We propose a novel online and adaptive truncation method for differentially
private Bayesian online estimation of a static parameter regarding a
population. We assume that sensitive information from individuals is collected
sequentially and the inferential aim is to estimate, on-the-fly, a static
parameter regarding the population to which those individuals belong. We
propose sequential Monte Carlo to perform online Bayesian estimation. When
individuals provide sensitive information in response to a query, it is
necessary to perturb it with privacy-preserving noise to ensure the privacy of
those individuals. The amount of perturbation is proportional to the
sensitivity of the query, which is determined usually by the range of the
queried information. The truncation technique we propose adapts to the
previously collected observations to adjust the query range for the next
individual. The idea is that, based on previous observations, we can carefully
arrange the interval into which the next individual's information is to be
truncated before being perturbed with privacy-preserving noise. In this way, we
aim to design predictive queries with small sensitivity, hence small
privacy-preserving noise, enabling more accurate estimation while maintaining
the same level of privacy. To decide on the location and the width of the
interval, we use an exploration-exploitation approach a la Thompson sampling
with an objective function based on the Fisher information of the generated
observation. We show the merits of our methodology with numerical examples.