  In this letter we revisit the problem of optimal design of quantum
tomographic experiments. In contrast to previous approaches where an optimal
set of measurements is decided in advance of the experiment, we allow for
measurements to be adaptively and efficiently re-optimised depending on data
collected so far. We develop an adaptive statistical framework based on
Bayesian inference and Shannon's information, and demonstrate a ten-fold
reduction in the total number of measurements required as compared to
non-adaptive methods, including mutually unbiased bases.
