Maximum Likelihood
Find β and θ that maximizes L(β, θ|data).
Partial Likelihood
If we can write the likelihood function as:
L(β, θ|data) = L1(β|data) L2(θ|data)
Then we simply maximize L1(β|data).
Profile Likelihood
If we can express θ as a function of β then we replace θ with the corresponding function.
Say, θ = g(β). Then, we maximize:
L(β, g(β)|data)
Marginal Likelihood
We integrate out θ from the likelihood equation by exploiting the fact that we can identify the probability distribution of θ conditional on β.
