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