## Resources

## Notation

variable | dimension | name |
---|---|---|

ith predictor |
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ith state |
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ith sample weight |
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weights | ||

all predictors | ||

all states | ||

identity matrix with columns entries being data point weights |

## Weighted Gaussian Linear regression

The log-likelihood of dataset with weighted samples which is modeled by a linear gaussian function is given by:

where is a Gaussian probability density function:

with parameters .

### Expansion of the log-likelihood

First without considering the weights we simplify

Simplifying with the weights:

## 1D Maximum likelihood

Given that we are in the 1D case ,

Set the derivaties with respect to the parameters to zero, and , and solve for and :