Gaussian Observation Model

A statistical approach for modelling data. This approach is assumed in the lm() function in R for linear modelling.

It assumes that:

1. All the observed data follows a Normal (Gaussian) distribution around an expected value
2. The residuals in the linear model are normally distributed around zero
3. The variance within the residuals is consistent between parameters

In this approach, the "maximum likelihood estimate" used to fit a model is actually the same as the Least squares estimation.

We use the $y_i\sim \text{N}(\mu ,{\sigma }^2)$ notation to describe this observation model.

You could use this in the glm() function as the distribution type of the response data, but it doesn't really make sense to and it's probably best to just use lm() for this.

E.g., these two models are basically equivalent, except the glm() function will give different and less useful parameters.

glm(y ~ x , family = gaussian())
lm(y ~ x)

!Lecture_week6, p.17