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 _ { 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 _ { SZ } \text { - - } M _ { 500 } , and the X-ray luminosity , L _ { x } \text { - - } M _ { 500 } , scaling relations .
We also account for a possible correlation between the intrinsic ( log-normal ) scatter of L _ { x } and Y _ { 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 _ { 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 .