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April 20th, 2021: created note.
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Independent means every (xi,yi)(x_i, y_i)(xi,yi) is independent of each (xj,yj)(x_j, y_j)(xj,yj).
Identically distributed means every (xi,yi)(x_i, y_i)(xi,yi) comes from the same distribution.
When it is Independent and Identically Distributed, p(D)=∏ip(xi,yi)=∏ip(xi)p(yi∣xi)p(\mathcal{D})=\prod_{i} p\left(x_{i}, y_{i}\right) = \prod_{i} p\left(x_{i}\right) p\left(y_{i} \mid x_{i}\right)p(D)=∏ip(xi,yi)=∏ip(xi)p(yi∣xi).
One assumption we need to make here is the Independent and Identically Distributed (i.i.d.) assumption.
Independent and Identically Distributed