  The quantum state of a superconducting transmon qubit inside a
three-dimensional cavity is monitored by reflection of a microwave field on the
cavity. The information inferred from the measurement record is incorporated in
a density matrix $\rho_t$, which is conditioned on probe results until $t$, and
in an auxiliary matrix $E_t$, which is conditioned on probe results obtained
after $t$. Here, we obtain these matrices from experimental data and we
illustrate their application to predict and retrodict the outcome of weak and
strong qubit measurements.
