  Interferometric visibilities, reflecting the complex correlations between
signals recorded at antennas in an interferometric array, carry information
about the angular structure of a distant source. While unknown antenna gains in
both amplitude and phase can prevent direct interpretation of these
measurements, certain combinations of visibilities called closure phases and
closure amplitudes are independent of antenna gains and provide a convenient
set of robust observables. However, these closure quantities have subtle noise
properties and are generally both linearly and statistically dependent. These
complications have obstructed the proper use of closure quantities in
interferometric analysis, and they have obscured the relationship between
analysis with closure quantities and other analysis techniques such as self
calibration. We review the statistics of closure quantities, noting common
pitfalls that arise when approaching low signal-to-noise due to the nonlinear
propagation of statistical errors. We then develop a strategy for isolating and
fitting to the independent degrees of freedom captured by the closure
quantities through explicit construction of linearly independent sets of
quantities along with their noise covariance in the Gaussian limit, valid for
moderate signal-to-noise, and we demonstrate that model fits have biased
posteriors when this covariance is ignored. Finally, we introduce a unified
procedure for fitting to both closure information and partially calibrated
visibilities, and we demonstrate both analytically and numerically the direct
equivalence of inference based on closure quantities to that based on self
calibration of complex visibilities with unconstrained antenna gains.
