  Estimation of uncertainty in deep learning models is of vital importance,
especially in medical imaging, where reliance on inference without taking into
account uncertainty could lead to misdiagnosis. Recently, the probabilistic
Variational AutoEncoder (VAE) has become a popular model for anomaly detection
in applications such as lesion detection in medical images. The VAE is a
generative graphical model that is used to learn the data distribution from
samples and then generate new samples from this distribution. By training on
normal samples, the VAE can be used to detect inputs that deviate from this
learned distribution. The VAE models the output as a conditionally independent
Gaussian characterized by means and variances for each output dimension. VAEs
can therefore use reconstruction probability instead of reconstruction error
for anomaly detection. Unfortunately, joint optimization of both mean and
variance in the VAE leads to the well-known problem of shrinkage or
underestimation of variance. We describe an alternative approach that avoids
this variance shrinkage problem by using quantile regression. Using estimated
quantiles to compute mean and variance under the Gaussian assumption, we
compute reconstruction probability as a principled approach to outlier or
anomaly detection. Results on simulated and Fashion MNIST data demonstrate the
effectiveness of our approach. We also show how our approach can be used for
principled heterogeneous thresholding for lesion detection in brain images.
