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Context: The strategy in this section can be used to derive a lower bound estimator (a stochastic objective function) for a variety of directed graphical models with continuous latent variables. We will restrict ourselves here to the common case where we have an i.i.d. dataset with latent variables per datapoint, and where we like to perform maximum likelihood (ML) or maximum a posteriori (MAP) inference on the (global) parameters, and variational inference on the latent variables. It is, for example, straightforward to extend this scenario to the case where we also perform variational inference on the global parameters; that algorithm is put in the appendix, but experiments with that case are left to future work. Note that our method can be applied to online, non-stationary settings, e.g. streaming data, but here we assume a fixed dataset for simplicity. The generative model (encoder) and variational approximation (decoder) from section 3 were used, where the described encoder and decoder have an equal number of hidden units. Since the Frey Face data are continuous, we used a decoder with Gaussian outputs, identical to the encoder, except that the means were constrained to the interval (0,1) using a sigmoidal activation function at the decoder output.Note that with hidden units we refer to the hidden layer of the neural networks of the encoder and decoder. Let us consider some dataset \mathbf{X}=\{\mathbf{x}^{(i)}\}_{i=1}^{N} consisting of N i.i.d. samples of some continuous or discrete variable \mathbf{x}. We assume that the data are generated by some random process, involving an unobserved continuous random variable \mathbf{z}. The process consists of two steps: (1) a value \mathbf{z}^{(i)} is generated from some prior distribution p_{\boldsymbol{\theta}^{*}}(\mathbf{z}); (2) a value \mathbf{x}^{(i)} is generated from some conditional distribution p_{\boldsymbol{\theta}^{*}}(\mathbf{x}|\mathbf{z}). We assume that the prior p_{\boldsymbol{\theta}^{*}}(\mathbf{z}) and likelihood p_{\boldsymbol{\theta}^{*}}(\mathbf{x}|\mathbf{z}) come from parametric families of distributions p_{\boldsymbol{\theta}}(\mathbf{z}) and p_{\boldsymbol{\theta}}(\mathbf{x}|\mathbf{z}), and that their PDFs are differentiable almost everywhere w.r.t. both \boldsymbol{\theta} and \mathbf{z}. Unfortunately, a lot of this process is hidden from our view: the true parameters \boldsymbol{\theta}^{*} as well as the values of the latent variables \mathbf{z}^{(i)} are unknown to us.

Question: As the data in Frey Face dataset is continuous, how did the authors process it?
They consider some dataset \mathbf{X}=\{\mathbf{x}^{(i)}\}_{i=1}^{N} consisting of N i.i.d. samples of some continuous or discrete variable \mathbf{x}. They assume that the data are generated by some random process, involving an unobserved continuous random variable \mathbf{z}. The process consists of two steps: (1) a value \mathbf{z}^{(i)} is generated from some prior distribution p_{\boldsymbol{\theta}^{*}}(\mathbf{z}); (2) a value \mathbf{x}^{(i)} is generated from some conditional distribution p_{\boldsymbol{\theta}^{*}}(\mathbf{x}|\mathbf{z}).