$p(x|\theta)$ in predictive distribution vs likelihood function in bayes I have doubts over the use of $p(x|\theta)$ in the bayesian context which I would like to clarify. In baye's theorem, the likelihood is considered a function of $\theta$, this means that $x$ is fixed and $\theta$ varies.
In the predictive distribution, $x$ is random and $\theta$ is fixed since the predictive distribution tries to average the distribution of $x$ over all possible values that $\theta$ can take.
Is my understanding of $p(x|\theta)$ in these 2 context correct ?
 A: The likelihood function $p(x|\theta)$ is a function of your data. This means that your $x$ is fixed and you want to check which value of $\theta$ is the most probable for your data, i.e. which value of $\theta$ maximized your $p(x|\theta)$.  $p(x|\theta)$ is the conditional distribution of $x$, and yes it can be seen as a function of $\theta$, but not as a distribution of $\theta$.
For the predictive distributions of $x$, you have two cases, the prior and the posterior predictive.
The posterior predictive is defined as
$$p(x|Data)=\int_{\Theta}p(x|\theta)p(\theta|Data)d\theta$$
In this case, you can consider your $x$ as a random quantity, because you define a (marginalized in terms of $\theta$) conditional distribution for $x$. However, I believe that you cannot consider $\theta$ fixed because it doesn't take certain values. You have to let $\theta$ take values all over the domain that the posterior distribution of $\theta$ is defined. As you said it tries to average over all possible values of $\theta$, so I assume that also $\theta$ has to be random.
