proportion of facet d who have an observed label value of 1. For example, the proportion of people outside the middleaged demographic who get approved for loans. Here n_{d}^{(1)} represents the number of members of the facet d who get a positive outcome and n_{d} the is number of members of the facet d.
If DPL is close enough to 0, then we say that demographic parity has been achieved.
For binary and multicategory facet labels, the DPL values range over the interval (1, 1). For continuous labels, we set a threshold to collapse the labels to binary.

Positive DPL values indicate that facet a is has a higher proportion of positive outcomes when compared with facet d.

Values of DPL near zero indicate a more equal proportion of positive outcomes between facets and a value of zero indicates perfect demographic parity.

Negative DPL values indicate that facet d has a higher proportion of positive outcomes when compared with facet a.
Whether or not a high magnitude of DPL is problematic varies from one situation to another. In a problematic case, a highmagnitude DPL might be a signal of underlying issues in the data. For example, a dataset with high DPL might reflect historical biases or prejudices against agebased demographic groups that would be undesirable for a model to learn.