Probabilistic classification

Machine learning and data mining

Problems Classification Clustering Regression Anomaly detection Association rules Reinforcement learning Structured prediction Feature engineering Feature learning Online learning Semi-supervised learning Unsupervised learning Learning to rank Grammar induction
Supervised learning (classification • regression) Decision trees Ensembles (Bagging, Boosting, Random forest) k-NN Linear regression Naive Bayes Neural networks Logistic regression Perceptron Relevance vector machine (RVM) Support vector machine (SVM)
Clustering BIRCH Hierarchical k-means Expectation-maximization (EM) DBSCAN OPTICS Mean-shift
Dimensionality reduction Factor analysis CCA ICA LDA NMF PCA t-SNE
Structured prediction Graphical models (Bayes net, CRF, HMM)
Anomaly detection k-NN Local outlier factor
Neural nets Autoencoder Deep learning Multilayer perceptron RNN Restricted Boltzmann machine SOM Convolutional neural network
Reinforcement Learning Q-Learning SARSA Temporal Difference (TD)
Theory Bias-variance dilemma Computational learning theory Empirical risk minimization Occam learning PAC learning Statistical learning VC theory
Machine learning venues NIPS ICML IJMLC ML JMLR ArXiv:cs.LG
Machine learning portal

In machine learning, a probabilistic classifier is a classifier that is able to predict, given a sample input, a probability distribution over a set of classes, rather than only outputting the most likely class that the sample should belong to. Probabilistic classifiers provide classification with a degree of certainty, which can be useful in its own right,^[1] or when combining classifiers into ensembles.

Types of classification

Formally, an "ordinary" classifier is some rule, or function, that assigns to a sample $x$ a class label $ŷ$ :

{\hat {y}}=f(x)

The samples come from some set $X$ (e.g., the set of all documents, or the set of all images), while the class labels form a finite set $Y$ defined prior to training.

Probabilistic classifiers generalize this notion of classifiers: instead of functions, they are conditional distributions $\Pr(Y\vert X)$ , meaning that for a given $x\in X$ , they assign probabilities to all $y\in Y$ (and these probabilities sum to one). "Hard" classification can then be done using the optimal decision rule^[2]^:39–40

{\hat {y}}=\operatorname {\arg \max }_{{y}}\Pr(Y=y\vert X)

or, in English, the predicted class is that which has the highest probability.

Binary probabilistic classifiers are also called binomial regression models in statistics. In econometrics, probabilistic classification in general is called discrete choice.

Some classification models, such as naive Bayes, logistic regression and multilayer perceptrons (when trained under an appropriate loss function) are naturally probabilistic. Other models such as support vector machines are not, but methods exist to turn them into probabilistic classifiers.

Generative and conditional training

Some models, such as logistic regression, are conditionally trained: they optimize the conditional probability $\Pr(Y\vert X)$ directly on a training set (see empirical risk minimization). Other classifiers, such as naive Bayes, are trained generatively: at training time, the class-conditional distribution $\Pr(X\vert Y)$ and the class prior $\Pr(Y)$ are found, and the conditional distribution $\Pr(Y\vert X)$ is derived using Bayes' rule.^[2]^:43

Probability calibration

Not all classification models are naturally probabilistic, and some that are, notably naive Bayes classifiers, decision trees and boosting methods, produce distorted class probability distributions.^[3] In the case of decision trees, where $Pr(y | x)$ is the proportion of training samples with label $y$ in the leaf where $x$ ends up, these distortions come about because learning algorithms such as C4.5 or CART explicitly aim to produce homogeneous leaves (giving probabilities close to zero or one, and thus high bias) while using few samples to estimate the relevant proportion (high variance).^[4]

For classification models that produce some kind of "score" on their outputs (such as a distorted probability distribution or the "signed distance to the hyperplane" in a support vector machine), there are several methods that turn these scores into properly calibrated class membership probabilities.

For the binary case, a common approach is to apply Platt scaling, which learns a logistic regression model on the scores.^[5] An alternative method using isotonic regression^[6] is generally superior to Platt's method when sufficient training data is available.^[3]

In the 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.^[7]

Evaluating probabilistic classification

Commonly used loss functions for probabilistic classification include log loss and the mean squared error between the predicted and the true probability distributions. The former of these is commonly used to train logistic models.

A method used to assign scores to pairs of predicted probabilities and actual discrete outcomes, so that different predictive methods can be compared, is called a scoring rule.

References

↑ Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome (2009). The Elements of Statistical Learning. p. 348. [I]n data mining applications the interest is often more in the class probabilities $p_{\ell }(x),\ell =1,\dots ,K$ themselves, rather than in performing a class assignment.
1 2 Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer.
1 2 Niculescu-Mizil, Alexandru; Caruana, Rich (2005). Predicting good probabilities with supervised learning (PDF). ICML. doi:10.1145/1102351.1102430.
↑ Zadrozny, Bianca; Elkan, Charles (2001). Obtaining calibrated probability estimates from decision trees and naive Bayesian classifiers (PDF). ICML. pp. 609–616.
↑ Platt, John (1999). "Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods" (PDF). Advances in large margin classifiers. 10 (3): 61–74.
↑ Zadrozny, Bianca; Elkan, Charles (2002). "Transforming classifier scores into accurate multiclass probability estimates". Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '02 (PDF). pp. 694–699. doi:10.1145/775047.775151. ISBN 1-58113-567-X. CiteSeerX: 10.1.1.13.7457.
↑ Hastie, Trevor; Tibshirani, Robert (1998). "Classification by pairwise coupling". The Annals of Statistics. 26 (2): 451–471. doi:10.1214/aos/1028144844. Zbl 0932.62071. CiteSeerX: 10.1.1.46.6032.

This article is issued from Wikipedia - version of the 11/30/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.