  Bayesian neural network models (BNN) have re-surged in recent years due to
the advancement of scalable computations and its utility in solving complex
prediction problems in a wide variety of applications. Despite the popularity
and usefulness of BNN, the conventional Markov Chain Monte Carlo based
implementation suffers from high computational cost, limiting the use of this
powerful technique in large scale studies. The variational Bayes inference has
become a viable alternative to circumvent some of the computational issues.
Although the approach is popular in machine learning, its application in
statistics is somewhat limited. This paper develops a variational Bayesian
neural network estimation methodology and related statistical theory. The
numerical algorithms and their implementational are discussed in detail. The
theory for posterior consistency, a desirable property in nonparametric
Bayesian statistics, is also developed. This theory provides an assessment of
prediction accuracy and guidelines for characterizing the prior distributions
and variational family. The loss of using a variational posterior over the true
posterior has also been quantified. The development is motivated by an
important biomedical engineering application, namely building predictive tools
for the transition from mild cognitive impairment to Alzheimer's disease. The
predictors are multi-modal and may involve complex interactive relations.
