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<li><a class="reference internal" href="#">1.16. Probability calibration</a><ul>
<li><a class="reference internal" href="#calibration-curves">1.16.1. Calibration curves</a></li>
<li><a class="reference internal" href="#calibrating-a-classifier">1.16.2. Calibrating a classifier</a></li>
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<section id="probability-calibration">
<span id="calibration"></span><h1><span class="section-number">1.16. </span>Probability calibration<a class="headerlink" href="#probability-calibration" title="Link to this heading">¶</a></h1>
<p>When performing classification you often want not only to predict the class
label, but also obtain a probability of the respective label. This probability
gives you some kind of confidence on the prediction. Some models can give you
poor estimates of the class probabilities and some even do not support
probability prediction (e.g., some instances of
<a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a>).
The calibration module allows you to better calibrate
the probabilities of a given model, or to add support for probability
prediction.</p>
<p>Well calibrated classifiers are probabilistic classifiers for which the output
of the <a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> method can be directly interpreted as a confidence
level.
For instance, a well calibrated (binary) classifier should classify the samples such
that among the samples to which it gave a <a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> value close to, say,
0.8, approximately 80% actually belong to the positive class.</p>
<p>Before we show how to re-calibrate a classifier, we first need a way to detect how
good a classifier is calibrated.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Strictly proper scoring rules for probabilistic predictions like
<a class="reference internal" href="generated/sklearn.metrics.brier_score_loss.html#sklearn.metrics.brier_score_loss" title="sklearn.metrics.brier_score_loss"><code class="xref py py-func docutils literal notranslate"><span class="pre">sklearn.metrics.brier_score_loss</span></code></a> and
<a class="reference internal" href="generated/sklearn.metrics.log_loss.html#sklearn.metrics.log_loss" title="sklearn.metrics.log_loss"><code class="xref py py-func docutils literal notranslate"><span class="pre">sklearn.metrics.log_loss</span></code></a> assess calibration (reliability) and
discriminative power (resolution) of a model, as well as the randomness of the data
(uncertainty) at the same time. This follows from the well-known Brier score
decomposition of Murphy <a class="footnote-reference brackets" href="#id12" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>. As it is not clear which term dominates, the score is
of limited use for assessing calibration alone (unless one computes each term of
the decomposition). A lower Brier loss, for instance, does not necessarily
mean a better calibrated model, it could also mean a worse calibrated model with much
more discriminatory power, e.g. using many more features.</p>
</div>
<section id="calibration-curves">
<span id="calibration-curve"></span><h2><span class="section-number">1.16.1. </span>Calibration curves<a class="headerlink" href="#calibration-curves" title="Link to this heading">¶</a></h2>
<p>Calibration curves, also referred to as <em>reliability diagrams</em> (Wilks 1995 <a class="footnote-reference brackets" href="#id13" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a>),
compare how well the probabilistic predictions of a binary classifier are calibrated.
It plots the frequency of the positive label (to be more precise, an estimation of the
<em>conditional event probability</em> <span class="math notranslate nohighlight">\(P(Y=1|\text{predict_proba})\)</span>) on the y-axis
against the predicted probability <a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> of a model on the x-axis.
The tricky part is to get values for the y-axis.
In scikit-learn, this is accomplished by binning the predictions such that the x-axis
represents the average predicted probability in each bin.
The y-axis is then the <em>fraction of positives</em> given the predictions of that bin, i.e.
the proportion of samples whose class is the positive class (in each bin).</p>
<p>The top calibration curve plot is created with
<a class="reference internal" href="generated/sklearn.calibration.CalibrationDisplay.html#sklearn.calibration.CalibrationDisplay.from_estimator" title="sklearn.calibration.CalibrationDisplay.from_estimator"><code class="xref py py-func docutils literal notranslate"><span class="pre">CalibrationDisplay.from_estimator</span></code></a>, which uses <a class="reference internal" href="generated/sklearn.calibration.calibration_curve.html#sklearn.calibration.calibration_curve" title="sklearn.calibration.calibration_curve"><code class="xref py py-func docutils literal notranslate"><span class="pre">calibration_curve</span></code></a> to
calculate the per bin average predicted probabilities and fraction of positives.
