We apply Bayesian inference to the issue of instrument calibration and experimental data uncertainty analysis for the specific application of measuring radiative intensity with a narrow-angle radiometer. We develop a physics-based instrument model that describes intensity, the indirectly-measured quantity of interest, as a function of scenario and uncertain model parameters. We identify a set of five uncertain parameters and find their probability distributions (the posterior or inverse problem) given the calibration data by applying Bayes' Theorem. We then employ the instrument model in a new scenario, a 1.5 MW coal-fired furnace. We obtain values for the five uncertain parameters in the model by sampling from the posterior distribution and then compute the intensity with quantifiable uncertainty at the measurement point of interest (the posterior predictive or forward problem).