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Modern machine learning algorithms excel in a great variety of tasks, but at the same time, it is also known that those complex models need to deal with uncertainty from different sources. Consequently, understanding if the model is indeed making accurate predictions or simply guessing at random is not trivial, and measuring the confidence bounds becomes very important. Bayesian machine learning seems to provide the solution, however, many of the state-of-the-art Bayesian algorithms use rigid parametric representations of the uncertainty where the learning process depends on the gradient of a predefined cost function. In this article, a new gradient-free training algorithm based on Approximate Bayesian Computation by Subset Simulation is proposed, where the likelihood function and the weights are defined by non-parametric formulations, resulting in a flexible and fairer representation of the uncertainty. The experiments, specially the engineering case study on composite materials subject to fatigue damage, show the ability of the proposed algorithm to consistently reach accurate predictions while avoiding gradient related instabilities, and most importantly, it provides a realistic and coherent quantification of the uncertainty represented by confidence bounds. All this may lead to a reduction of safety factors in engineering problems, and in general, allows us to make well-informed decisions in situations with a high degree of uncertainty and risk. A comparison with the state-of-the-art Bayesian Neural Networks is also carried out. • Neural networks trained with approximate Bayesian computation. • Accurate and flexible representation of the uncertainty in the observed data. • Stability of predictions thanks to the gradient-free nature of the algorithm. • Non-parametric weights and likelihood function provide adaptability to data. • Appropriate when decisions are dependent on the level of uncertainty. [ABSTRACT FROM AUTHOR]