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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.

Sampling of Bayesian posteriors with a non-Gaussian probabilistic learning on manifolds from a small dataset