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This paper is about a new Bayesian method for multi label learning. The goal is to classify accurately in settings where there are many potential labels but only a few of them apply to each data point. The basis of the new results is a new generative model for the label vector of each example. Specifically the label vector y_n of the n-th example is generated as y_n f(V(\sigma(Wx_n)), where Wx_n is a lower dimensional projection of the n-th instance x_n, followed by an element-wise sigmoid activation \sigma. The final operation f corresponds to drawing Poisson random variables with rates given by V(\sigma(Wx_n)) and thresholding these so-called latent counts by taking the minimum with 1.