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# Uncertainty Metrics

The goal of this library is to provide an easy-to-use interface for measuring uncertainty across Google and the open-source community.

Machine learning models often produce incorrect (over or under confident) probabilities. In real-world decision making systems, classification models must not only be accurate, but also should indicate when they are likely to be incorrect. For example, one important property is calibration: the idea that a model's predicted probabilities of outcomes reflect true probabilities of those outcomes. Intuitively, for class predictions, calibration means that if a model assigns a class with 90% probability, that class should appear 90% of the time.

## Installation

```pip install uncertainty_metrics
```

To install the latest development version, run

```pip install "git+https://github.com/google/uncertainty_metrics.git#egg=uncertainty_metrics"
```

There is not yet a stable version (nor an official release of this library). All APIs are subject to change.

## Getting Started

Here are some examples to get you started.

Expected Calibration Error.

```import uncertainty_metrics.numpy as um

probabilities = ...
labels = ...
ece = um.ece(labels, probabilities, num_bins=30)
```

Reliability Diagram.

```import uncertainty_metrics.numpy as um

probabilities = ...
labels = ...
diagram = um.reliability_diagram(labels, probabilities)
```

Brier Score.

```import uncertainty_metrics as um

tf_probabilities = ...
labels = ...
bs = um.brier_score(labels=labels, probabilities=tf_probabilities)
```

How to diagnose miscalibration. Calibration is one of the most important properties of a trained model beyond accuracy. We demonsrate how to calculate calibration measure and diagnose miscalibration with the help of this library. One typical measure of calibration is Expected Calibration Error (ECE) (Guo et al., 2017). To calculate ECE, we group predictions into M bins (M=15 in our example) according to their confidence, which in ECE is the value of the max softmax output, and compute the accuracy in each bin. Let B_m be the set of examples whose predicted confidence falls into the m th interval. The Acc and the Conf of bin B_m is

ECE is defined to be the sum of the absolute value of the difference of Acc and Conf in each bin. Thus, we can see that ECE is designed to measure the alignment between accuracy and confidence. This provides a quantitative way to measure calibration. The better calibration leads to lower ECE.

In this example, we also need to introduce mixup (Zhang et al., 2017). It is a data-augmentation technique in image classification, which improves both accuracy and calibration in single model. Mixup applies the following only in the training,

We focus on the calibration (measured by ECE) of Mixup + BatchEnsemble (Wen et al., 2020). We first calculate the ECE of some fully trained models using this library.

```import tensorflow as tf
import uncertainty_metrics.numpy as um

test_images, test_labels = ...
model = ...

# Obtain predictive probabilities.
probs = model(test_images, training=False) # probs is of shape [4, testset_size, num_classes] if the model is an ensemble of 4 individual models.
ensemble_probs = tf.reduce_mean(model, axis=0)

# Calculate individual calibration error.
individual_eces = []
for i in range(ensemble_size):
individual_eces.append(um.ece(labels, probs[i], num_bins=15))

ensemble_ece = um.ece(labels, ensemble_probs, num_bins=15)
```

We collect the ECE in the following table.

Method/Metric CIFAR-10 CIFAR-100
Acc ECE Acc ECE
BatchEnsemble In 95.88 2.3% 80.64 8.7%
En 96.22 1.8% 81.85 2.8%
Mixup0.2 BE In 96.43 0.8% 81.44 1.5%
En 96.75 1.5% 82.79 3.9%
Mixup1 BE In 96.67 5.5% 81.32 6.6%
En 96.98 6.4% 83.12 9.7%

In the above table, In stands for individual model; En stands for ensemble models. Mixup0.2 stands for small mixup augmentation while mixup1 stands for strong mixup augmentation. Ensemble typically improves both accuracy and calibration, but this does not apply to mixup. Scalars obsure useful information, so we try to understand more insights by examining the per-bin result.

```ensemble_metric = um.GeneralCalibrationError(
num_bins=15,
binning_scheme='even',
class_conditional=False,
max_prob=True,
norm='l1')
ensemble_metric.update_state(labels, ensemble_probs)

individual_metric = um.GeneralCalibrationError(
num_bins=15,
binning_scheme='even',
class_conditional=False,
max_prob=True,
norm='l1')
for i in range(4)
individual_metric.update_state(labels, probs[i])

ensemble_reliability = ensemble_metric.accuracies - ensemble_metric.confidences
individual_reliability = (
individual_metric.accuracies - individual_metric.confidences)
```

Now we can plot the reliability diagram which demonstrates more details of calibration. The backbone model in the following figure is BatchEnsemble with ensemble size 4. The plot has 6 lines: we trained three independent BatchEnsemble models with large, small, and no Mixup; and for each model, we compute the calibration of both ensemble and individual predictions. The plot shows that only Mixup models have positive (Acc - Conf) values on the test set, which suggests that Mixup encourages underconfidence. Mixup ensemble's positive value is also greater than Mixup individual's. This suggests that Mixup ensembles compound in encouraging underconfidence, leading to worse calibration than when not ensembling. Therefore, we successfully find the reason why Mixup+Ensemble leads to worse calibration, by leveraging this library.

