Probabilistic classification: Difference between revisions

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==Generative and conditional training==

Some models, such as [[logistic regression]], are conditionally trained: they optimize the conditional probability <math>\Pr(Y \vert X)</math> directly on a training set (see [[empirical risk minimization]]). Other than classifiers, such as [[naive Bayes]], are trained [[Generative model|generatively]]: at training time, the class-conditional distribution <math>\Pr(X \vert Y)</math> and the class [[Prior probability|prior]] <math>\Pr(Y)</math> are found, and the conditional distribution <math>\Pr (Y \vert X)</math> is derived using [[Bayes' theorem|Bayes' rule]].<ref name="bishop"/>{{rp|43}}

==Probability calibration==

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Not all classification models are naturally probabilistic, and some that are, notably naive Bayes classifiers, [[decision tree learning|decision trees]] and [[Boosting (machine learning)|boosting]] methods, produce distorted class probability distributions.<ref name="Niculescu">{{cite conference |last1=Niculescu-Mizil |first1=Alexandru |first2=Rich |last2=Caruana |title=Predicting good probabilities with supervised learning |conference=ICML |year=2005 |url=http://machinelearning.wustl.edu/mlpapers/paper_files/icml2005_Niculescu-MizilC05.pdf |doi=10.1145/1102351.1102430 |url-status=dead |archive-url=https://web.archive.org/web/20140311005243/http://machinelearning.wustl.edu/mlpapers/paper_files/icml2005_Niculescu-MizilC05.pdf |archive-date=2014-03-11 }}</ref> In the case of decision trees, where {{math|Pr(''y''{{!}}'''x''')}} is the proportion of training samples with label {{mvar|y}} in the leaf where {{math|'''x'''}} ends up, these distortions come about because learning algorithms such as [[C4.5]] or [[Predictive analytics#Classification and regression trees|CART]] explicitly aim to produce homogeneous leaves (giving probabilities close to zero or one, and thus high [[Bias of an estimator|bias]]) while using few samples to estimate the relevant proportion (high [[Bias–variance tradeoff|variance]]).<ref>{{cite conference |first1=Bianca |last1=Zadrozny |first2=Charles |last2=Elkan |title=Obtaining calibrated probability estimates from decision trees and naive Bayesian classifiers |url=http://cseweb.ucsd.edu/~elkan/calibrated.pdf |year=2001 |conference=ICML |pages=609–616}}</ref>

[[File:Calibration plot.png|thumb|An example calibration plot]] Calibration can be assessed using a '''calibration plot''' (also called a '''reliability diagram''').<ref name="Niculescu" /><ref>{{Cite web|url=https://jmetzen.github.io/2015-04-14/calibration.html|title=Probability calibration|website=jmetzen.github.io|access-date=2019-06-18}}</ref> A standardizationcalibration plot shows the proportion of items in each class for bands of predicted probability or score (such as a distorted chanceprobability distribution or the "signed distance to the hyperplane" in a support vector machine). Deviations from the identity function indicate a poorly-calibrated classifier for which the predicted probabilities or scores can not be used as probabilities. In this case one can use a method to turn these scores into properly [[Calibration (statistics)|calibrated]] class membership probabilities.

For the [[binary classification|binary]] case, a common approach is to apply [[Platt scaling]], which learns a [[logistic regression]] model on the scores.<ref name="platt99">{{cite journal |last=Platt |first=John |author-link=John Platt (computer scientist) |title=Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods |journal=Advances in Large Margin Classifiers |volume=10 |issue=3 |year=1999 |pages=61–74 |url=https://www.researchgate.net/publication/2594015}}</ref>

An alternative method using [[isotonic regression]]<ref>{{Cite book | last1 = Zadrozny | first1 = Bianca| last2 = Elkan | first2 = Charles| doi = 10.1145/775047.775151 | chapter = Transforming classifier scores into accurate multiclass probability estimates | chapter-url = httphttps://www.cs.cornell.edu/courses/cs678/2007sp/ZadroznyElkan.pdf| title = Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '02 | pages = 694–699| year = 2002 | isbn = 978-1-58113-567-1| id = [[CiteSeerX]]: {{URL|1=citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.7457|2=10.1.1.13.7457}}| citeseerx = 10.1.1.164.8140| s2cid = 3349576}}</ref> is generally superior to Platt's method when sufficient training data is available.<ref name="Niculescu"/>

In the [[multiclass classification|multiclass]] case, one can use a reduction to binary tasks, followed by univariate calibration with an algorithm as described above and further application of the pairwise coupling algorithm by Hastie and Tibshirani.<ref>{{Cite journal | last1 = Hastie | first1 = Trevor| last2 = Tibshirani | first2 = Robert| doi = 10.1214/aos/1028144844 | title = Classification by pairwise coupling | journal = [[The Annals of Statistics]] | volume = 26 | issue = 2 | pages = 451–471| year = 1998 | zbl = 0932.62071| id = [[CiteSeerX]]: {{URL|1=citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.6032|2=10.1.1.46.6032}}| citeseerx = 10.1.1.309.4720 }}</ref>