  We present a learning theory for the training of a linear system operator
having an input compositional variable and propose a Bayesian inversion method
for inferring the unknown variable from an output of a noisy linear system. We
assume that we have partial or even no knowledge of the operator but have
training data of input and ouput. A compositional variable satisfies the
constraints that the elements of the variable are all non-negative and sum to
unity. We quantified the uncertainty in the trained operator and present the
convergence rates of training in explicit forms for several interesting cases
under stochastic compositional models. The trained linear operator with the
covariance matrix, estimated from the training set of pairs of ground-truth
input and noisy output data, is further used in evaluation of posterior
uncertainty of the solution. This posterior uncertainty clearly demonstrates
uncertainty propagation from noisy training data and addresses possible
mismatch between the true operator and the estimated one in the final solution.