## Background & API

Uncertainty Metrics provides several types of measures of probabilistic error:

• Calibration error
• Proper scoring rules
• Information critera
• Diversity
• AUC/Rejection
• Visualization tools

We support the following calibration metrics:

• Expected Calibration Error [3]
• Root-Mean-Squared Calibration Error [14]
• Static Calibration Error [2]
• General Calibration Error (a space of calibration metrics) [2]
• Class-conditional / Class-conflationary versions of all of the above. [2]
• Bayesian Expected Calibration Error
• Semiparametric Calibration Error
```features, labels = ...  # get from minibatch
probs = model(features)
ece = um.ece(labels=labels, probs=probs, num_bins=10)
```

Example: Bayesian Expected Calibration Error. ECE is a scalar summary statistic of miscalibration evaluated on a finite sample of validation data. Resulting in a single scalar, the sampling variation due to the limited amount of validation data is hidden, and this can result in significant over- or under-estimation of the ECE as well as wrongly concluding significant differences in ECE between multiple models.

To address these issues, a Bayesian estimator of the ECE can be used. The resulting estimate is not a single scalar summary but instead a probability distribution over possible ECE values.

The Bayesian ECE can be used like the normal ECE, as in the following code:

```# labels_true is a tf.int32 Tensor
logits = model(validation_data)
ece_samples = um.bayesian_expected_calibration_error(
10, logits=logits, labels_true=labels_true)

ece_quantiles = tfp.stats.percentile(ece_samples, [10,50,90])
```

The above code also includes an example of using the samples to infer 10%/50%/90% quantiles of the distribution of possible ECE values.

### Proper Scoring Rules

Here is an example of how to use the Brier score as loss function for a classifier. Suppose you have a classifier implemented in Tensorflow and your current training code looks like

```per_example_loss = tf.nn.sparse_softmax_cross_entropy_with_logits(
labels=target_labels, logits=logits)
loss = tf.reduce_mean(per_example_loss)
```

Then you can alternatively use the API-compatible Brier loss as follows:

```per_example_loss = um.brier_score(labels=target_labels, logits=logits)
loss = tf.reduce_mean(per_example_loss)
```

The Brier score penalizes low-probability predictions which do occur less than the cross-entropy loss.

Example: Brier score's decomposition. Here is an example of how to compute calibration metrics for a classifier. Suppose you evaluate the accuracy of your classifier on the validation set,

```logits = model(validation_data)
class_prediction = tf.argmax(logits, 1)
accuracy = tf.metrics.accuracy(validation_labels, class_prediction)
```

You can compute additional metrics using the so called Brier decomposition that quantify prediction uncertainty, resolution, and reliability by appending the following code,

```uncert, resol, reliab = um.brier_decomposition(labels=labels, logits=logits)
```

In particular, the reliability (`reliab` in the above line) is a measure of calibration error, with a value between 0 and 2, where zero means perfect calibration.

Example: Continuous Ranked Probability Score (CRPS). The continuous ranked probability score (CRPS) has several equivalent definitions.

CRPS has two desirable properties:

1. It generalizes the absolute error loss and recovers the absolute error if a predicted distribution F is deterministic.
2. It is reported in the same units as the predicted quantity.

To compute CRPS we either need to make an assumption regarding the form of F or need to approximate the expectations over F using samples from the predictive model. In the current code we implement one analytic solution to CRPS for predictive models with univariate Normal predictive distributions, and one generic form for univariate predictive regression models that uses a sample approximation.

For a regression model which predicts Normal distributions with mean `means` and standard deviation `stddevs`, we can compute CRPS as follows:

```squared_errors = tf.square(target_labels - pred_means)
per_example_crps = um.crps_normal_score(
labels=target_labels,
means=pred_mean,
stddevs=pred_stddevs)
```

For non-Normal models, as long as we can sample predictions, we can construct a Tensor `predictive_samples` of size `(ninstances, npredictive_samples)` and evaluate the Monte Carlo CRPS against the true targets `target_labels` using the following code,

```per_example_crps = um.crps_score(
labels=target_labels,
predictive_samples=predictive_samples)
```

### Information Criteria

Information criteria are used after or during model training to estimate the predictive performance on future holdout data. They can be useful for selecting among multiple possible models or to perform hyperparameter optimization. There are also strong connections between cross validation estimates and some information criteria.

We estimate information criteria using log-likelihood values on training samples. In particular, for both the WAIC and the ISCV criteria we assume that we have an ensemble of models with equal weights, such that the average predictive distribution over the ensemble is a good approximation to the true Bayesian posterior predictive distribution.

Both nWAIC criteria have comparable properties. To estimate the negative WAIC, we use the following code.

```# logp has shape (n,m), n instances, m ensemble members
neg_waic, neg_waic_sem = um.negative_waic(logp, waic_type="waic1")
```

You can select the type of nWAIC to estimate using `waic_type="waic1"` or `waic_type="waic2"`. The method returns the scalar estimate as well as the standard error of the mean of the estimate.

