Generalized quantum measurements (also known as positive operator-valued measures or POVMs) are of great
importance in quantum information and quantum foundations, but often difficult to perform. We present an
experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context.
One of the most interesting POVMs, the symmetric, informationally complete-POVM (or SIC-POVM), is the
most compact set of measurements that can be used to fully describe a quantum state. We use our technique
to carry out the first experimental characterization of the state of a qutrit using SIC-POVMs. Because of the
highly symmetric nature of this measurement, such a representation has the unique property that it permits all
other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making
no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states
encoded in single photons.