Uncertainty quantification is an essential task in machine learning - a task in which neural networks (NNs) have traditionally not excelled. This can be a limitation for safety-critical applications, where uncertainty-aware methods like Gaussian processes or Bayesian linear regression are often preferred. Bayesian neural networks are an approach to address this limitation. They assume probability distributions for all parameters and yield distributed predictions. However, training and inference are typically intractable and approximations must be employed. A promising approximation is NNs with Bayesian last layer (BLL). They assume distributed weights only in the last linear layer and yield a normally distributed prediction. NNs with BLL can be seen as a Bayesian linear regression model with learned nonlinear features. To approximate the intractable Bayesian neural network, point estimates of the distributed weights in all but the last layer should be obtained by maximizing the marginal likelihood. This has previously been challenging, as the marginal likelihood is expensive to evaluate in this setting and prohibits direct training through backpropagation. We present a reformulation of the log-marginal likelihood of a NN with BLL which allows for efficient training using backpropagation. Furthermore, we address the challenge of quantifying uncertainty for extrapolation points. We provide a metric to quantify the degree of extrapolation and derive a method to improve the uncertainty quantification for these points. Our methods are derived for the multivariate case and demonstrated in a simulation study, where we compare Bayesian linear regression applied to a previously trained neural network with our proposed algorithm
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