All-sky observations show both Galactic and non-Galactic diffuse emission, for example from interstellar matter or the cosmic microwave background (CMB). The decomposition of the emission into different underlying radiative components is an important signal reconstruction problem. We aim to reconstruct radiative all-sky components using spectral data, without incorporating knowledge about physical or spatial correlations. We built a self-instructing algorithm based on variational autoencoders following three steps: (1) We stated a forward model describing how the data set is generated from a smaller set of features, (2) we used Bayes' theorem to derive a posterior probability distribution, and (3) used variational inference and statistical independence of the features to approximate the posterior. From this, we derived a loss function and optimized it with neural networks. The resulting algorithm contains a quadratic error norm with a self-adaptive variance estimate to minimize the number of hyperparameters. We trained our algorithm on independent pixel vectors, each vector representing the spectral information of the same pixel in 35 Galactic all-sky maps ranging from the radio to the gamma-ray regime. The algorithm calculates a compressed representation of the input data. We find the feature maps derived in the algorithm's latent space show spatial structures that can be associated with all-sky representations of known astrophysical components. Our resulting feature maps encode (1) the dense interstellar medium (ISM), (2) the hot and dilute regions of the ISM, and (3) the CMB, without being informed about these components a priori. We conclude that Bayesian signal reconstruction with independent Gaussian latent space statistics is sufficient to reconstruct the dense and the dilute ISM, as well as the CMB, from spectral correlations only. The approximation of the posterior can be performed computationally efficient using variational inference and neural networks, making them a suitable approach to probabilistic data analysis.