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Currently, the minimize step to obtain $x^*$ which maximises $\textnormal{logp}(x \mid y, \theta)$ simply randomly initializes the initial value for the minimizer - this can be problematic if the optimum is far from this point.
Ideally, the initializer should produce a random point around some point of interest - the mean/mode of the prior on the latent $x$ is a good choice (note that $x$ is Gaussian and thus mean = mode).