  This paper tackles the challenge presented by small-data to the task of
Bayesian inference. A novel methodology, based on manifold learning and
manifold sampling, is proposed for solving this computational statistics
problem under the following assumptions: 1) neither the prior model nor the
likelihood function are Gaussian and neither can be approximated by a Gaussian
measure; 2) the number of functional input (system parameters) and functional
output (quantity of interest) can be large; 3) the number of available
realizations of the prior model is small, leading to the small-data challenge
typically associated with expensive numerical simulations; the number of
experimental realizations is also small; 4) the number of the posterior
realizations required for decision is much larger than the available initial
dataset. The method and its mathematical aspects are detailed. Three
applications are presented for validation: The first two involve mathematical
constructions aimed to develop intuition around the method and to explore its
performance. The third example aims to demonstrate the operational value of the
method using a more complex application related to the statistical inverse
identification of the non-Gaussian matrix-valued random elasticity field of a
damaged biological tissue (osteoporosis in a cortical bone) using ultrasonic
waves.
