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Official statistics is dipping its toe in the ocean of big data, and leaders are emphasizing the need for a major paradigm change. One aspect is the increased volume of data that are not collected on probability samples of the target population. Making full use of these data requires a fundamental change, not only in data collection and dissemination, but also in the methods of statistical inference. The classical design-based'' approach to survey inference, developed from the seminal work of Neyman \cite{41}, is simply not applicable to these data. Rather, statistical models are needed that potentially reflect selection bias from the lack of random sampling. I suggest that Calibrated Bayes'' is the appropriate statistical paradigm for addressing the analysis. Under this paradigm, inferences for a particular data set are Bayesian, but models are sought that yield inferences with robust repeated sampling properties. Probability sampling remains a powerful tool under this paradigm, since by ensuring that the selection mechanism is ignorable it enhances robust modeling, but it is not essential for the inference. I outline two applications of Calibrated Bayes to data collected by the U.S. Census Bureau.