  The method of classical shadows heralds unprecedented opportunities for
quantum estimation with limited measurements [H.-Y. Huang, R. Kueng, and J.
Preskill, Nat. Phys. 16, 1050 (2020)]. Yet its relationship to established
quantum tomographic approaches, particularly those based on likelihood models,
remains unclear. In this article, we investigate classical shadows through the
lens of Bayesian mean estimation (BME). In direct tests on numerical data, BME
is found to attain significantly lower error on average, but classical shadows
prove remarkably more accurate in specific situations -- such as high-fidelity
ground truth states -- which are improbable in a fully uniform Hilbert space.
We then introduce an observable-oriented pseudo-likelihood that successfully
emulates the dimension-independence and state-specific optimality of classical
shadows, but within a Bayesian framework that ensures only physical states. Our
research reveals how classical shadows effect important departures from
conventional thinking in quantum state estimation, as well as the utility of
Bayesian methods for uncovering and formalizing statistical assumptions.
