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Stochastic models with global parameters $\bm{\theta}$ and latent variables$\bm{z}$ are common, and variational inference (VI) is popular for theirestimation. This paper uses a variational approximation (VA) that comprises aGaussian with factor covariance matrix for the marginal of $\bm{\theta}$, andthe exact conditional posterior of $\bm{z}|\bm{\theta}$. Stochasticoptimization for learning the VA only requires generation of $\bm{z}$ from itsconditional posterior, while $\bm{\theta}$ is updated using the naturalgradient, producing a hybrid VI method. We show that this is a well-definednatural gradient optimization algorithm for the joint posterior of$(\bm{z},\bm{\theta})$. Fast to compute expressions for the Tikhonov dampedFisher information matrix required to compute a stable natural gradient updateare derived. We use the approach to estimate probabilistic Bayesian neuralnetworks with random output layer coefficients to allow for heterogeneity.Simulations show that using the natural gradient is more efficient than usingthe ordinary gradient, and that the approach is faster and more accurate thantwo leading benchmark natural gradient VI methods. In a financial applicationwe show that accounting for industry level heterogeneity using the deep modelimproves the accuracy of probabilistic prediction of asset pricing models.

Natural Gradient Hybrid Variational Inference with Application to Deep Mixed Models