It is sometimes referred to as the likelihood of the data, and sometimes referred to as a statistical model. The difference is whether we are looking at $$p(x | \theta)$$ as

## a function of $$x$$, where $$\theta$$ is known 🔗

If $$\theta$$ is a known model parameter, then $$p_x(x|\theta) = p(x; \theta) = p_\theta(x)$$ is the probability of $$x$$ according to a model parameterized by $$\theta$$, also known as a model/statistical model/observation model measuring uncertainty about $$x$$ given $$\theta$$.

(If $$\theta$$ is a known random variable, $$p(x|\theta)$$ is just a conditional probability, $$\frac{p(x, \theta)}{p(\theta)}$$.)

## a function of $$\theta$$, where $$x$$ is known 🔗

Unlike the above, the emphasis is on investigating the unknown $$\theta$$.

$$p(x|\theta)$$ is the probability of some observed data $$x$$, that resulted from the random variable $$\theta$$ taking on different values.

When doing MLE to find the assignment $$\hat{\theta}$$ for $$\theta$$ that maximizes likelihood $$p(x|\theta)$$, $$p(x|\hat{\theta})$$ is also called the maximum likelihood of $$\theta$$ given $$x$$, $$\mathcal L(\hat\theta|x)$$.

In other words, it’s a function of $$\theta$$ (written more explicitly as $$p_\theta(x|\theta)$$) that measures the extent to which observed $$x$$ supports particular values of $$\theta$$ in a parametric model.