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You might be interested in TensorFlow Probability. It has a Python API, and has been chosen to replace Theano as the PyMC3 backend at some point in the future. Tensorflow Probability can also be used for MCMC directly, and it has dedicated functionality for Bayesian structural time series modelling. There is a nice blog post which provides an introduction.


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You can see that \begin{align} p(\sigma \vert Y) &\propto p(\sigma)p(Y\vert\sigma) %\\&= %p(\sigma) \int_{0}^1\frac{1}{\sigma}\phi\left(\frac{x - y}{\sigma}\right)dx\\ %&= p(\sigma)\left[\Phi\left(\frac{y}{\sigma}\right) - \Phi\left(\frac{y-1}{\sigma}\right)\right], \end{align} and that \begin{align*} p(\sigma \vert Y, Z) &\propto p(\sigma)...


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This does not seem to be necessarily wrong. If the process that generates Y given X is the same underlying physical phenomenon that generates Z given X - in other words, if Z is another view on Y when both are caused by the same X - then observing more data through Z wouldn't necessarily change what the model knows about X. Of course, it is difficult to say ...


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