  This paper studies the problem of testing whether a function is monotone from
a nonparametric Bayesian perspective. Two new families of tests are
constructed. The first uses constrained smoothing splines, together with a
hierarchical stochastic-process prior that explicitly controls the prior
probability of monotonicity. The second uses regression splines, together with
two proposals for the prior over the regression coefficients. The finite-sample
performance of the tests is shown via simulation to improve upon existing
frequentist and Bayesian methods. The asymptotic properties of the Bayes factor
for comparing monotone versus non-monotone regression functions in a Gaussian
model are also studied. Our results significantly extend those currently
available, which chiefly focus on determining the dimension of a parametric
linear model.
