PPL Mixture Workflow¶

The PPL layer is useful when the statistical formula is clearer than an estimator tree. This tutorial fits a two-component Gaussian mixture with free parameter slots and then inspects the concrete distribution that the expression lowered into.

The important idea is that PPL expressions are not a separate modeling world. They lower back to Mixle distributions, estimators, targets, and inference routes.

1. Build Synthetic Data¶

import numpy as np

rng = np.random.RandomState(1)
data = list(
    np.concatenate(
        [
            rng.normal(-5.0, 1.0, 8000),
            rng.normal(5.0, 1.0, 8000),
        ]
    )
)

The data has two clear modes. A one-Gaussian model would blur the structure; a mixture can represent the latent group.

2. Declare The Model¶

from mixle.ppl import Mix, Normal, free

expr = Mix([Normal(free, free), Normal(free, free)])

Normal(free, free) means both the mean and scale are estimated. Mix adds a latent component assignment, so fit routes to the same EM machinery used by mixle.stats.MixtureEstimator.

3. Fit The Expression¶

model = expr.fit(
    data,
    max_its=80,
    rng=np.random.RandomState(7),
)

The returned object keeps the PPL-facing wrapper, while model.dist is the underlying fitted distribution.

4. Inspect The Bound Distribution¶

means = sorted(component.mu for component in model.dist.components)
weights = model.dist.w

print(means)
print(weights)
print(model.dist.log_density(0.0))

After fitting, use ordinary distribution methods for scoring, sampling, and capability inspection.

5. Ask Posterior Questions¶

responsibilities = model.posterior([-5.0, 0.0, 5.0])
print(responsibilities)

For a mixture, the posterior is responsibility mass over components. For an HMM, the same idea becomes state posterior mass through time.

6. Add Observed Covariates¶

PPL expressions can also reference observed fields. The expression below is a Poisson regression-style model whose rate depends on an observed feature.

from mixle.ppl import Field, Poisson

regression = Poisson(free * Field("x") + free)
fitted = regression.fit(counts, given={"x": x})

Use the PPL layer when it clarifies the model declaration. Use the estimator API directly when you need exact control over every child estimator, initialization, or backend detail.

Validation Checklist¶

For mixture-style PPL models:

run more than one random seed or use a restart strategy for difficult data;
inspect the lowered distribution, not only the wrapper;
compare against a simpler baseline on held-out log score;
check whether component labels are identifiable enough for the intended interpretation;
record the lowered model structure in provenance.

Read Probabilistic Programming for the full expression language and Inference for the lower-level fitting controls.