A bivariate distribution with continuous margins can be uniquely decomposed
via a copula and its marginal distributions. We consider the problem of
estimating the copula function and adopt a nonparametric Bayesian approach. On
the space of copula functions, we construct a finite dimensional approximation
subspace which is parameterized by a doubly stochastic matrix. A major problem
here is the selection of a prior distribution on the space of doubly stochastic
matrices also known as the Birkhoff polytope. The main contributions ofthis
paper are the derivation of a simple formula for the Jeffreys prior and showing
that it is proper. It is known in the literature that for a complex problem
like the one treated here, the latter result is difficult to show. The Bayes
estimator resulting from the Jeffreys prior is then evaluated numerically via
Markov chain Monte Carlo methodology. A rather extensive simulation experiment
is carried out. In many cases, the results favor the Bayes estimator over
frequentist estimators such as the standard kernel estimator and the Deheuvels
estimator in terms of mean integrated squared error.