  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 Bayesian approach. On the space of
copula functions, we construct a finite-dimensional approximation subspace that
is parametrized 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 of this 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 above results are difficult to obtain. 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 favour the Bayes estimator over
frequentist estimators such as the standard kernel estimator and Deheuvels'
estimator in terms of mean integrated squared error.
