We have introduced a Bayesian neural network in quantitative log prediction studies with the goal of improving the petrophysical characterization and quantifying the uncertainty of model predictions. Neural network (NN) methods are gaining popularity in the petrophysics and geophysics communities; however, uncertainty quantification in model predictions is often neglected in the available literature, where the prediction is frequently performed in a deterministic setting. Determination of the uncertainty of the petrophysical model requires the estimation of the posterior distribution of the neural parameters that is generally mathematically intractable; for this reason, we adopt a variational approach to approximate the posterior model of the Bayesian network. To represent the uncertainty, we randomly draw samples from the posterior distribution of neural parameters to predict the model variables given the input data, leading to a learned-ensemble predictor. The proposed approach combines the ability of the NN of extracting hidden relations within the data set that physical relations cannot describe and the probabilistic framework for uncertainty quantification. We apply the proposed method to a well-log data set from the Volve Field, offshore Norway, to predict well logs in intervals where the data are incomplete or missing due to operational issues in the drilling procedure. The proposed approach is validated in intervals where the true data are available but not included in the training process. In the proposed application, the correlation coefficient between predictions and true data is greater than 0.9. In terms of accuracy, the results are comparable to those obtained using a traditional NN approach; however, the proposed method also provides a quantification of the uncertainty in the results, which offers additional information on the confidence in the predictions.