Sample selection bias

Question

In Learning and Evaluating Classifiers under Sample Selection Bias, we suppose that

examples $(x, y, s)$ are drawn independently from a distribution $D$ with domain $X × Y × S$ where $X$ is the feature space, $Y$ is the label space and S is a binary space. The variable $s$ controls the selection of examples ($1$ means the example is selected, $0$ means the example is not selected). We only have access to the examples that have $s = 1$, which we call the selected sample. If the selected sample (ignoring $s$) is not a random sample of $D$ we say that the selected sample is biased.

In the text, the author argues that:

If $s$ is independent of $y$ given $x$ (that is $P (s|x, y) = P (s|x))$, the selected sample is biased but the biasedness only depends on the feature vector $x$.

Could someone explain this last quote? Does this means that the dataset we choose for training a learning algorithm is biased by the features of the samples?

Answer 1

The fact that the selected sample is biased, but the latter only depends on the feature vector x means that the distribution P(x) of the sample is different than the real one and it doesn't depend on the label space Y (e.g it is not mandatory to observe class imbalance that can cause biased sample).