  Differential privacy is a rigorous privacy standard that has been applied to
a range of data analysis tasks. To broaden the application scenarios of
differential privacy when data records have dependencies, the notion of
Bayesian differential privacy has been recently proposed. However, it is
unknown whether Bayesian differential privacy preserves three nice properties
of differential privacy: sequential composability, parallel composability, and
post-processing. In this paper, we provide an affirmative answer to this
question; i.e., Bayesian differential privacy still have these properties. The
idea behind sequential composability is that if we have $m$ algorithms $Y_1,
Y_2, \ldots, Y_m$, where $Y_{\ell}$ is independently $\epsilon_{\ell}$-Bayesian
differential private for ${\ell}=1,2,\ldots,m$, then by feeding the result of
$Y_1$ into $Y_2$, the result of $Y_2$ into $Y_3$, and so on, we will finally
have an $\sum_{\ell=1}^m \epsilon_{\ell}$-Bayesian differential private
algorithm. For parallel composability, we consider the situation where a
database is partitioned into $m$ disjoint subsets. The $\ell$-th subset is
input to a Bayesian differential private algorithm $Y_{\ell}$, for
${\ell}=1,2,\ldots,m$. Then the parallel composition of $Y_1$, $Y_2$, $\ldots$,
$Y_m$ will be $\max_{\ell=1}^m \epsilon_{\ell}$-Bayesian differential private.
The post-processing property means that a data analyst, without additional
knowledge about the private database, cannot compute a function of the output
of a Bayesian differential private algorithm and reduce its privacy guarantee.
