# Relevance vector machine

## Machine learning technique / From Wikipedia, the free encyclopedia

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In mathematics, a **Relevance Vector Machine (RVM)** is a machine learning technique that uses Bayesian inference to obtain parsimonious solutions for regression and probabilistic classification.^{[1]}
The RVM has an identical functional form to the support vector machine, but provides probabilistic classification.

It is actually equivalent to a Gaussian process model with covariance function:

- $k(\mathbf {x} ,\mathbf {x'} )=\sum _{j=1}^{N}{\frac {1}{\alpha _{j}}}\varphi (\mathbf {x} ,\mathbf {x} _{j})\varphi (\mathbf {x} ',\mathbf {x} _{j})$

where $\varphi$ is the kernel function (usually Gaussian), $\alpha _{j}$ are the variances of the prior on the weight vector
$w\sim N(0,\alpha ^{-1}I)$, and $\mathbf {x} _{1},\ldots ,\mathbf {x} _{N}$ are the input vectors of the training set.^{[2]}

Compared to that of support vector machines (SVM), the Bayesian formulation of the RVM avoids the set of free parameters of the SVM (that usually require cross-validation-based post-optimizations). However RVMs use an expectation maximization (EM)-like learning method and are therefore at risk of local minima. This is unlike the standard sequential minimal optimization (SMO)-based algorithms employed by SVMs, which are guaranteed to find a global optimum (of the convex problem).

The relevance vector machine was patented in the United States by Microsoft (patent expired September 4, 2019).^{[3]}

- Kernel trick
- Platt scaling: turns an SVM into a probability model

- Tipping, Michael E. (2001). "Sparse Bayesian Learning and the Relevance Vector Machine".
*Journal of Machine Learning Research*.**1**: 211–244. - Candela, Joaquin Quiñonero (2004). "Sparse Probabilistic Linear Models and the RVM".
*Learning with Uncertainty - Gaussian Processes and Relevance Vector Machines*(PDF) (Ph.D.). Technical University of Denmark. Retrieved April 22, 2016. - US 6633857, Michael E. Tipping, "Relevance vector machine"

- dlib C++ Library
- The Kernel-Machine Library
- rvmbinary: R package for binary classification
- scikit-rvm
- fast-scikit-rvm, rvm tutorial