# `density_metrics`

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> Axis ``density_metrics`` on sub-layer ``L5_A_metric_specification`` (layer ``l5``).

## Sub-layer

**L5_A_metric_specification**

## Axis metadata

- Default: `['log_score', 'crps']`
- Sweepable: False
- Status: operational

## Operational status summary

- Operational: 4 option(s)
- Future: 0 option(s)

## Options

### `coverage_rate`  --  operational

Empirical coverage rate -- share of OOS observations falling within the nominal-α interval.

See [coverage_rate function page](../density_metrics/coverage_rate.md) for full documentation + parameters + standalone usage. Standalone: ``mf.functions.coverage_rate``.

### `crps`  --  operational

Continuous ranked probability score -- generalisation of MAE to densities.

Density-forecast metric ``crps``. ``CRPS = ∫ (F̂(y) - 1{y ≥ y_obs})² dy``. Strictly-proper, expressed in the same units as the target. Reduces to MAE when the predictive distribution is a point mass at the predicted value. Standard density-score in weather / macro forecasting (Gneiting-Katzfuss 2014).

**When to use**

Distributional forecasts; comparing point and density forecasts on a common scale.

**References**

* macroforecast design Part 3, L5: 'evaluation = (metric × benchmark × aggregation × decomposition × ranking).'
* Gneiting & Raftery (2007) 'Strictly Proper Scoring Rules, Prediction, and Estimation', JASA 102(477): 359-378. (doi:10.1198/016214506000001437)
* Gneiting & Katzfuss (2014) 'Probabilistic Forecasting', Annual Review of Statistics and Its Application 1: 125-151.

**Related options**: [`log_score`](#log-score), [`interval_score`](#interval-score), [`coverage_rate`](#coverage-rate)

_Last reviewed 2026-05-05 by macroforecast author._

### `interval_score`  --  operational

Winkler (1972) interval score -- jointly penalises miscoverage + interval width.

See [interval_score function page](../density_metrics/interval_score.md) for full documentation + parameters + standalone usage. Standalone: ``mf.functions.interval_score``.

### `log_score`  --  operational

Logarithmic predictive density score -- ``log f̂(y_t)``.

Density-forecast metric ``log_score``. The strictly-proper scoring rule recommended by Gneiting & Raftery (2007). Equivalent to the Bayesian predictive log-likelihood. Larger = better. Requires ``forecast_object = density / quantile`` from L4.

When the predictive density is parametric (e.g. Gaussian) the score reduces to a closed-form involving the predictive mean / variance.

**When to use**

Default scoring rule for Bayesian forecasts; probabilistic horse-race ranking.

**References**

* macroforecast design Part 3, L5: 'evaluation = (metric × benchmark × aggregation × decomposition × ranking).'
* Gneiting & Raftery (2007) 'Strictly Proper Scoring Rules, Prediction, and Estimation', JASA 102(477): 359-378. (doi:10.1198/016214506000001437)
* Gneiting & Katzfuss (2014) 'Probabilistic Forecasting', Annual Review of Statistics and Its Application 1: 125-151.

**Related options**: [`crps`](#crps), [`interval_score`](#interval-score), [`coverage_rate`](#coverage-rate)

_Last reviewed 2026-05-05 by macroforecast author._