# Bayesian Classifier for Normally Distributed classes

## Reference

[Click here for Lecture PPT: ](https://people.engr.tamu.edu/rgutier/web_courses/csce666_s20/l5.pdf)TAMU

## Introduction

Bayes classifier for Normally distributed classes.

### Case 1: If the same covariance for each class: Linear Classifier

### Case 2: If different covariance: Quadratic Classifier

When the likelihood is expressed as a normal distribution:

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-9ba3ae89d4eaec6e34fc116317d59019e06e6c42%2Fimage.png?alt=media)

## Linear Classifier: Linear Discriminant Analysis (LDA)

The boundary function can be expressed as a linear function $$g_i(x) = \bf{w_i^Tx } + b_i$$

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-338abe9b6924bfc37854614b8441a3b256170779%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-a579dbf05634ed99f9cfe30fba5e377d1a48b5f8%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-28526720e574fc39d01d4cb038cd28a149d2691d%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-2819bfbd53b7bc8a0dab5e3ed991b03f0f0c006b%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-ad478ee330ed918dddff8d36da349effa412e829%2Fimage.png?alt=media)

## Quadratic Classifier

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-024e71082568b22b22be7e44705e8e5020bacc5e%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-4c89cb2d30b513bdbd6efade24f68ef52e0c8701%2Fimage.png?alt=media)

## Example

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-51dbce976751c926fc7aa5b95042f4f503d2df6b%2Fimage.png?alt=media)

![](https://3698175758-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MAwtzMy_pbrChIExFtN%2Fuploads%2Fgit-blob-8a3bea8ee6c2b5f89c2522a3f3a6ebda26eb5bcc%2Fimage.png?alt=media)
