Bayesian Linear Regression

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The aim of Bayesian Linear Regression is not to find the single “best” value of the model parameters, but rather to determine the posterior distribution for the model parameters. Not only is the response generated from a probability distribution, but the model parameters are assumed to come from a distribution as well. Introduction to Bayesian Linear Regression | Towards Data Science

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In the Bayesian viewpoint, we formulate linear regression using probability distributions rather than point estimates. The response, y, is not estimated as a single value, but is assumed to be drawn from a probability distribution. The output, y is generated from a normal (Gaussian) Distribution characterized by a mean and variance. The mean for linear regression is the transpose of the weight matrix multiplied by the predictor matrix. The variance is the square of the standard deviation σ (multiplied by the Identity matrix because this is a multi-dimensional formulation of the model).

Bayesian methods have a highly desirable quality: they avoid overfitting. They do this by making some assumptions beforehand about the likely distribution of the answer. Another byproduct of this approach is that they have very few parameters. Machine Learning has both Bayesian algorithms for both classification (Two-class Bayes' point machine) and regression (Bayesian linear regression). Note that these assume that the data can be split or fit with a straight line. - Dinesh Chandrasekar