  In the streaming data setting, where data arrive continuously or in frequent
batches and there is no pre-determined amount of total data, Bayesian models
can employ recursive updates, incorporating each new batch of data into the
model parameters' posterior distribution. Filtering methods are currently used
to perform these updates efficiently, however, they suffer from eventual
degradation as the number of unique values within the filtered samples
decreases. We propose Generative Filtering, a method for efficiently performing
recursive Bayesian updates in the streaming setting. Generative Filtering
retains the speed of a filtering method while using parallel updates to avoid
degenerate distributions after repeated applications. We derive rates of
convergence for Generative Filtering and conditions for the use of sufficient
statistics instead of fully storing all past data. We investigate the
alleviation of filtering degradation through simulation and Ecological species
count data.
