What do we want to compute and why?
I thought you are mostly interested in the inference issues here. Once we have the trained model we can inference any distribution using the model. For instance:
1. We can do sampling using the joint distribution $P(X,Z)$.
Why? It is a method for use to generate new data points.
How? We first sample from the latent variables $Z$ and then sample from the conditional distribution $P(X|Z)$. If $X$ are all binary varialbes we can just sample according to a binomial distribution.
2. We can plot the conditional expectation $E(X|Z)$.
Why? We can intuit the role of the latent variables by checking the expectation of the manifest variables.
How? We just sample from the latent variables and then get the conditional distribution and that is the expectation of the manifest variables. We then can plot the images on a N-D grid where the grid axes correspond to $z_1$...$z_n$ respectively.
For instance, if we set two latent variables(discretized here) for the MNIST task and we may get such a grid:
We can see that most likel the vertical coyordinate controls the thickness and the horizental coordinate the curve.
3. We can calculate the joint distribution of the manifest varialbes
Why? We can use this distribution to check the validity of some new data. If the probability of a new sample is too low we can say that it an invalid sample and otherwise it would most like a similar sample.
How? We can use a validation dataset and a test dataset. Firstly over the validation dataset(all valid samples) we calculate the joint probability(of the manifest variables by summing out the latent variables) of each sample and then find the 75 percentile and the 25 percentile. Then we try the test samples(may contain invalid samples), if the probability falls in the range(25-75 percentile) we say that it is valid otherwise invalid. In this way we can filter out some bad samples from the test dataset.
Actually once we know the parameters of the Bayes model we can get any distribution and we can use them to find many interesting things.