arviz_stats.loo_score

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arviz_stats.loo_score#

arviz_stats.loo_score(data, var_name=None, kind='crps', pointwise=False, round_to=None, log_weights=None, pareto_k=None)[source]#

Compute PWM-based CRPS/SCRPS with PSIS-LOO-CV weights.

Implements the probability-weighted-moment (PWM) identity for the continuous ranked probability score (CRPS) with Pareto-smoothed importance sampling leave-one-out (PSIS-LOO-CV) weights, but returns its negative as a maximization score (larger is better). This assumes that the PSIS-LOO-CV approximation is working well.

Specifically, the PWM identity used here is

\[\operatorname{CRPS}_{\text{loo}}(F, y) = E_{\text{loo}}\left[|X - y|\right] + E_{\text{loo}}[X] - 2\cdot E_{\text{loo}} \left[X\,F_{\text{mid}}(X) \right],\]

where \(F_{\text{mid}}\) is the midpoint CDF estimator defined as \(F_{\text{mid}}(x_{(i)}) := F^-_{(i)} + w_{(i)}/2\). This midpoint formulation provides improved accuracy for weighted samples compared to the left-continuous CDF.

The PWM identity is described in [3], traditional CRPS and SCRPS are described in [1] and [2], and the PSIS-LOO-CV method is described in [4] and [5].

Parameters:
dataxarray.DataTree or InferenceData

Input data. It should contain the posterior_predictive, observed_data and log_likelihood groups.

var_namestr, optional

The name of the variable in the log_likelihood group to use. If None, the first variable in observed_data is used and assumed to match log_likelihood and posterior_predictive names.

kindstr, default “crps”

The kind of score to compute. Available options are:

  • ‘crps’: continuous ranked probability score. Default.

  • ‘scrps’: scale-invariant continuous ranked probability score.

pointwisebool, default False

If True, include per-observation score values in the return object.

round_toint or str, optional

If integer, number of decimal places to round the result. If string of the form ‘2g’ number of significant digits to round the result. Defaults to ‘2g’. Use None to return raw numbers.

log_weightsxarray.DataArray, optional

Pre-computed smoothed log weights from PSIS. Must be provided together with pareto_k. If not provided, PSIS will be computed internally.

pareto_kxarray.DataArray, optional

Pre-computed Pareto k-hat diagnostic values. Must be provided together with log_weights.

Returns:
collections.namedtuple

If pointwise is False (default), a namedtuple named CRPS or SCRPS with fields mean and se. If pointwise is True, the namedtuple also includes pointwise and pareto_k fields.

Notes

For a single observation with posterior-predictive draws \(x_1, \ldots, x_S\) and PSIS-LOO-CV weights \(w_i \propto \exp(\ell_i)\) normalized so that \(\sum_{i=1}^S w_i = 1\), define the PSIS-LOO-CV expectation as

\[E_{\text{loo}}[g(X)] := \sum_{i=1}^S w_i\, g(x_i).\]

For weighted samples, we use the midpoint CDF estimator rather than the left-continuous CDF. Given sorted values \(x_{(1)} \leq \cdots \leq x_{(S)}\) with corresponding weights \(w_{(i)}\), define the left-cumulative weight \(F^-_{(i)} = \sum_{j<i} w_{(j)}\) and the midpoint CDF as

\[F_{\text{mid}}(x_{(i)}) := F^-_{(i)} + \frac{w_{(i)}}{2}.\]

The first probability-weighted moment using the midpoint CDF is \(b_1 := \sum_{i=1}^S w_{(i)}\, x_{(i)}\, F_{\text{mid}}(x_{(i)})\). With this, the nonnegative CRPS under PSIS-LOO-CV is

\[\operatorname{CRPS}_{\text{loo}}(F, y) = E_{\text{loo}}\left[\,|X-y|\,\right] + E_{\text{loo}}[X] - 2\,b_1.\]

For the scale term for the SCRPS, we use the PSIS-LOO-CV weighted Gini mean difference given by \(\Delta_{\text{loo}} := E_{\text{loo}}\left[\,|X - X'|\,\right]\). This admits the PWM representation given by

\[\Delta_{\text{loo}} = 2\,E_{\text{loo}}\left[\,X\,\left(2F_{\text{loo}}(X') - 1\right)\,\right].\]

A finite-sample weighted order-statistic version of this is used in the function and is given by

\[\Delta_{\text{loo}} = 2 \sum_{i=1}^S w_{(i)}\, x_{(i)} \left\{\,2 F^-_{(i)} + w_{(i)} - 1\,\right\},\]

where \(x_{(i)}\) are the values sorted increasingly, \(w_{(i)}\) are the corresponding normalized weights, and \(F^-_{(i)} = \sum_{j<i} w_{(j)}\).

The locally scale-invariant score returned for kind="scrps" is

\[S_{\text{SCRPS}}(F, y) = -\frac{E_{\text{loo}}\left[\,|X-y|\,\right]}{\Delta_{\text{loo}}} - \frac{1}{2}\log \Delta_{\text{loo}}.\]

When PSIS weights are highly variable (large Pareto \(k\)), Monte-Carlo noise can increase. This function surfaces PSIS-LOO-CV diagnostics via pareto_k and warns when tail behavior suggests unreliability.

References

[1]

Bolin, D., & Wallin, J. (2023). Local scale invariance and robustness of proper scoring rules. Statistical Science, 38(1), 140–159. https://doi.org/10.1214/22-STS864 arXiv preprint https://arxiv.org/abs/1912.05642

[2]

Gneiting, T., & Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction, and Estimation. Journal of the American Statistical Association, 102(477), 359–378. https://doi.org/10.1198/016214506000001437

[3]

Taillardat M, Mestre O, Zamo M, Naveau P (2016). Calibrated ensemble forecasts using quantile regression forests and ensemble model output statistics. Mon Weather Rev 144(6):2375–2393. https://doi.org/10.1175/MWR-D-15-0260.1

[4]

Vehtari, A., Gelman, A., & Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5), 1413–1432. https://doi.org/10.1007/s11222-016-9696-4 arXiv preprint https://arxiv.org/abs/1507.04544

[5]

Vehtari, A., et al. (2024). Pareto Smoothed Importance Sampling. Journal of Machine Learning Research, 25(72). https://jmlr.org/papers/v25/19-556.html arXiv preprint https://arxiv.org/abs/1507.02646

Examples

Compute scores and return the mean and standard error:

In [1]: from arviz_stats import loo_score
   ...: from arviz_base import load_arviz_data
   ...: dt = load_arviz_data("centered_eight")
   ...: loo_score(dt, kind="crps")
   ...: 
Out[1]: CRPS(mean=-6.249941511055281, se=1.4913883904505707)

In [2]: loo_score(dt, kind="scrps")
Out[2]: SCRPS(mean=-2.2638824796689585, se=0.09528725284940921)
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