In the Bayes formula, $$ p(\theta|D) = \frac{ p(D|\theta) p(\theta) }{ p(D) } $$
which of $D, \theta$ should we regard as given and which as variables,
which of the factors $p(\theta|D), p(D|\theta), p(\theta), p(D)$ are valid probabilities
which of $D, \theta$ can refer to more than one data point?
This is not for a class.
To explain the question:
$P(D|\theta)$ is the likelihood, which is described as proportional to a probability, but not itself a probability. I believe it is true that for fixed $\theta$, $\int p(D|\theta) d D = 1$, seems like this must be true. But with $D$ fixed and $\theta$ variable, the integral $\int p(D|\theta) d\theta$ is not necessarily one.
I assume that $p(\theta|D)$ is a probability, otherwise the normalizing constant in the denominator woudl not be necessary. I assume that $\int p(\theta|D) d\theta = 1$, but integrating the posterior with respect to $D, \int p(\theta|D) d D$, is not necessarily onee. Is that true?
Often $D$ is written in font $\mathcal{D}$ (look closely), which I believe is a hint that $D$ can describe one data point or multiple data points or all the data. For example, the likelihood of the data is often assumed to factor across individuald data points. However I believe $\theta$ is a single thing (though possibly a vector) regardless of the data.