Separability metric

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Created: September 29, 2019 / Updated: April 24, 2020 / Status: in progress / 4 min read (~797 words)

Given a tabular dataset, what is the theoretical limit on the number of entries we can accurately predict? Many real world problems are not functions, that is to say that the same inputs may produce different outputs. Given a problem that should return the same output given the same inputs, what should be returned when an input has a set of different possible outputs?

In machine learning, we try to minimize the concept of loss. When an input is known to generate different potential outputs, such a system will tend to choose the output with the most probability.

We are given a dataset of features, where we want to predict a feature Y given a set of features X. For some X, we may have different values of Y.

X Y
1 3
1 4
1 3

In order to reflect on this problem, we use an iterative approach.

When a dataset is composed of 1 point, then we have 100% separability.

If X provides two different values, then we can separate the dataset into two distinct set, which leads us to 100% separability.

However, if X contains the same value, then the separability will depend on the target value:

  • If the target value is the same for both points, then we have 100% separability, since both X lead to the same Y
  • If the target value is different, then we only have 50% separability. This is due to the fact that, given no additional information, the best we can do is to randomly pick one of the two options for Y.

Number of unique input X/Number of points/rows
Given a tabular dataset, compute the separability metric as follow:

  • 1 point: 100% separable
  • 2 points: given the target feature to predict, if the other attributes can separate two distinct targets, then 100%, if not, 50%
  • x points: if the target is different for all points, yet the attributes are all the same, we should expect the metric to be 1/x
  • x points with y, z similar targets: in the case that we have x points with similar attributes, but for which there are y and z similar targets (two groups with the same target), we can at best hope for ...
    For a dataset with 1 B and 3 C as output values, we expect the metric to be between 0.25 and 0.75 (1/4 and 3/4)
  • sum(count(points for target x with attributes X)/count(points with attributes X))/count(points)
  • It may make sense to have a separability metric per target output (when those are categorical)
  • In the case of numerical targets, minimizing for distance between all the targets (clustering) would turn the problem into the categorical form
  • Datasets with defined inputs and target are often not functions (i.e, the same inputs may produce different outputs)