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Bayesian Linear Regression: What is, Function & Real Life Applications in 2024

Bayesian Linear Regression: What is, Function & Real Life Applications in 2024

Q: 1. What is the Bayesian inference?

Bayesian inferences are a group of mathematical operations based on the Bayes theorem. It is a mathematical method for determining conditional probability. The possibility of a result occurring dependent on the likelihood of a primary outcome occurring is known as conditional probability. Professionals can use Bayes theorem to alter previous forecasts or hypotheses. It is used in finance to assess the risk of providing money to potential borrowers. By considering how probable each specific person is to have an illness and the general accuracy of the test, Bayes' theorem may be used to evaluate the accuracy of medical test findings.

Q: 2. How is bayesian linear regression different from ordinary linear regression?

The ordinary linear regression is a frequentist method, which implies that there are sufficient measurements to make a valid statement. The data is augmented with extra information in the form of a prior probability distribution in the Bayesian method. The posterior belief about the parameters is obtained by combining the previous knowledge about the parameters with the data's likelihood function using Bayes theorem. The Bayesian interpretation of both linear and logistic regression employs statistical analysis within the context of Bayesian hypothesis.

Q: 3. How does Bayesian linear regression work?

We define linear regression using probability distributions rather than point estimates from a Bayesian perspective. The answer, y, is supposed to be chosen from a probability distribution rather than being evaluated as a single number. The result, y, is produced by a normal (Gaussian) distribution with a mean and variance. In linear regression, the mean is calculated by multiplying the weight matrix by the predictor matrix. Since this is a multi-dimensional version of the model, the variance is the square of the standard deviation multiplied by the Identity matrix. The goal of Bayesian Linear Regression is to identify the posterior distribution for the model parameters, not to find a particular perfect result for the model parameters.

By Pavan Vadapalli

Updated on Feb 24, 2025 | 5 min read | 8.5k views

Table of Contents

What Is A Linear Regression?

Linear regression attempts to show the connection between two factors by fitting a direct condition to noticed information. One variable is viewed as an illustrative variable, and the other is viewed as a needy variable. For instance, a modeller should relate loads of people to their statues utilising a straight relapse model.

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Now the next step is to know what the Bayesian linear equation is and how it can be calculated to get the desired result.

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Now, What Is The Bayesian Linear Equation?

So, when we talk about Bayesian regression methods, we know that it is a very powerful method because they provide us with the entire distribution over regression parameters. In order to calculate inadequate data or unequal distributed data, Bayesian Linear Regression provides a natural mechanism.

You can place a prior on the coefficients so that if the data is absent, the prior can take the place of data. Statistical analysis is conducted under the conditions of Bayesian interface in Bayesian linear regression in statistics.

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We use probability distribution instead of point estimates to devise linear regression.

The output is achieved from a probability distribution, rather than usual regression techniques. The goal of Bayesian linear regression is to find Posterior instead of model parameters.

Model parameters are supposed to occur from a distribution.

The posterior expression is

Posterior= (Likelihood*Prior)/Normalization

The above equation is similar to Bayes’ Theorem, which is

Source

Real-life Application Of Bayesian Linear Regression

According to the following graphs, Linear regression and Bayesian regression can generate the same predictions.

Source

In the last graph, we can assume that if the predictive distribution is inside the thick colour, then the density of data is high and if it is in the scarce area the density can be considered as low.

Let’s talk about the advantage of Bayesian Regression:

With the help of Bayesian processing, we can retrieve the complete variety of inferential solutions instead of a point estimate.
It works efficiently with the small size of the dataset.
It is very suitable for the online form of learning, whereas, in the form of batch learning, we have the whole dataset.
It is a very powerful and tested approach.

Now, let’s talk about the disadvantage of Bayesian linear regression.

It does not work efficiently if the dataset contains a huge amount of data.
The conjecture of the model can be time-consuming.

Read: Linear Project Ideas & Topics

Conclusion

So, we can see what a powerful method Bayesian linear regression is. Many advantages can be taken with Bayesian linear regression and is one of the natural mechanism to calculate insufficient or poorly distributed data.

A Bayesian viewpoint is an instinctive form of seeing the world. To its frequent counterpart, Bayesian Inference can embellish a very convenient substitute. It is used in various fields like data-science, machine learning, and many more. It helps in building various models with the help of which we can solve many problems.

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