The precision of estimates of system performance and of parameters that affect the performance is often based upon the standard deviation obtained from the usual equation for the propagation of variances derived from a Taylor series expansion. With ever increasing computing power it is now possible to utilize the Bayesian hierarchical approach to yield improved estimates of the precision. Although quite popular in the statistical community, the Bayesian approach has not been widely used in the heat transfer and fluid mechanics communities because of its complexity and subjectivity. The paper develops the necessary equations and applies them to two typical heat transfer problems, measurement of conductivity with heat losses and heat transfer from a fin. Because of the heat loss the probability distribution of the conductivity is far from Gaussian. Using this conductivity distribution for the fin gives a very long tailed distribution for the heat transfer from the fin.