This thesis presents two new techniques for analyzing data from impurity transport experiments in magnetically confined plasmas, with specific applications to the Alcator C-Mod tokamak. The objective in developing these new techniques is to improve the quality of the experimental results used to test simulations of turbulent transport: better characterization of the uncertainty in the experimental results will yield a better test of the simulations. Transport codes are highly sensitive to the gradients of the background temperature and density profiles, so the first half of this thesis presents a new approach to fitting tokamak profiles using nonstationary Gaussian process regression. This powerful technique overcomes many of the shortcomings of previous spline-based data smoothing techniques, and can even handle more complicated cases such as line-integrated measurements, computation of second derivatives, and 2d fitting of spatially- and temporally-resolved measurements. The second half of this thesis focuses on experimental measurements of impurity transport coefficients. It is shown that there are considerable shortcomings in existing point estimates of these quantities. Next, a linearized model of impurity transport data is constructed and used to estimate diagnostic requirements for impurity transport measurements. It is found that spatial resolution is more important than temporal resolution. Finally, a fully Bayesian approach to inferring experimental impurity transport coefficient profiles which overcomes the shortcomings of the previous approaches through use of multimodal nested sampling is developed and benchmarked using synthetic data. These tests reveal that uncertainties in the transport coefficient profiles previously attributed to uncertainties in the temperature and density profiles are in fact entirely explained by changes in the spline knot positions. Appendices are provided describing the extensive work done to determine the derivatives of stationary and nonstationary covariance kernels and the open source software developed as part of this thesis work. The techniques developed here will enable more rigorous benchmarking of turbulent transport simulations, with the ultimate goal of developing a predictive capability.