This paper introduces Least Squares Support Vector Machines (LS-SVMs) and compares them with Vapnik's SVM Regression and Relevance Vector Machines (RVMs). LS-SVMs are a variant of SVMs that use a least squares approach to minimize the squared error, leading to a simpler computational algorithm. The paper discusses the computational methods for LS-SVMs, including the Nystrom Method and Incomplete Cholesky Factorization, which are used to handle large datasets efficiently. It also covers feature selection techniques, such as information gain, and how they can be used to identify relevant features for classification tasks. The Bayesian approach to learning is explored, where maximum likelihood and hierarchical Bayesian methods are used to estimate model parameters. This approach allows for probabilistic predictions and provides a distribution over the model parameters, which can be used for uncertainty quantification. The paper also discusses Bayesian inference and how it can be used to estimate hyperparameters, such as the noise level and regularization parameters. The Bayesian approach to feature selection is introduced, where relevance vectors are used to identify the most important features. This method replaces traditional feature selection techniques, including support vectors, and provides a more efficient way to approximate complex functions with fewer features. The results section presents the performance of RVMs on two datasets: noisy 'sinc' data for regression and Ripley's synthetic data for classification. The results show that RVMs outperform Support Vector Machines (SVMs) in terms of test error, with RVMs requiring fewer relevance vectors to achieve similar performance. The paper concludes that LS-SVMs and RVMs offer efficient and effective methods for regression and classification tasks, with RVMs providing better performance in terms of feature selection and predictive accuracy.This paper introduces Least Squares Support Vector Machines (LS-SVMs) and compares them with Vapnik's SVM Regression and Relevance Vector Machines (RVMs). LS-SVMs are a variant of SVMs that use a least squares approach to minimize the squared error, leading to a simpler computational algorithm. The paper discusses the computational methods for LS-SVMs, including the Nystrom Method and Incomplete Cholesky Factorization, which are used to handle large datasets efficiently. It also covers feature selection techniques, such as information gain, and how they can be used to identify relevant features for classification tasks. The Bayesian approach to learning is explored, where maximum likelihood and hierarchical Bayesian methods are used to estimate model parameters. This approach allows for probabilistic predictions and provides a distribution over the model parameters, which can be used for uncertainty quantification. The paper also discusses Bayesian inference and how it can be used to estimate hyperparameters, such as the noise level and regularization parameters. The Bayesian approach to feature selection is introduced, where relevance vectors are used to identify the most important features. This method replaces traditional feature selection techniques, including support vectors, and provides a more efficient way to approximate complex functions with fewer features. The results section presents the performance of RVMs on two datasets: noisy 'sinc' data for regression and Ripley's synthetic data for classification. The results show that RVMs outperform Support Vector Machines (SVMs) in terms of test error, with RVMs requiring fewer relevance vectors to achieve similar performance. The paper concludes that LS-SVMs and RVMs offer efficient and effective methods for regression and classification tasks, with RVMs providing better performance in terms of feature selection and predictive accuracy.