  We propose a Bayesian pseudo posterior mechanism to generate record-level
synthetic databases equipped with an $(\epsilon,\delta)-$ probabilistic
differential privacy (pDP) guarantee, where $\delta$ denotes the probability
that any observed database exceeds $\epsilon$. The pseudo posterior mechanism
employs a data record-indexed, risk-based weight vector with weight values $\in
[0, 1]$ that surgically downweight the likelihood contributions for high-risk
records for model estimation and the generation of record-level synthetic data
for public release. The pseudo posterior synthesizer constructs a weight for
each data record using the Lipschitz bound for that record under a log-pseudo
likelihood utility function that generalizes the exponential mechanism (EM)
used to construct a formally private data generating mechanism. By selecting
weights to remove likelihood contributions with non-finite log-likelihood
values, we guarantee a finite local privacy guarantee for our pseudo posterior
mechanism at every sample size. Our results may be applied to \emph{any}
synthesizing model envisioned by the data disseminator in a computationally
tractable way that only involves estimation of a pseudo posterior distribution
for parameters, $\theta$, unlike recent approaches that use naturally-bounded
utility functions implemented through the EM. We specify mild conditions that
guarantee the asymptotic contraction of $\delta$ to $0$ over the space of
databases. We illustrate our pseudo posterior mechanism on the sensitive family
income variable from the Consumer Expenditure Surveys database published by the
U.S. Bureau of Labor Statistics. We show that utility is better preserved in
the synthetic data for our pseudo posterior mechanism as compared to the EM,
both estimated using the same non-private synthesizer, due to our use of
targeted downweighting.
