  We consider the problem of purchasing data for machine learning or
statistical estimation. The data analyst has a budget to purchase datasets from
multiple data providers. She does not have any test data that can be used to
evaluate the collected data and can assign payments to data providers solely
based on the collected datasets. We consider the problem in the standard
Bayesian paradigm and in two settings: (1) data are only collected once; (2)
data are collected repeatedly and each day's data are drawn independently from
the same distribution. For both settings, our mechanisms guarantee that
truthfully reporting one's dataset is always an equilibrium by adopting
techniques from peer prediction: pay each provider the mutual information
between his reported data and other providers' reported data. Depending on the
data distribution, the mechanisms can also discourage misreports that would
lead to inaccurate predictions. Our mechanisms also guarantee individual
rationality and budget feasibility for certain underlying distributions in the
first setting and for all distributions in the second setting.
