Modern machine learning (ML) techniques, in conjunction with simulation-based methods, present remarkable potential for various scientific and engineering applications. Within the materials science field, these data-based methods can be used to build efficient structure–property linkages that can be further integrated within multi-scale simulations, or guide experiments in a materials discovery setting. However, a critical shortcoming of state-of-the-art ML techniques is their lack of reliable uncertainty/error estimates, which severely limits their use for materials or other engineering applications where data is often scarce and uncertainties are substantial. This paper presents methods for Bayesian learning of neural networks (NN) that allow consideration of both aleatoric uncertainties that account for the inherent stochasticity of the data-generating process, and epistemic uncertainties, which arise from consideration of limited amounts of data. In particular, algorithms based on approximate variational inference and (pseudo-)Bayesian model averaging achieve an appropriate trade-off between accuracy of the uncertainty estimates and accessible computational cost. The performance of these algorithms is first presented on simple 1D examples to illustrate their behavior in both extrapolation and interpolation settings. The approach is then applied for the prediction of homogenized and localized properties of a composite material. In this setting, data is generated from a finite element model, which permits a study of the behavior of the probabilistic learning algorithms under various amounts of aleatoric and epistemic uncertainties. •Bayesian neural networks for data-driven materials modeling using small data.•Algorithms based on variational inference and probabilistic model averaging.•Example of surrogate modeling of an elastoplastic composite material.•Aleatoric uncertainties arise from random placement of fibers in microstructure.•Epistemic uncertainties are quantified and can be reduced by gathering more data.