  In surveys, it is typically up to the individuals to decide if they want to
participate or not, which leads to participation bias: the individuals willing
to share their data might not be representative of the entire population.
Similarly, there are cases where one does not have direct access to any data of
the target population and has to resort to publicly available proxy data
sampled from a different distribution. In this paper, we present Differentially
Private Propensity Scores for Bias Correction (DiPPS), a method for
approximating the true data distribution of interest in both of the above
settings. We assume that the data analyst has access to a dataset $\tilde{D}$
that was sampled from the distribution of interest in a biased way. As
individuals may be more willing to share their data when given a privacy
guarantee, we further assume that the analyst is allowed locally differentially
private access to a set of samples $D$ from the true, unbiased distribution.
Each data point from the private, unbiased dataset $D$ is mapped to a
probability distribution over clusters (learned from the biased dataset
$\tilde{D}$), from which a single cluster is sampled via the exponential
mechanism and shared with the data analyst. This way, the analyst gathers a
distribution over clusters, which they use to compute propensity scores for the
points in the biased $\tilde{D}$, which are in turn used to reweight the points
in $\tilde{D}$ to approximate the true data distribution. It is now possible to
compute any function on the resulting reweighted dataset without further access
to the private $D$. In experiments on datasets from various domains, we show
that DiPPS successfully brings the distribution of the available dataset closer
to the distribution of interest in terms of Wasserstein distance. We further
show that this results in improved estimates for different statistics.
