  The use of galaxy clusters as precision cosmological probes relies on an
accurate determination of their masses. However, inferring the relationship
between cluster mass and observables from direct observations is difficult and
prone to sample selection biases. In this work, we use weak lensing as the best
possible proxy for cluster mass to calibrate the Sunyaev-Zel'dovich (SZ) effect
measurements from the APEX-SZ experiment. For a well-defined (ROSAT) X-ray
complete cluster sample, we calibrate the integrated Comptonization parameter,
$Y_{\rm SZ}$, to the weak-lensing derived total cluster mass, $M_{500}$. We
employ a novel Bayesian approach to account for the selection effects by
jointly fitting both the SZ Comptonization, $Y_{\rm SZ}\text{--}M_{500}$, and
the X-ray luminosity, $L_{\rm x}\text{--}M_{500}$, scaling relations. We also
account for a possible correlation between the intrinsic (log-normal) scatter
of $L_{\rm x}$ and $Y_{\rm SZ}$ at fixed mass. We find the corresponding
correlation coefficient to be $r= 0.47_{-0.35}^{+0.24}$, and at the current
precision level our constraints on the scaling relations are consistent with
previous works. For our APEX-SZ sample, we find that ignoring the covariance
between the SZ and X-ray observables biases the normalization of the $Y_{\rm
SZ}\text{--}M_{500}$ scaling high by $1\text{--}2\sigma$ and the slope low by
$\sim 1\sigma$, even when the SZ effect plays no role in the sample selection.
We conclude that for higher-precision data and larger cluster samples, as
anticipated from on-going and near-future cluster cosmology experiments,
similar biases (due to intrinsic covariances of cluster observables) in the
scaling relations will dominate the cosmological error budget if not accounted
for correctly.
