I am aware that there is a plethora of deep generative models out there (e.g. variational autoencoders (VAE), GANs) that can model high-dimensional data as the images of latent variables under a non-linear mapping (typically neural network).
In more traditional methods such as probabilistic PCA, the latent variables can be marginalised analytically. In Bayesian PCA (BPCA), we can additionally integrate out the linear mapping, from the latent space to the observation space, by adopting the variational lower bound that leads to closed form updates of the parameters. The Gaussian Process Latent Variable (GPLVM) model adopts a non-linear probabilistic mapping (a Gaussian process) that can be marginalised. These two models enjoy to a certain degree analytical solutions concerning the inference of the latent variables and the mapping.
I have been wondering whether there is any research into more "complex" models (perhaps I should call them deep) that are capable of modelling more complex data distributions than the GPVLM and BPCA, but retain analytical solutions when inferring the posterior of the latent variables (like BPCA) or the mapping (like GPLVM)?
What I like about the GPLVM and BPCA is that they possess an objective function that can be analytically optimised, as opposed to the intractable objective of VAEs that necessitates Monte-Carlo averages and stochastic gradient. Could somebody please point me to such examples of more complex generative models that admit analytical inference for working out the posterior of the latent variables or the mapping?
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