  We study Bayesian histograms for distribution estimation on $[0,1]^d$ under
the Wasserstein $W_v, 1 \leq v < \infty$ distance in the i.i.d sampling regime.
We newly show that when $d < 2v$, histograms possess a special \textit{memory
efficiency} property, whereby in reference to the sample size $n$, order
$n^{d/2v}$ bins are needed to obtain minimax rate optimality. This result holds
for the posterior mean histogram and with respect to posterior contraction:
under the class of Borel probability measures and some classes of smooth
densities. The attained memory footprint overcomes existing minimax optimal
procedures by a polynomial factor in $n$; for example an $n^{1 - d/2v}$ factor
reduction in the footprint when compared to the empirical measure, a minimax
estimator in the Borel probability measure class. Additionally constructing
both the posterior mean histogram and the posterior itself can be done
super--linearly in $n$. Due to the popularity of the $W_1,W_2$ metrics and the
coverage provided by the $d < 2v$ case, our results are of most practical
interest in the $(d=1,v =1,2), (d=2,v=2), (d=3,v=2)$ settings and we provide
simulations demonstrating the theory in several of these instances.
