Build a classifier out of a single (mostly false) assumption: that the features are independent. What makes this model distinctive is that training is one pass of aggregation — no iterations, no learning rate.

Input

X  # (n, d) real-valued features
y  # (n,)  integer labels

If X arrives one-dimensional, treat it as (n, 1).

Training

Handle classes in ascending sorted order. For class c:

• prior $P(c) = n_c / n$
• per-feature mean $\mu_{c,i}$
• per-feature population variance (ddof=0, i.e. divide by n_c) $\sigma^2_{c,i}$

A zero variance would divide by zero during prediction, so add 1e-6 to every variance.

The usual choice is 1e-9, and there is a reason it is 1e-6 here. Rounding to 6 decimal places — as the return rule below requires — turns 1e-9 straight into 0.0, the smoothing disappears, and prediction blows up on log(0). The constant has to survive the rounding.

Prediction

Pick the class with the largest log posterior.

$\log P(c \mid x) \propto \log P(c) + \sum_i \log \mathcal{N}(x_i; \mu_{c,i}, \sigma^2_{c,i})$

$\log \mathcal{N}(x; \mu, \sigma^2) = -\frac{1}{2}\log(2\pi\sigma^2) - \frac{(x - \mu)^2}{2\sigma^2}$

On a tie, pick the smaller class value.

Return

fit_gaussian_nb(X, y) → a dictionary with these four keys. Round every float to 6 decimal places.

{"classes": [...], "priors": [...], "means": [[...], ...], "vars": [[...], ...]}

Implement fit_gaussian_nb(X, y).

• classes is the class list in ascending sorted order.
• priors[j] = n_c / n
• means[j][i] and vars[j][i] — per-class, per-feature mean and population variance (ddof=0). numpy's .var() already defaults to ddof=0, so it works as-is.
• Add 1e-6 to every variance — with 1e-9 the smoothing vanishes under round(v, 6).
• Round every float with round(v, 6).