Predicted Target ≠ Sum of Contributions (Case Study Notebook)

[RamiFisherTW]

While working with the case study notebook in the PyMC Marketing Mix Model repository, I observed a consistent discrepancy between the predicted target (y) values and the sum of individual contribution components. These values should mathematically align, but they differ significantly across the time series.

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🧪 Reproduction Steps

# Step 1: Sample predictions and contributions
y_pred_train = mmm.sample_posterior_predictive(
    X_train,
    include_last_observations=False,
    original_scale=True,
    var_names=[
        "y",
        "channel_contribution",
        "intercept",
        "control_contribution",
        "yearly_seasonality_contribution"
    ],
    extend_idata=False,
    random_seed=42,
)

# Step 2: Compare predictions and total contributions
diff1 = (
    mmm.idata.posterior_predictive["y"].mean(dim=["chain", "draw"]) * 
    mmm.get_target_transformer()["scaler"].scale_[0]
    - mmm.compute_mean_contributions_over_time(original_scale=True).sum(axis=1)
)

diff2 = (
    y_pred_train["y"].mean(dim=["sample"]).values
    - mmm.compute_mean_contributions_over_time(original_scale=True).sum(axis=1)
)

print(f"Mean difference 1: {diff1.mean().values}")
print(f"Mean difference 2: {diff2.mean()}")

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✅ Expected Behavior

The sum of all contributions—channel, intercept, control, seasonality—should match the predicted target values when using original_scale=True.
Specifically:
y_t ≈ channel_t + intercept_t + control_t + seasonality_t

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❌ Actual Behavior

Even with consistent use of original_scale=True and the same scaling factors:

• posterior_predictive["y"] and y_pred_train["y"] both differ from the sum of contributions.
• The difference is not negligible and appears structured across time.

Possible causes:

• Missing components in compute_mean_contributions_over_time()
• Mismatch in scaling logic or inverse transformation
• Implementation-level inconsistency in the contribution pipeline

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🙏 Request for Guidance

Would love clarification on the following:

1. Is an additional component (e.g. noise/sigma) expected in the sum for it to match y?
2. Is this numerical mismatch expected behavior under the current model assumptions?
3. Could this be a bug in compute_mean_contributions_over_time() or the transformation pipeline?

[shuaTW]

Yes, I'm seeing the same problem by me as well.