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Suppose I have some quantity I want to forecast, like the traffic at a particular intersection or the sales volume at a particular store. I have three sources of data to make use of:

  1. Broad-scale historical data that assumes the distribution is stationary/normal and gives only a mean and standard deviation. Both of those assumptions are clearly false, but empirical knowledge of the historical average and variance should have at least some value.

  2. The outputs from an existing predictive model. I can query this model and recieve a time series of what it thinks the quantity should be over the recent past and/or near future. In an ideal world this is all that's needed, but it's unclear how trustworthy this model is, and unlike (1) it says nothing about variance.

  3. An actual observation of the quantity's value right now (with nominal measurement noise).

I can think of (1) as a predictive model that does nothing but draw samples from that normal distribution, so intuitively, model (1) gives me one prediction for the current value, model (2) gives me a different prediction for the current value, and the actual observed value (3) should give me a way of intelligently interpolating between the two. It seems like I should be able to use those three values to define a combination of models (1) and (2) that's "optimal" in some Bayesian sense, and then make my final predictions using that. How would I actually do that in practice, though?

Obtaining confidence bounds around the predictions from (2) is also desirable.

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