<a class="reference internal" href="generated/sklearn.calibration.CalibrationDisplay.html#sklearn.calibration.CalibrationDisplay.from_estimator" title="sklearn.calibration.CalibrationDisplay.from_estimator"><code class="xref py py-func docutils literal notranslate"><span class="pre">CalibrationDisplay.from_estimator</span></code></a>
takes as input a fitted classifier, which is used to calculate the predicted
probabilities. The classifier thus must have <a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> method. For
the few classifiers that do not have a <a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> method, it is
possible to use <a class="reference internal" href="generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV" title="sklearn.calibration.CalibratedClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">CalibratedClassifierCV</span></code></a> to calibrate the classifier
outputs to probabilities.</p>
<p>The bottom histogram gives some insight into the behavior of each classifier
by showing the number of samples in each predicted probability bin.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/calibration/plot_compare_calibration.html"><img alt="../_images/sphx_glr_plot_compare_calibration_001.png" src="../_images/sphx_glr_plot_compare_calibration_001.png" /></a>
</figure>
<p><a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LogisticRegression</span></code></a> returns well calibrated predictions by default as it has a
canonical link function for its loss, i.e. the logit-link for the <a class="reference internal" href="model_evaluation.html#log-loss"><span class="std std-ref">Log loss</span></a>.
This leads to the so-called <strong>balance property</strong>, see <a class="footnote-reference brackets" href="#id19" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>8<span class="fn-bracket">]</span></a> and
<a class="reference internal" href="linear_model.html#logistic-regression"><span class="std std-ref">Logistic regression</span></a>.
In contrast to that, the other shown models return biased probabilities; with
different biases per model.</p>
<p><a class="reference internal" href="generated/sklearn.naive_bayes.GaussianNB.html#sklearn.naive_bayes.GaussianNB" title="sklearn.naive_bayes.GaussianNB"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianNB</span></code></a> (Naive Bayes) tends to push probabilities to 0 or 1 (note the counts
in the histograms). This is mainly because it makes the assumption that
features are conditionally independent given the class, which is not the
case in this dataset which contains 2 redundant features.</p>
<p><a class="reference internal" href="generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier" title="sklearn.ensemble.RandomForestClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RandomForestClassifier</span></code></a> shows the opposite behavior: the histograms
show peaks at probabilities approximately 0.2 and 0.9, while probabilities
close to 0 or 1 are very rare. An explanation for this is given by
Niculescu-Mizil and Caruana <a class="footnote-reference brackets" href="#id14" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a>: “Methods such as bagging and random
forests that average predictions from a base set of models can have
difficulty making predictions near 0 and 1 because variance in the
underlying base models will bias predictions that should be near zero or one
away from these values. Because predictions are restricted to the interval
[0,1], errors caused by variance tend to be one-sided near zero and one. For
example, if a model should predict p = 0 for a case, the only way bagging
can achieve this is if all bagged trees predict zero. If we add noise to the
trees that bagging is averaging over, this noise will cause some trees to
predict values larger than 0 for this case, thus moving the average
prediction of the bagged ensemble away from 0. We observe this effect most
strongly with random forests because the base-level trees trained with
random forests have relatively high variance due to feature subsetting.” As
a result, the calibration curve shows a characteristic sigmoid shape, indicating that
the classifier could trust its “intuition” more and return probabilities closer
to 0 or 1 typically.</p>
<p><a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> (SVC) shows an even more sigmoid curve than the random forest, which
is typical for maximum-margin methods (compare Niculescu-Mizil and Caruana <a class="footnote-reference brackets" href="#id14" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a>), which
focus on difficult to classify samples that are close to the decision boundary (the
support vectors).</p>
</section>
<section id="calibrating-a-classifier">
<h2><span class="section-number">1.16.2. </span>Calibrating a classifier<a class="headerlink" href="#calibrating-a-classifier" title="Link to this heading">¶</a></h2>
<p>Calibrating a classifier consists of fitting a regressor (called a
<em>calibrator</em>) that maps the output of the classifier (as given by
<a class="reference internal" href="../glossary.html#term-decision_function"><span class="xref std std-term">decision_function</span></a> or <a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a>) to a calibrated probability
in [0, 1]. Denoting the output of the classifier for a given sample by <span class="math notranslate nohighlight">\(f_i\)</span>,
the calibrator tries to predict the conditional event probability
<span class="math notranslate nohighlight">\(P(y_i = 1 | f_i)\)</span>.</p>
<p>Ideally, the calibrator is fit on a dataset independent of the training data used to
fit the classifier in the first place.
