Nearly Consistent Finite Particle Estimates in Streaming Importance Sampling

In Bayesian inference, we seek to compute information about random variables such as moments or quantiles on the basis of {available data} and prior information. When distribution is {intractable}, Monte Carlo (MC) sampling usually required. {Importance a standard MC tool that approximates this unavailable with set weighted samples.} This procedure asymptotically consistent number samples (particles) go infinity. However, retaining infinitely many particles intractable. Thus, propose way only keep \emph{finite representative subset} their augmented importance weights \emph{nearly consistent}\blue{, i.e., they converge close neighborhood population ground truth in large sample limit.} To do so {an online manner}, (1) embed posterior density estimate reproducing kernel Hilbert space (RKHS) through its mean embedding; (2) sequentially project RKHS element onto lower-dimensional subspace using maximum discrepancy, an integral probability metric. Theoretically, establish scheme results bias determined by compression parameter, which yields tunable tradeoff between consistency memory. experiments, observe compressed estimates achieve comparable performance dense ones substantial reductions representational complexity.