Regression using Bayes’ Theorem

What is Bayesian regression?

Bayesian regression is a statistical method that uses Bayes’ theorem to estimate the probability distribution of model parameters given the observed data. The process involves defining a prior distribution for the parameters, updating this distribution using the observed data through Bayes’ theorem, and obtaining the posterior distribution of the parameters.

The following steps outline the process of developing a Bayesian linear regression model:

1. Defining the model: We define the linear regression model, where the dependent variable is a linear combination of the independent variables with Gaussian noise.

2. Specifying prior distributions: We assign prior distributions for the model parameters. This can be done using a conjugate prior if available; otherwise, a noninformative prior can be used.

3. Calculating the posterior distribution: We use Bayes’ theorem to calculate the posterior distribution of the parameters given the observed data. This involves multiplying the likelihood of the observed data presented in the parameters by the prior distribution of the parameters.

4. Obtaining point estimates: Once we calculate the posterior distribution, we obtain point estimates for the parameters using the MAP estimator or the mean of the posterior distribution.

5. Making predictions: We make predictions for new data by obtaining the predictive distribution, the distribution of the dependent variable given the independent variables, and the parameters.

6. Comparing models: Finally, we compare multiple models using model selection criteria such as the Bayesian information criteria (BIC) or the deviance information criterion (DIC) to choose the best-fitting model.

The process can be presented in the following form: