A big data service is any data-originated resource that is offered over the Internet. The performance of a big data service depends on the data bought from the data collectors. However, the problem of optimal pricing and data allocation in big data services is not well-studied. In this paper, we propose an auction-based big data market model. We first define the data cost and utility based on the impact of data size on the performance of big data analytics, e.g., machine learning algorithms. The big data services are considered as digital goods and uniquely characterized with ''unlimited supply'' compared to conventional goods which are limited. We therefore propose a Bayesian profit maximization auction which is truthful, rational, and computationally efficient. The optimal service price and data size are obtained by solving the profit maximization auction. Finally, experimental results on a real-world taxi trip dataset show that our big data market model and auction mechanism effectively solve the profit maximization problem of the service provider.