feat: Add Bayesian Structured Time Series (BSTS) Demo Notebook

[albertpod]

This PR adds a new Jupyter notebook demonstrating a complete end-to-end workflow for Bayesian Structured Time Series (BSTS).

The notebook serves as a practical example of:

1. Dynamic Data Ingestion: Fetching real-time historical and forecast weather data via the Open-Meteo API.
2. Bayesian Inference: Using a State Space Model to decompose demand into Trend, Seasonality, and Exogenous variables (Temperature/Rain).

[Nimrais]

Very fun example! Good job! I did spend some time trying to make a very similiar model to work.

I see for you prediction rule you use

@rule AugmentedPoisson(:out, Marginalisation) (q_η::GaussianDistributionsFamily,) = begin
    return Poisson(exp(mean(q_η)))
end

In the literature your AugmentedPoisson is called Poisson–lognormal distribution, I found a nice cheatsheet for it https://jchiquet.github.io/assets/talks/pln_semstat_11-22.pdf.

More or less it means in this model you can easily improve your prediction quality changing this rule into

@rule AugmentedPoisson(:out, Marginalisation) (q_η::GaussianDistributionsFamily,) = begin
    _mean = mean(q_η)+ 0.5*var(q_η)
    return Poisson(exp(_mean))
end

[albertpod]

@Nimrais, looks cool! Feel free to file a PR inside this PR with any suggestion!

[albertpod]

@ismailsenoz I integrated both the Poisson and BinomialPolya code into this notebook, but I think I made some mistakes with the scaling or data pre-processing before feeding it to the model. Consequently, the inference is worse than what we saw previously. Could you take a look?

Main to-dos before merging:

• Check the narrative and explanations.

• Fix the inference for the bonus Poisson and BinomialPolya nodes. (@ismailsenoz I can't see the mistake, it worked in script you shared for Poisson, for BinomialPolya, I haven't ran on tacos dataset, but should work imo)

• Some weird warning when running inference for Poisson and Binomial about constraints.. I don't see why

┌ Warning: Some variables in factorization constraint q(z, z₁, z₂, zprev, F, Q, η, μ, β, s) = q(z, z₁, z₂, zprev, β, η, μ, s)q(F)q(Q) are not present in the context.

• Rerun the notebook one last time before NeurIPS to get a more up-to-date forecast!