  We consider data release protocols for data $X=(S,U)$, where $S$ is
sensitive; the released data $Y$ contains as much information about $X$ as
possible, measured as $\operatorname{I}(X;Y)$, without leaking too much about
$S$. We introduce the Robust Local Differential Privacy (RLDP) framework to
measure privacy. This framework relies on the underlying distribution of the
data, which needs to be estimated from available data. Robust privacy
guarantees are ensuring privacy for all distributions in a given set
$\mathcal{F}$, for which we study two cases: when $\mathcal{F}$ is the set of
all distributions, and when $\mathcal{F}$ is a confidence set arising from a
$\chi^2$ test on a publicly available dataset. In the former case we introduce
a new release protocol which we prove to be optimal in the low privacy regime.
In the latter case we present four algorithms that construct RLDP protocols
from a given dataset. One of these approximates $\mathcal{F}$ by a polytope and
uses results from robust optimisation to yield high utility release protocols.
However, this algorithm relies on vertex enumeration and becomes
computationally inaccessible for large input spaces. The other three algorithms
are low-complexity and build on randomised response. Experiments verify that
all four algorithms offer significantly improved utility over regular LDP.
