This paper addresses the duality between the deterministic feed-forward neural networks (FF-NNs) and linear Bayesian networks (LBNs), which are the generative stochastic models representing probability distributions over the visible data based on a linear function of a set of latent (hidden) variables. The maximum entropy principle is used to define a unique generative model corresponding to each FF-NN, called projected belief network (PBN). The FF-NN exactly recovers the hidden variables of the dual PBN. The large- N asymptotic approximation to the PBN has the familiar structure of an LBN, with the addition of an invertible nonlinear transformation operating on the latent variables. It is shown that the exact nature of the PBN depends on the range of the input (visible) data details for the three cases of input data range are provided. The likelihood function of the PBN is straightforward to calculate, allowing it to be used as a generative classifier. An example is provided in which a generative classifier based on the PBN has comparable performance to a deep belief network in classifying handwritten characters. In addition, several examples are provided that demonstrate the duality relationship, for example, by training networks from either side of the duality.