We determine the mass scale of Planck galaxy clusters using gravitational lensing mass measurements from the Cluster Lensing And Supernova survey with Hubble (CLASH). We have compared the lensing masses to the Planck Sunyaev-Zeldovich (SZ) mass proxy for 21 clusters in common, employing a Bayesian analysis to simultaneously fit an idealized CLASH selection function and the distribution between the measured observables and true cluster mass. We used a tiered analysis strategy to explicitly demonstrate the importance of priors on weak lensing mass accuracy. In the case of an assumed constant bias, $b_{SZ}$, between true cluster mass, $M_{500}$, and the Planck mass proxy, $M_{PL}$, our analysis constrains $1- b_{SZ} = 0.73pm 0.10$ when moderate priors on weak lensing accuracy are used, including a zero-mean Gaussian with standard deviation of 8% to account for possible bias in lensing mass estimations. Our analysis explicitly accounts for possible selection bias effects in this calibration sourced by the CLASH selection function. Our constraint on the cluster mass scale is consistent with recent results from the Weighing the Giants program and the Canadian Cluster Comparison Project. It is also consistent, at 1.34$sigma$, with the value needed to reconcile the Planck SZ cluster counts with Plancks base $Lambda$CDM model fit to the primary cosmic microwave background anisotropies.
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