Distributional Inference: Predict a Distribution, Not a Point

A ModelTask::Regression head returns a predictive distribution per row — a Gaussian (predicted_mean, predicted_std) or a set of quantiles — instead of a single number. Where conformal prediction wraps any predictor with a distribution-free interval, a regression head is trained to emit calibrated uncertainty directly, with proper-scoring objectives that make that uncertainty honest.

Two output forms, both standard:

• Parametric Gaussian — the head predicts μ and a raw scale; serving maps the scale to a positive σ = floor + softplus(raw). Smooth, cheap, a closed- form density. The default.
• Quantile — the head predicts a set of levels (e.g. 0.05, 0.5, 0.95) directly. Distribution-free in shape, robust to non-Gaussian outcomes, and the input to conformal CQR. The serving adapter sorts each row’s quantiles so they never cross.

Choose the objective by your label

The four regression objectives are all proper scores — minimising them rewards a calibrated distribution, not merely an accurate mean. (MSE on the predicted mean is not proper for a distribution and is only a secondary point- accuracy diagnostic.)

• β-NLL (default) — Seitzer’s variance-weighted Gaussian NLL. The plain joint μ,σ² NLL has a well-documented pathology: it down-weights high-error points by inflating their variance, starving the mean’s gradient and collapsing to overconfidence elsewhere. β-NLL re-weights each row’s NLL by a detached σ^{2β}, restoring the mean’s gradient and removing the collapse. β = 0.5 is the recommended default.
• CRPS — the closed-form Gaussian continuous ranked probability score, the other collapse-resistant choice: strictly proper, in the outcome’s units, and far more stable under joint μ,σ² training than NLL.
• Gaussian NLL — the classic mean-variance objective, provided for completeness and as the pathology baseline. Prefer β-NLL or CRPS.
• Pinball — the quantile objective; trains each quantile to its level, with a non-crossing penalty that discourages crossing during training.

The same CRPS / NLL math headlines the calibration eval — one source of truth for the score, used as both the training loss and the eval metric.

Calibrated, not merely accurate

A regression head is not done until its coverage is verified. Two models can share a mean (identical MSE) yet one be badly miscalibrated. The calibration eval is the gate: the central interval should cover at ≈ its nominal level, and the head’s proper score (CRPS/NLL) should beat a constant-variance baseline. Verify coverage; never ship on NLL alone.

Aleatoric, not epistemic

A parametric Gaussian head models aleatoric (irreducible data) noise only. It does not know what it has not seen: off-distribution it can be confidently wrong. For uncertainty about the unseen, reach for the rest of the spectrum:

• distribution-free coverage with no model assumption → conformal prediction;
• amortized epistemic posteriors → a future neural-process head.

Do not read this head’s σ as epistemic.

The uncertainty evidence channel

A served distribution rides the uncertainty evidence channel (predicted_mean, predicted_std, quantiles, context_ref) — the same additive substrate as vector, inference, and conformal. When the prediction was conditioned on an assembled context set, the channel records which rows informed it in context_ref — data-driven provenance applied to prediction. Register it like any custom channel (see Declare a Custom Provenance Channel); the distribution columns then merge into the result and are SQL-reachable.