In the previous chapters, we discussed latent variable models that map a latent variable z to an observed variable x using (a series of) stochastic transformations pZ|X(z|x). By considering the map to be stochastic, methods such as the discussed variational autoencoder (Kingma and Welling, 2014, 2019) can easily change the dimensionality of the latent space: adding dimensions is done by mixing in noise random variables, while reducing dimensions is achieved through contractive maps like pooling. The probabilistic framework makes this easy to handle.