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I am trying to understand the structure and, in particular, normality properties of the predictive distribution of a Bayesian structural time-series model. My reasoning is as follows.

The posterior distribution of model parameters (such as various standard deviations) is non-Gaussian in general. The inference via MCMC yields posterior draws from this distribution, and variational inference yields posterior draws from a surrogate, which is a collection of independent Gaussians. For each draw, regardless of the inference method used, the Kalman filter gives a multivariate Gaussian. The final posterior is a mixture of the multivariate Gaussians corresponding to the posterior draws of the model parameters. This implies that the predictive posterior is non-Gaussian; it is a mixture of Gaussians.

Does the above hold? If not, what is the correct explanation?

For some context, I am studying tfp.sts.forecast in TensorFlow Probability:

https://github.com/tensorflow/probability/blob/v0.12.1/tensorflow_probability/python/sts/forecast.py#L173-L375

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The question has been answered on GitHub by the author of that forecast as follows:

What you've said sounds correct to me. The predictive distribution over multiple timesteps is created explicitly in the code as a mixture of LinearGaussianStateSpaceModel distributions (each of which is multivariate Gaussian). … [S]o the marginal distribution at any given step is a mixture of Gaussians.

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