  This paper proposes a new nonparametric Bayesian bootstrap for a mixture
model, by developing the traditional Bayesian bootstrap. We first reinterpret
the Bayesian bootstrap, which uses the P\'olya-urn scheme, as a gradient ascent
algorithm which associated one-step solver. The key then is to use the same
basic mechanism as the Bayesian bootstrap with the switch from a point mass
kernel to a continuous kernel. Just as the Bayesian bootstrap works solely from
the empirical distribution function, so the new Bayesian bootstrap for mixture
models works off the nonparametric maximum likelihood estimator for the mixing
distribution. From a theoretical perspective, we prove the convergence and
exchangeability of the sample sequences from the algorithm and also illustrate
our results with different models and settings and some real data.