This is because performance of the classifier on its training data would be
better than for novel data. Using the classifier output of training data
to fit the calibrator would thus result in a biased calibrator that maps to
probabilities closer to 0 and 1 than it should.</p>
</section>
<section id="usage">
<h2><span class="section-number">1.16.3. </span>Usage<a class="headerlink" href="#usage" title="Link to this heading">¶</a></h2>
<p>The <a class="reference internal" href="generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV" title="sklearn.calibration.CalibratedClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">CalibratedClassifierCV</span></code></a> class is used to calibrate a classifier.</p>
<p><a class="reference internal" href="generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV" title="sklearn.calibration.CalibratedClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">CalibratedClassifierCV</span></code></a> uses a cross-validation approach to ensure
unbiased data is always used to fit the calibrator. The data is split into k
<code class="docutils literal notranslate"><span class="pre">(train_set,</span> <span class="pre">test_set)</span></code> couples (as determined by <code class="docutils literal notranslate"><span class="pre">cv</span></code>). When <code class="docutils literal notranslate"><span class="pre">ensemble=True</span></code>
(default), the following procedure is repeated independently for each
cross-validation split: a clone of <code class="docutils literal notranslate"><span class="pre">base_estimator</span></code> is first trained on the
train subset. Then its predictions on the test subset are used to fit a
calibrator (either a sigmoid or isotonic regressor). This results in an
ensemble of k <code class="docutils literal notranslate"><span class="pre">(classifier,</span> <span class="pre">calibrator)</span></code> couples where each calibrator maps
the output of its corresponding classifier into [0, 1]. Each couple is exposed
in the <code class="docutils literal notranslate"><span class="pre">calibrated_classifiers_</span></code> attribute, where each entry is a calibrated
classifier with a <a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> method that outputs calibrated
probabilities. The output of <a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> for the main
<a class="reference internal" href="generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV" title="sklearn.calibration.CalibratedClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">CalibratedClassifierCV</span></code></a> instance corresponds to the average of the
predicted probabilities of the <code class="docutils literal notranslate"><span class="pre">k</span></code> estimators in the <code class="docutils literal notranslate"><span class="pre">calibrated_classifiers_</span></code>
list. The output of <a class="reference internal" href="../glossary.html#term-predict"><span class="xref std std-term">predict</span></a> is the class that has the highest
probability.</p>
<p>When <code class="docutils literal notranslate"><span class="pre">ensemble=False</span></code>, cross-validation is used to obtain ‘unbiased’
predictions for all the data, via
<a class="reference internal" href="generated/sklearn.model_selection.cross_val_predict.html#sklearn.model_selection.cross_val_predict" title="sklearn.model_selection.cross_val_predict"><code class="xref py py-func docutils literal notranslate"><span class="pre">cross_val_predict</span></code></a>.
These unbiased predictions are then used to train the calibrator. The attribute
<code class="docutils literal notranslate"><span class="pre">calibrated_classifiers_</span></code> consists of only one <code class="docutils literal notranslate"><span class="pre">(classifier,</span> <span class="pre">calibrator)</span></code>
couple where the classifier is the <code class="docutils literal notranslate"><span class="pre">base_estimator</span></code> trained on all the data.
In this case the output of <a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> for
<a class="reference internal" href="generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV" title="sklearn.calibration.CalibratedClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">CalibratedClassifierCV</span></code></a> is the predicted probabilities obtained
from the single <code class="docutils literal notranslate"><span class="pre">(classifier,</span> <span class="pre">calibrator)</span></code> couple.</p>
<p>The main advantage of <code class="docutils literal notranslate"><span class="pre">ensemble=True</span></code> is to benefit from the traditional
ensembling effect (similar to <a class="reference internal" href="ensemble.html#bagging"><span class="std std-ref">Bagging meta-estimator</span></a>). The resulting ensemble should
both be well calibrated and slightly more accurate than with <code class="docutils literal notranslate"><span class="pre">ensemble=False</span></code>.
