> For the complete documentation index, see [llms.txt](https://docs.algenta.ai/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.algenta.ai/machine-learning/ml-mixture-models.md).

# Gaussian mixture models

Gaussian mixture models fit by log-space EM with BIC/AIC model selection.

`mixture_models` fits Gaussian mixture models via expectation-maximization — replacing `sklearn.mixture.GaussianMixture` (diagonal covariance) with pure Mojo: seeded k-means++ initialization, log-space E-step (log-sum-exp), variance-floored M-step, and BIC for choosing the number of components. Zero Python imports.

## The modules

| Module           | What it does                                                                                                                                                                                                                                  |
| ---------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `mixture_models` | Gaussian mixture models fit by log-space EM — seeded k-means++ initialization, log-sum-exp E-step, variance-floored M-step, and BIC for choosing the number of components — replacing `sklearn.mixture.GaussianMixture` (diagonal covariance) |

## Functions

| Function                                                       | What it does                                                                                                                                          |
| -------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------- |
| `GMMResult`                                                    | 1D fit result: `weights` / `means` / `vars` (components sorted by mean ascending), `loglik`, per-iteration `loglik_trace`, `iterations`, `converged`. |
| `GMMResultND`                                                  | N-D diagonal-covariance fit result; `means` / `vars` are flat row-major (k × cols), components sorted by first mean coordinate.                       |
| `gmm_fit(x, k, max_iter, tol, seed)`                           | Fit a k-component 1D Gaussian mixture via EM. Empty result if `len(x) == 0`, `k <= 0`, or `k > len(x)`.                                               |
| `gmm_fit_nd(X, rows, cols, k, max_iter, tol, seed)`            | Fit on flat row-major `X` (rows × cols) with diagonal covariance. Empty result on shape violations (`len(X) != rows * cols`, `k > rows`, ...).        |
| `gmm_responsibilities(x, weights, means, vars)`                | Posterior component probabilities per 1D point, flat (n × k); each row sums to 1. Empty list on inconsistent lengths.                                 |
| `gmm_responsibilities_nd(X, rows, cols, weights, means, vars)` | Posterior component probabilities per N-D row, computed in log space.                                                                                 |
| `gmm_bic(loglik, n, k, dims)`                                  | Bayesian Information Criterion, `p·ln(n) − 2·loglik` with `p = k(2·dims + 1) − 1`; lower is better.                                                   |

## Guarantees

* **EM monotonicity**: the E-step runs in log space via log-sum-exp, so `loglik_trace` is non-decreasing — the tests assert this on every fit (slack 1e-9). Convergence is `|Δloglik| < tol` before `max_iter`.
* **Determinism**: k-means++ initialization uses a seeded xorshift64 RNG; refitting with the same seed reproduces `loglik`, `iterations`, and means exactly (tested at seed 42).
* **No component collapse**: variances are floored at `VAR_FLOOR = 1e-9` and weights at `1e-12`; a single-point fit returns the exact mean with a floored variance, and all-equal data with k=2 stays finite (no NaN/inf).

## Choosing k with BIC

Fit each candidate k, then compare `gmm_bic(result.loglik, n, k, dims)` — lowest wins. On the two-cluster test data (100 points near 0, 100 near 10), k=2 beats k=1 by BIC, and the fitted model recovers means within 0.5, weights 0.5/0.5 within 0.1 (0.75/0.25 for a 150/50 split).

## Verification

Test reference values are hand-computed closed forms, checked to tight tolerances:

* k=1 on `[1, 2, 3]` is the exact MLE: mean 2, population variance 2/3 (to 1e-12), log-likelihood **−3.6486179374517715** (to 1e-9).
* Responsibilities under `w=[0.5,0.5], means=[0,10], vars=[1,1]`: the midpoint x=5 gives exactly 0.5/0.5; x=4 gives `1/(1+e^{−10}) = 0.9999546021312976` (to 1e-9).
* BIC: `gmm_bic(-100, 100, 2, 1) = 223.02585092994047` and `gmm_bic(-50, 200, 3, 2) = 174.1764431316725` (to 1e-6).

**Next:** promote what you trained with [statistical model comparison](/machine-learning/ml-model-comparison.md), or browse the [engine catalog](/engine-runtime/catalog.md).


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