I want to operate a Bayesian Neural Network classification where I have the output as y as element of (honest and malicious) and theta is the weight and bias from the neural network and D is the data comprising of (x,y). I am lacking knowledge on the mathematical formulation of the output given other variables.

there’s documentation for a bayes NN:

http://edwardlib.org/tutorials/bayesian-neural-network

but if I’ve correctly gathered from your post that you might be lacking exposure to these types of things, playing around with a standard feedforward neural net and separately implementing linear and logistic regression (along with their bayesian versions) might be good practice before hand.