R2D2M2 prior for Saturated MMM

[ferrine]

The existing MMM can be improved with variance decomposition framework, r2d2m2, which is plug and play into the existing code

r2 = pm.Beta("r2", mu=0.8, sigma=0.5*0.8)
total_sigma = pm.LogNormal("total_sigma", np.log(np.std(data.y)), 0.1)
model_sigma = r2 ** .5 * total_sigma
error_sigma = (1-r2) ** .5 * total_sigma
control_fourier_features_split = pm.Dirichlet("control_features_split", np.ones(K), dims="control_fourier")
control_features_split = model_sigma * control_fourier_features_split[:-1] ** .5
fourier_split = model_sigma * control_fourier_features_split[-1] ** .5

Reference https://arxiv.org/abs/2208.07132

[juanitorduz]

Thanks! yo you recommend this for the seasonality (Fourier modes) components? Or more generally, on the whole model?

[wd60622]

Are the coefficients suppose to be for all betas in the model or just for a subset like the control / or fourier? @ferrine

[ferrine]

For control and Fourier. Each Fourier component is normalised the same way as control.

For marketing variables it would be another treatment, ideally via interactive prior predictive analysis

[wd60622]

Gotcha. So in this example, the control_fourier dim is the combination of fourier and controls? i.e. [*controls_dims, *fourier_dims]

And then the error_sigma would be used as the prior of sigma of likelihood, right?

Think this could fit into the generalization of the Prior class I linked. I.e

gamma_control, gamma_fourier, sigma = create_r2d2_priors(
    r2=Prior("Beta", mu=0.8, sigma=0.8 * 0.5), 
    total_sigma=Prior("LogNormal", mu=np.log(np.std(y)), sigma=0.1),
    # These are the dim names in the model that will be combined
    dims=["control", "fourier_mode"], 
)

model_config = {
    "likelihood": Prior("Normal", sigma=sigma), 
    "gamma_control": gamma_control, 
    "gamma_fourier": gamma_fourier,
}
mmm = MMM(..., model_config=model_config, ...)

gamma_control, gamma_fourier, and sigma would have to work like our priors class.

Also, can you expand on the "interactive prior predictive analysis"?

[wd60622]

Also, do people tend to put priors on K as well?

[ferrine]

What is k

[ferrine]

Ah, I see, k is better set manually, is is problematic to infer

[AlfredoJF]

Hey @wd60622 , @ferrine

This R2D2M2 prior will enable extra dims like geo for geo-hierarchical modelling? #648 (comment)

[wd60622]

Hi @AlfredoJF, Good question. The first step might not include geo, but it might be possible to extend. Maybe @ferrine knows better about this in practice.

In your mind, @AlfredoJF, would the user specify the r2 by geo? Or would it be global r2? Or better yet, what prior information would you have? Do you have an example API specifying the input and output so we can understand what the user would provide?

In my mind, the function just has to return sigma and various betas but adding geo becomes complex. Where does (geo, ) dim get included? All variables? sigmas? all betas? some betas?

Let me know your thoughts!