# Which one is better maximum likelihood or marginal likelihood and why?

While performing regression if we go by the definition from: What is the difference between a partial likelihood, profile likelihood and marginal likelihood?

that, Maximum Likelihood
Find β and θ that maximizes L(β, θ|data).

While, 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.

Which is the better methodology to maximize and why?

Each of these will give different results with a different interpretation. The first finds the pair $\beta$,$\theta$ which is most probable, while the second finds the $\beta$ which is (marginally) most probable. Imagine that your distribution looks like this:

$\beta=1$$\beta=2$
$\theta=1$0.0 0.2
$\theta=2$0.1 0.2
$\theta=3$0.3 0.2

Then the maximum likelihood answer is $\beta=1$ ($\theta=3$), while the maximum marginal likelihood answer is $\beta=2$ (since, marginalizing over $\theta$, $P(\beta=2)=0.6$).

I’d say that in general, the marginal likelihood is often what you want – if you really don’t care about the values of the $\theta$ parameters, then you should just collapse over them. But probably in practice these methods will not yield very different results – if they do, then it may point to some underlying instability in your solution, e.g. multiple modes with different combinations of $\beta$,$\theta$ that all give similar predictions.