Suppose I observe a random variable $x$ drawn from a non-central t- or F- distribution, and would like to perform inference on the non-centrality parameter. How would a Bayesian approach this problem? Are there known conjugate priors for these?
For the t-distribution I have seen some reference to LeCoutre's 'lambda-prime' distribution, but I believe this is for a non-informative prior (and I know very little about Bayesian analysis). Is there a parallel result for the F-distribution? How are these used?