I often see the posterior predictive distribution in ML defined as follows:
$$p(y^* \mid x^*, X, Y) = \int p(y^* \mid x^*, \omega)p(\omega, X, Y) d\omega$$
where $\omega$ are all parameters, $x^*$ is a new input point and $X, Y$ is the training dataset.
What confuses me is the lower case $y^*$ and $x^*$, because I am not sure whether it is a random variable and where it comes from.
Without knowing a lot about Bayesian statistics, I would first define the posterior $P(W \mid X, Y)$ (with $W$ being the parameters). Then use the law of total probability to obtain
$$P(Y \mid X) = \int P(Y \mid X, W)P(W)dW$$
Next, when I get a new point $x^*$, I would set $P(Y= y^* \mid X = x^*)$. Are $y^*$ and $x^*$ as random variables necessary?