The main advantage of using <code class="docutils literal notranslate"><span class="pre">ensemble=False</span></code> is computational: it reduces the
overall fit time by training only a single base classifier and calibrator
pair, decreases the final model size and increases prediction speed.</p>
<p>Alternatively an already fitted classifier can be calibrated by setting
<code class="docutils literal notranslate"><span class="pre">cv="prefit"</span></code>. In this case, the data is not split and all of it is used to
fit the regressor. It is up to the user to
make sure that the data used for fitting the classifier is disjoint from the
data used for fitting the regressor.</p>
<p><a class="reference internal" href="generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV" title="sklearn.calibration.CalibratedClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">CalibratedClassifierCV</span></code></a> supports the use of two regression techniques
for calibration via the <code class="docutils literal notranslate"><span class="pre">method</span></code> parameter: <code class="docutils literal notranslate"><span class="pre">"sigmoid"</span></code> and <code class="docutils literal notranslate"><span class="pre">"isotonic"</span></code>.</p>
<section id="sigmoid">
<span id="sigmoid-regressor"></span><h3><span class="section-number">1.16.3.1. </span>Sigmoid<a class="headerlink" href="#sigmoid" title="Link to this heading">¶</a></h3>
<p>The sigmoid regressor, <code class="docutils literal notranslate"><span class="pre">method="sigmoid"</span></code> is based on Platt’s logistic model <a class="footnote-reference brackets" href="#id15" id="id6" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a>:</p>
<div class="math notranslate nohighlight">
\[p(y_i = 1 | f_i) = \frac{1}{1 + \exp(A f_i + B)} \,,\]</div>
<p>where <span class="math notranslate nohighlight">\(y_i\)</span> is the true label of sample <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(f_i\)</span>
is the output of the un-calibrated classifier for sample <span class="math notranslate nohighlight">\(i\)</span>. <span class="math notranslate nohighlight">\(A\)</span>
and <span class="math notranslate nohighlight">\(B\)</span> are real numbers to be determined when fitting the regressor via
maximum likelihood.</p>
<p>The sigmoid method assumes the <a class="reference internal" href="#calibration-curve"><span class="std std-ref">calibration curve</span></a>
can be corrected by applying a sigmoid function to the raw predictions. This
assumption has been empirically justified in the case of <a class="reference internal" href="svm.html#svm"><span class="std std-ref">Support Vector Machines</span></a> with
common kernel functions on various benchmark datasets in section 2.1 of Platt
1999 <a class="footnote-reference brackets" href="#id15" id="id7" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a> but does not necessarily hold in general. Additionally, the
logistic model works best if the calibration error is symmetrical, meaning
the classifier output for each binary class is normally distributed with
the same variance <a class="footnote-reference brackets" href="#id18" id="id8" role="doc-noteref"><span class="fn-bracket">[</span>7<span class="fn-bracket">]</span></a>. This can be a problem for highly imbalanced
classification problems, where outputs do not have equal variance.</p>
<p>In general this method is most effective for small sample sizes or when the
un-calibrated model is under-confident and has similar calibration errors for both
high and low outputs.</p>
</section>
<section id="isotonic">
<h3><span class="section-number">1.16.3.2. </span>Isotonic<a class="headerlink" href="#isotonic" title="Link to this heading">¶</a></h3>
<p>The <code class="docutils literal notranslate"><span class="pre">method="isotonic"</span></code> fits a non-parametric isotonic regressor, which outputs
a step-wise non-decreasing function, see <a class="reference internal" href="classes.html#module-sklearn.isotonic" title="sklearn.isotonic"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.isotonic</span></code></a>. It minimizes:</p>
<div class="math notranslate nohighlight">
\[\sum_{i=1}^{n} (y_i - \hat{f}_i)^2\]</div>
<p>subject to <span class="math notranslate nohighlight">\(\hat{f}_i \geq \hat{f}_j\)</span> whenever
<span class="math notranslate nohighlight">\(f_i \geq f_j\)</span>. <span class="math notranslate nohighlight">\(y_i\)</span> is the true
label of sample <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(\hat{f}_i\)</span> is the output of the
calibrated classifier for sample <span class="math notranslate nohighlight">\(i\)</span> (i.e., the calibrated probability).
This method is more general when compared to ‘sigmoid’ as the only restriction
is that the mapping function is monotonically increasing. It is thus more
powerful as it can correct any monotonic distortion of the un-calibrated model.
However, it is more prone to overfitting, especially on small datasets <a class="footnote-reference brackets" href="#id17" id="id9" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>.</p>
<p>Overall, ‘isotonic’ will perform as well as or better than ‘sigmoid’ when
there is enough data (greater than ~ 1000 samples) to avoid overfitting <a class="footnote-reference brackets" href="#id14" id="id10" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Impact on ranking metrics like AUC</p>
<p>It is generally expected that calibration does not affect ranking metrics such as
ROC-AUC. However, these metrics might differ after calibration when using
<code class="docutils literal notranslate"><span class="pre">method="isotonic"</span></code> since isotonic regression introduces ties in the predicted
probabilities. This can be seen as within the uncertainty of the model predictions.
