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I have been looking at recurrent neural networks and LSTM models for time series, and it is interesting that they can predict multiple days ahead. For example, I can take inputs of 100 days and predict 7 or 10 days into the future.

I was wondering if there is a way to do this type of multi-step time series prediction using a Bayesian time-series model? Now let me be a little more precise about what I mean. A bayesian model can obviously predict one day ahead, based upon the previous 1, 2 or 3 days. That is a simple bayesian auto-regressive process. But is it possible to simultaneously predict 2, 3, 4 or more days ahead, given training data for previous days or weeks or months?

Now, it is fine if the usual models and methods can only output a single day at a time, and hence cannot predict multiple days in advance. In those cases, I would predict one day in advance, append the prediction to the dataset and predict again iteratively. So that would work. If that is the case, then someone can simply indicate that. BUT I was not sure if this multiple outputs idea was possible with bayesian models?

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Yes it is possible to do this with a Bayesian time series model. Predictions from Bayesian models are distributions rather than single values. Usually, these distribution are represented by samples (when using MCMC for example). So in practice, we can apply your "multiple output" idea for each sample, that is for a given sample, make a prediction one step ahead and then use this prediction to make another prediction one step ahead (so two steps ahead), and repeat this process as many times as you want. Doing that, each sample actually represents a predicted trajectory and the collection of samples gives you the predictive distribution of future trajectories.

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