Stretch: Gaussian Naive Bayes
The problem, from first principles
02-naive-bayes-bernoulli modeled each feature as a coin flip — present or absent, a fine model for binary features but useless for real-valued ones. Naive Bayes's core assumption (features are conditionally independent given the class) doesn't care what distribution each feature follows, only that some per-feature probability model is chosen. For continuous features, the classic choice is a Gaussian: assume each feature, within each class, follows its own normal distribution.
From theory to code
Implement gaussian_nb_fit, gaussian_log_likelihood, and gaussian_nb_predict. input is real-valued this time (02-naive-bayes-bernoulli's features were binary), but the overall shape of fitting and prediction is unchanged — only the per-feature probability model itself changes from Bernoulli to Gaussian.
Constraints
gaussian_nb_fit(input, labels, var_smoothing=1e-9)returns a dict of per-class log-priors, per-feature means, and per-feature variances.var_smoothingfloors every variance above0, scaled by the largest per-feature variance across the entire dataset (not per class).gaussian_log_likelihood(x, mean, variance)returns a single float: the summed log-density across all features.gaussian_nb_predict(model, queries)returns one predicted class per row ofqueries, by argmax oflog_prior + log_likelihoodover classes.
Hints
Open one at a time. Each gives away a little more than the last.
Hint 1
The independence assumption is identical to 02-naive-bayes-bernoulli's — only the per-feature term inside the sum changes, from a Bernoulli log-probability to a Gaussian log-density.
Hint 2
epsilon = var_smoothing * np.var(input, axis=0).max() computed once over the whole dataset, then added to every class's own per-feature variance, prevents a constant-within-class feature from ever producing a literal variance = 0.
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