In case, you strictly want to keep the ranking and thus AUC scores, use
<code class="docutils literal notranslate"><span class="pre">method="sigmoid"</span></code> which is a strictly monotonic transformation and thus keeps
the ranking.</p>
</div>
</section>
<section id="multiclass-support">
<h3><span class="section-number">1.16.3.3. </span>Multiclass support<a class="headerlink" href="#multiclass-support" title="Link to this heading">¶</a></h3>
<p>Both isotonic and sigmoid regressors only
support 1-dimensional data (e.g., binary classification output) but are
extended for multiclass classification if the <code class="docutils literal notranslate"><span class="pre">base_estimator</span></code> supports
multiclass predictions. For multiclass predictions,
<a class="reference internal" href="generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV" title="sklearn.calibration.CalibratedClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">CalibratedClassifierCV</span></code></a> calibrates for
each class separately in a <a class="reference internal" href="multiclass.html#ovr-classification"><span class="std std-ref">OneVsRestClassifier</span></a> fashion <a class="footnote-reference brackets" href="#id16" id="id11" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a>. When
predicting
probabilities, the calibrated probabilities for each class
are predicted separately. As those probabilities do not necessarily sum to
one, a postprocessing is performed to normalize them.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/calibration/plot_calibration_curve.html#sphx-glr-auto-examples-calibration-plot-calibration-curve-py"><span class="std std-ref">Probability Calibration curves</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/calibration/plot_calibration_multiclass.html#sphx-glr-auto-examples-calibration-plot-calibration-multiclass-py"><span class="std std-ref">Probability Calibration for 3-class classification</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/calibration/plot_calibration.html#sphx-glr-auto-examples-calibration-plot-calibration-py"><span class="std std-ref">Probability calibration of classifiers</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/calibration/plot_compare_calibration.html#sphx-glr-auto-examples-calibration-plot-compare-calibration-py"><span class="std std-ref">Comparison of Calibration of Classifiers</span></a></p></li>
</ul>
</aside>
<aside class="topic">
<p class="topic-title">References:</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id12" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">1</a><span class="fn-bracket">]</span></span>
<p>Allan H. Murphy (1973).
<a class="reference external" href="https://doi.org/10.1175/1520-0450(1973)012%3C0595:ANVPOT%3E2.0.CO;2">“A New Vector Partition of the Probability Score”</a>
Journal of Applied Meteorology and Climatology</p>
</aside>
<aside class="footnote brackets" id="id13" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">2</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://journals.ametsoc.org/waf/article/5/4/640/40179">On the combination of forecast probabilities for
consecutive precipitation periods.</a>
Wea. Forecasting, 5, 640–650., Wilks, D. S., 1990a</p>
</aside>
<aside class="footnote brackets" id="id14" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id4">1</a>,<a role="doc-backlink" href="#id5">2</a>,<a role="doc-backlink" href="#id10">3</a>)</span>
<p><a class="reference external" href="https://www.cs.cornell.edu/~alexn/papers/calibration.icml05.crc.rev3.pdf">Predicting Good Probabilities with Supervised Learning</a>,
A. Niculescu-Mizil & R. Caruana, ICML 2005</p>
</aside>
<aside class="footnote brackets" id="id15" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id6">1</a>,<a role="doc-backlink" href="#id7">2</a>)</span>
<p><a class="reference external" href="https://www.cs.colorado.edu/~mozer/Teaching/syllabi/6622/papers/Platt1999.pdf">Probabilistic Outputs for Support Vector Machines and Comparisons
to Regularized Likelihood Methods.</a>
J. Platt, (1999)</p>
</aside>
<aside class="footnote brackets" id="id16" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id11">5</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://dl.acm.org/doi/pdf/10.1145/775047.775151">Transforming Classifier Scores into Accurate Multiclass
Probability Estimates.</a>
B. Zadrozny & C. Elkan, (KDD 2002)</p>
</aside>
<aside class="footnote brackets" id="id17" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id9">6</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4180410/">Predicting accurate probabilities with a ranking loss.</a>
Menon AK, Jiang XJ, Vembu S, Elkan C, Ohno-Machado L.
Proc Int Conf Mach Learn. 2012;2012:703-710</p>
</aside>
<aside class="footnote brackets" id="id18" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">7</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://projecteuclid.org/euclid.ejs/1513306867">Beyond sigmoids: How to obtain well-calibrated probabilities from
binary classifiers with beta calibration</a>
Kull, M., Silva Filho, T. M., & Flach, P. (2017).</p>
</aside>
<aside class="footnote brackets" id="id19" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">8</a><span class="fn-bracket">]</span></span>
<p>Mario V. Wüthrich, Michael Merz (2023).
<a class="reference external" href="https://doi.org/10.1007/978-3-031-12409-9">“Statistical Foundations of Actuarial Learning and its Applications”</a>
Springer Actuarial</p>
</aside>
</aside>
</aside>
</section>
</section>
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