Errata

In this section, we used the function perm_fun() which was defined in "Resampling" Section. The original function calculates the mean difference between samples, but in the example presented here, we need to calculate the difference between proportions. To solve this issue, a proposed solution is to create a new function that only changes this specific line of code:

return x.loc[list(idx_B)].mean() - x.loc[list(idx_A)].mean()

with this:

return x.loc[list(idx_B)].sum()/ nB - x.loc[list(idx_A)].sum()/ nA

(The code is in Python)

**Figure 4-10 — LOWESS target in partial-residual plot**

In the associated code in the GitHub repository for Figure 4-10, the LOWESS smoother is applied to the component (`results.ypartial`) rather than to the **partial residuals**. Replace:

```python
smoothed = sm.nonparametric.lowess(results.ypartial, results.feature, frac=1/3)
```

with:

```python
smoothed = sm.nonparametric.lowess(
results.ypartial + results.residual, # PR_i(x)
results.feature,
frac=1/3
)
```

**Rationale.** The gray dashed line is intended to be a LOWESS smooth of the **partial residuals** to show the empirical relationship without imposing a polynomial. Smoothing the component (the black line) instead does not match the textbook definition of a partial-residual plot.

Note from the Author or Editor:
------------------
Hello,

I looked at this more. In contrast to the R implementation, GaussianNB treats categorical features as numerical. This is not correct. We can see this if we build a model with categorical features only. With the MultinomialNB, we get this prediction:
array([[0.65369619, 0.34630381]]))
while the GaussianNB results in this prediction:
(array([[9.99994372e-01, 5.62841090e-06]])
so essentially [1, 0].

It will be necessary to build two separate models and then combine the predictions.
------------------
My Response:

Thanks for taking a closer look.
You’re absolutely right that GaussianNB treats all inputs as continuous. If we feed it one-hot encoded categoricals, those 0/1 dummies are modeled as Gaussians with very small within-class variances, which can drive extreme posteriors (the near [1, 0] you observed when using only categoricals). That behavior is consistent with a mismatch between the model family (Gaussian) and the data type (categorical).

That said, if most of the signal lies in the continuous features and the goal is ranking by P(Y=1), a GaussianNB-centric approach can work quite well even if the categorical piece is handled imperfectly. As the book notes, Naive Bayes often provides decent ranking but biased probability estimates; calibration (e.g., CalibratedClassifierCV with isotonic or sigmoid) helps if calibrated probabilities matter.

Best,