Clinical patient records are an example of high-dimensional data that is typically collected from disparate sources and comprises of multiple likelihoods with noisy as well as missing values. In this work, we propose an unsupervised generative model that can learn a low-dimensional representation among the observations in a latent space, while making use of all available data in a heterogeneous data setting with missing values. We improve upon the existing Gaussian process latent variable model (GPLVM) by incorporating multiple likelihoods and deep neural network parameterised back-constraints to create a non-linear dimensionality reduction technique for heterogeneous data. In addition, we develop a variational inference method for our model that uses numerical quadrature. We establish the effectiveness of our model and compare against existing GPLVM methods on a standard benchmark dataset as well as on clinical data of Parkinson's disease patients treated at the HUS Helsinki University Hospital.