What is the correct way to refer to the Likelihood $L(\theta, \alpha | X)$ where $\theta$ and $\alpha$ are parameters?

Question

I am currently studying some latent variable models.

In many works, I found the following equation:

$L( \theta, \boldsymbol\pi | x ) = \sum_{c=1}^{C} f(x| \theta, \alpha = \alpha_c) \cdot \pi_c$

where $\theta$ is a continuous parameter, $\alpha$ is latent variable (categorical, in the case), and $\boldsymbol\pi = (\pi_1, \ldots, \pi_C)$ where $\pi_c = P(\alpha = \alpha_c)$.

In those works, $L( \theta, \boldsymbol\pi | x )$ is usually called of "marginalized likelihood" or "marginal likelihood".

My question is how can I refer to the likelihood function BEFORE the marginalization, that is, $L(\theta, \alpha |x) = f(x | \theta, \alpha)$ ?

I intend to mention both functions, so I think calling the latter only "likelihood" would be a bit vague.

Can someone tell me what is the most correct term to refer to the likelihood $L(\theta, \alpha |x)$?

Answer 1

Here some answers explain clearly that the marginal likelihood is not a type of likelihood function so it should be no problem to just refer to both with correct names (and probably add a coment pointing to the difference clearly).

Another way to express the difference would be to use different notation (letters). The wikipedia page reffers to marginal likelihood with what would be in your case

$p(x|\pi)=\sum_{c} f(x|\theta, \alpha=\alpha_c).\pi_c$