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Maximum Likelihood approach
We now need to find θ. Maximum likelihood seeks the value of θ that
maximizes the likelihood function:
θ^ML := arg max
θ
p(x1; : : : ; xnjθ);
This value best explains the data according to the chosen distribution family.
Maximum Likelihood equation
The analytic criterion for this maximum likelihood estimator is:
rθ
nYi=1
p(xijθ) = 0:
Simply put, the maximum is at a peak. There is no “upward” direction.

Maximum likelihood and the logarithm trick
θ^ML = arg max
θ
nYi=1
p(xijθ) = arg max
θ
ln
nYi=1
p(xijθ) = arg max θ
nXi=1
ln p(xijθ)
To then solve for θ^ML, find
rθ
nXi=1
ln p(xijθ) =
nXi=1
rθ ln p(xijθ) = 0:
Depending on the choice of the model, we will be able to solve this
1. analytically (via a simple set of equations)
2. numerically (via an iterative algorithm using different equations)
3. approximately (typically when #2 converges to a local optimal solution)
#maximum #maximumlikelihood #linearregression #linear #regression

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