  Reliable estimation and approximation of probability density functions is
fundamental for their further processing. However, their specific properties,
i.e. scale invariance and relative scale, prevent the use of standard methods
of spline approximation and have to be considered when building a suitable
spline basis. Bayes Hilbert space methodology allows to account for these
properties of densities and enables their conversion to a standard Lebesgue
space of square integrable functions using the centered log-ratio
transformation. As the transformed densities fulfill a zero integral
constraint, the constraint should likewise be respected by any spline basis
used. Bayes Hilbert space methodology also allows to decompose bivariate
densities into their interactive and independent parts with univariate
marginals. As this yields a useful framework for studying the dependence
structure between random variables, a spline basis ideally should admit a
corresponding decomposition. This paper proposes a new spline basis for
(transformed) bivariate densities respecting the desired zero integral
property. We show that there is a one-to-one correspondence of this basis to a
corresponding basis in the Bayes Hilbert space of bivariate densities using
tools of this methodology. Furthermore, the spline representation and the
resulting decomposition into interactive and independent parts are derived.
Finally, this novel spline representation is evaluated in a simulation study
and applied to empirical geochemical data.
