Lost in the introduction discussing probability/likelihood [duplicate]

Question

I have some questions about the following paragraph which introduces a masters level course.

In this unit we consider the Frequentist (i.e. counting) approach to statistical inference and computing the probability/likelihood of the data $y$ given the parameters $\theta$, $P(y|\theta)$. Under this approach we use Maximum Likelihood Estimation (MLE) to estimate the parameters of probabilistic/likelihood models, i.e. $\hat{\theta} = > arg max_{\theta}P(y|\theta)$. If we are dealing with a known probability distribution of the data then we use $P(y|\theta)$, but in general we are interested in the probability $P(y|x, u, \theta)$ where $u$ can be 'predictor' variables and $x=x(u,\theta)$ can be intermediate variables dependent on $u$ and $\theta$. In this unit we primarily focus on the cases of $P(y|\theta)$ and $P(y|u=x, \theta)=P(y|x, \theta)$.

Q1) I understand that big P means a discrete distribution. Does this statement only apply to discrete distributions?

Q2) Why is it talking about likelihood of data? I understood likelihood to pertain to parameters as in MLE?

Answer 1

I think big P in this notation is just for probability in general. That is, $P(y|\theta)$ is just probability of y given $\theta$.

Likelihood is used to refer to the likelihood function, which is the same in form as the probability density/mass function, but with the parameters as the unknown. Thus, while we can have $f(x|\theta)$ as the density/mass function of $X$ given $\theta$ value of parameters, we can have the corresponding likelihood function $L(\theta|x)$, where we are interested in $\theta$ and have the dataset $x$ to use.