Example: Importance Sampling Cross Validation Criterion (ISCV).

We can estimate the ISCV using the following code:

```# logp has shape (n,m), n instances, m ensemble members
iscv, iscv_sem = um.importance_sampling_cross_validation(logp)
```

## To add a new metric

1. Add the paper reference to the `References` section below.
2. Add the metric definition to the numpy/ dir for a numpy based metric or to the tensorflow/ dir for a tensorflow based metric.s
3. Add the metric class or function to the corresponding init.py file.
4. Add a test that at a minimum implements the metric using 'import uncertainty_metrics as um' and um.your metric and checks that the value is in the appropriate range.

## References

[1] Guo, C., Pleiss, G., Sun, Y., & Weinberger, K. Q. (2017, August). On calibration of modern neural networks. In Proceedings of the 34th International Conference on Machine Learning. Paper Link.

[2] Nixon, J., Dusenberry, M. W., Zhang, L., Jerfel, G., & Tran, D. (2019). Measuring Calibration in Deep Learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops (pp. 38-41). Paper Link.

[3] Naeini, Mahdi Pakdaman, Gregory Cooper, and Milos Hauskrecht. "Obtaining well calibrated probabilities using bayesian binning." Twenty-Ninth AAAI Conference on Artificial Intelligence. 2015. Paper Link.

[4] Naeini, Mahdi Pakdaman, Gregory F. Cooper, and Milos Hauskrecht. "Binary classifier calibration using a Bayesian non-parametric approach." Proceedings of the 2015 SIAM International Conference on Data Mining. Society for Industrial and Applied Mathematics, 2015. Paper Link.

[5] J. Platt. Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods. Advances in Large Margin Classifiers, 10(3):61â€“74, 1999. Paper Link.

[6] Kumar, A., Liang, P. S., & Ma, T. (2019). Verified uncertainty calibration. In Advances in Neural Information Processing Systems (pp. 3787-3798). Paper Link.

[7] Kumar, A., Sarawagi, S., & Jain, U. (2018, July). Trainable calibration measures for neural networks from kernel mean embeddings. In International Conference on Machine Learning (pp. 2805-2814). Paper Link. [8] Calibrating Neural Networks Documentation. Link.

[9] Zadrozny, Bianca, and Charles Elkan. "Transforming classifier scores into accurate multiclass probability estimates." Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining. 2002. Paper Link.

[10] MÃ¼ller, Rafael, Simon Kornblith, and Geoffrey E. Hinton. "When does label smoothing help?." Advances in Neural Information Processing Systems. 2019. Paper Link.

[11] Pereyra, Gabriel, et al. "Regularizing neural networks by penalizing confident output distributions." arXiv preprint arXiv:1701.06548 (2017). Paper Link.

[12] Lakshminarayanan, B., Pritzel, A., and Blundell, C. Simple and scalable predictive uncertainty estimation using deep ensembles. In NIPS, pp. 6405â€“6416. 2017. Paper Link.

[13] Louizos, C. and Welling, M. Multiplicative normalizing flows for variational Bayesian neural networks. In ICML, volume 70, pp. 2218â€“2227, 2017. Paper Link.

[14] Nguyen, K. & Oâ€™Connor, B. (2015) Posterior calibration and exploratory analysis for natural language processing models. Empirical Methods in Natural Language Processing. (PDF)

[15] Jochen Brocker, "Reliability, sufficiency, and the decomposition of proper scores", Quarterly Journal of the Royal Meteorological Society, 2009. (PDF)

[16] Stefan Depeweg, JosÃ© Miguel HernÃ¡ndez-Lobato, Finale Doshi-Velez, and Steffen Udluft, "Decomposition of uncertainty for active learning and reliable reinforcement learning in stochastic systems", stat 1050, 2017. (PDF)

[17] Alan E. Gelfand, Dipak K. Dey, and Hong Chang. "Model determination using predictive distributions with implementation via sampling-based methods", Technical report No. 462, Department of Statistics, Stanford university, 1992. [(PDF)](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.860.3702&rep=re p1&type=pdf)

[18] Tilmann Gneiting and Adrian E. Raftery, "Strictly Proper Scoring Rules, Prediction, and Estimation", Journal of the American Statistical Association (JASA), 2007. [(PDF)](https://www.stat.washington.edu/raftery/Research/PDF/Gneiting2007jasa.pd f)

[19] Andrey Malinin, Bruno Mlodozeniec and Mark Gales, "Ensemble Distribution Distillation.", arXiv:1905.00076, 2019. (PDF)

[20] Aki Vehtari, Andrew Gelman, and Jonah Gabry. "Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC", arXiv:1507.04544, (PDF)

[21] Sumio Watanabe, "Mathematical Theory of Bayesian Statistics", CRC Press, 2018.

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