From Wikipedia
The prior distribution is the distribution of the parameter(s) before any data is observed, i.e. $p(\theta \mid \alpha )$. ...
The sampling distribution is the distribution of the observed data conditional on its parameters, i.e. $p(\mathbf {X}\mid \theta )$ . This is also termed the likelihood,...
The marginal likelihood (sometimes also termed the evidence) is the distribution of the observed data marginalized over the parameter(s), i.e. $$p(\mathbf {X}\mid \alpha )=\int_{\theta }p(\mathbf {X}\mid \theta )p(\theta \mid \alpha )\operatorname {d}\!\theta .$$
The posterior distribution is the distribution of the parameter(s) after taking into account the observed data. This is determined by Bayes' rule, which forms the heart of Bayesian inference: $$ p(\theta \mid \mathbf {X},\alpha )={\frac {p(\mathbf {X}\mid\theta )p(\theta \mid \alpha )}{p(\mathbf {X} \mid \alpha )}}\propto p(\mathbf {X} \mid \theta )p(\theta \mid \alpha ) $$
In the calculation of the marginal likelihod and posterior distribution, I wonder what is the reason that $p(\mathbf {X }\mid \theta )$ is not $p(\mathbf {X} \mid \theta, \alpha )$ instead?
The posterior predictive distribution is the distribution of a new data point, marginalized over the posterior: $$ p(\tilde {x} \mid \mathbf {X},\alpha )=\int_{\theta}p(\tilde {x} \mid \theta )p(\theta \mid \mathbf {X},\alpha )\operatorname {d}\!\theta $$
Why is $p(\tilde{x} \mid \theta )$ not $p(\tilde {x} \mid \theta, X, \alpha )$ instead?
Thanks!