Title: Hidden Markov models are recurrent neural networks: A disease progression modeling application. Abstract: Hidden Markov models (HMMs) are commonly used for disease progression modeling when the true state of a patient is not fully known. Since HMMs may have multiple local optima, performance can be improved by incorporating additional patient covariates to inform parameter estimation. To allow for this, we formulate a special case of recurrent neural networks (RNNs), which we name hidden Markov recurrent neural networks (HMRNNs), and prove that each HMRNN has the same likelihood function as a corresponding discrete-observation HMM. As a neural network, the HMRNN can also be combined with any other predictive neural networks that take patient covariate information as input. We first show that parameter estimates from HMRNNs are numerically close to those obtained from HMMs via the Baum-Welch algorithm, thus empirically validating their theoretical equivalence. We then demonstrate how the HMRNN can be combined with other neural networks to improve parameter estimation and prediction, using an Alzheimer's disease dataset. The HMRNN yields parameter estimates that improve disease forecasting performance and offer a novel clinical interpretation compared with a standard HMM.