  Probability density functions (PDFs) can be understood as continuous
compositions by the theory of Bayes spaces. The origin of a Bayes space is
determined by a given reference measure. This can be easily changed through the
well-known chain rule which has an impact on the geometry of the Bayes space.
This work provides a mathematical framework for setting a reference measure. It
is used to develop a weighting scheme on the bounded domain of distributional
data. The impact on statistical analysis is shown from the perspective of
simplicial functional principal component analysis. Moreover, a novel centered
log-ratio transformation is proposed to map a weighted Bayes spaces into an
unweighted $L^2$ space, enabling to use most tools developed in functional data
analysis (e.g. clustering, regression analysis, etc.) while accounting for the
weighting strategy. The potential of our proposal is shown through simulation
and on a real case study using Italian income data.
