I had a doubt while doing Andrew Ng's CS 229.
For reference: https://see.stanford.edu/materials/aimlcs229/cs229-notes1.pdf
In linear regression, while using MLE to calculate parameters, we assume that y has a univariate Gaussian distribution p(⃗y|X; θ). This makes sense to me when probability of the data is viewed as a function of y, for a fixed value of theta.
Then in the notes it says: "When we wish to explicitly view this as a function of θ, we will instead call it the likelihood function [given by]: L(θ) = L(θ; X, ⃗y) = p(⃗y|X; θ)" Here's my doubt - Since our probability density function ( (which we are calling likelihood function) is now a function of θ, does that mean we are saying θ has a Gaussian distribution?
If that is the case, since θ is multivariable, we are essentially giving our probability density function a multivariable input. Then shouldn't our probability distribution be a multivariate gaussian distribution now? I don't understand how changing a single variable input to a multivariable input is changing nothing about our function?