Simple question here about deriving posteriors.
Suppose I have some likelihood in mind for my data, $p(y|\theta)$, and I also have a particular conjugate prior in mind $p(\theta)$.
Now, I have obtained a single observation, $y_1$, which I use to update my prior into a posterior: $p(\theta|y_1) \propto p(y_1|\theta) p(\theta)$.
Now suppose I have more data points, say three in total $\{y_1, y_2, y_3\}$. I'm now interested in the posterior $p(\theta| y_1, y_2, y_3)$ which might commonly be written as $\prod_{i=1}^3 p(y_i|\theta) p(\theta)$ if each data point were drawn independently.
My question is how to carry out this product - is it just over the likelihood or also the prior? That is, should I think of the posterior as
(i) $p(\theta|y_N) \propto p(\theta) \prod_{i=1}^3 p(y_i|\theta) $
or
(ii) $ p(\theta|y_N) \propto p(\theta)^3 \prod_{i=1}^3 p(y_i|\theta) $ ?
That is, do I multiply by the prior probability 3 times?