\begin{align} p(x\mid y) = \frac{p(y \mid x) p(x)}{p(y)} \end{align}
this is a direct result of the probability chain rule. Typically, we name p(y|x) the “likelihood”, p(x) the “prior”. Better normalization What if you don’t fully know p(y), say it was parameterized over x?
\begin{align} p(x|y) &= \frac{p(y|x) \cdot p(x)}{p(y)} \\ &= \frac{p(y|x) \cdot p(x)}{\sum_{X_{i}} p(y|X_{i})} \end{align}
just apply law of total probability! taad