Bayes Theorem

From Conditional Probability, we know that
$$
P(x|y) = \frac{P(x, y)}{P(y)}
$$
Now,
$$
\begin{align*}
P(x|y) &= \frac{P(x, y)}{P(y)} \\P(x,y) &= P(x|y) P(y) .... (i)
\end{align*}
$$
And.
$$
\begin{align*}
P(y|x) &= \frac{P(x, y)}{P(x)} \\P(x,y) &= P(y|x) P(x) .... (ii)
\end{align*}
$$
From (i) and (ii),
$$
\begin{align*}
P(x|y) p(y) &= p(y|x) p(x) \\p(x|y) &= \frac{p(y|x) p(x)} {p(y)}
\end{align*}
$$

Here,

• $p(x|y)$ is the Posterior Probability
• $p(y)$ is the Marginal Probability
• $p(y|x)$ is the Likelihood
• $p(x)$ is the Prior Probability

[!def] Bayes Theorem
$$
p(x|y) = \frac{p(y|x) p(x)} {p(y)}
$$