This paper introduces Gaussian Processes (GPs) as a Bayesian framework for regression. GPs are flexible, non-parametric models that define a distribution over functions using a stochastic process. The paper explains how GPs can be used to model regression problems, how to incorporate training data, and how to learn hyperparameters using the marginal likelihood. It also discusses the practical advantages of GPs, such as their ability to provide uncertainty estimates and their flexibility in modeling complex data. GPs are defined by a mean function and a covariance function. The mean function describes the average behavior of the function, while the covariance function determines the similarity between function values at different points. The paper illustrates how GPs can be used to generate samples from a distribution over functions and how these samples can be used to make predictions. The paper then discusses how to update the prior distribution of a GP with training data to obtain the posterior distribution. This involves computing the posterior mean and covariance functions, which allow for predictions of new data points. The paper also addresses the issue of noise in the training data and shows how it can be incorporated into the GP model. The paper then discusses how to train a GP model by optimizing the marginal likelihood with respect to the hyperparameters of the mean and covariance functions. This process allows the model to learn the most appropriate hyperparameters that best fit the data. The paper also discusses the trade-off between model complexity and data fit in GP models, noting that the marginal likelihood automatically balances these two factors. Finally, the paper concludes by discussing the current trends in GP research, including the development of approximate methods for larger datasets and the use of sparse approximations to improve computational efficiency. The paper also highlights the importance of understanding the properties of functions drawn from GPs with particular covariance functions, as this can help in choosing appropriate covariance functions that reflect prior knowledge.This paper introduces Gaussian Processes (GPs) as a Bayesian framework for regression. GPs are flexible, non-parametric models that define a distribution over functions using a stochastic process. The paper explains how GPs can be used to model regression problems, how to incorporate training data, and how to learn hyperparameters using the marginal likelihood. It also discusses the practical advantages of GPs, such as their ability to provide uncertainty estimates and their flexibility in modeling complex data. GPs are defined by a mean function and a covariance function. The mean function describes the average behavior of the function, while the covariance function determines the similarity between function values at different points. The paper illustrates how GPs can be used to generate samples from a distribution over functions and how these samples can be used to make predictions. The paper then discusses how to update the prior distribution of a GP with training data to obtain the posterior distribution. This involves computing the posterior mean and covariance functions, which allow for predictions of new data points. The paper also addresses the issue of noise in the training data and shows how it can be incorporated into the GP model. The paper then discusses how to train a GP model by optimizing the marginal likelihood with respect to the hyperparameters of the mean and covariance functions. This process allows the model to learn the most appropriate hyperparameters that best fit the data. The paper also discusses the trade-off between model complexity and data fit in GP models, noting that the marginal likelihood automatically balances these two factors. Finally, the paper concludes by discussing the current trends in GP research, including the development of approximate methods for larger datasets and the use of sparse approximations to improve computational efficiency. The paper also highlights the importance of understanding the properties of functions drawn from GPs with particular covariance functions, as this can help in choosing appropriate covariance functions that reflect prior knowledge.