Bayes Theorem

#probability #interview

From Conditional Probability, we know that

P (x | y) = \frac{P (x, y)}{P (y)}

Now,

\begin{aligned} P (x | y) & = \frac{P (x, y)}{P (y)} \\ P (x, y) & = P (x | y) P (y) . . . . (i) \end{aligned}

And.

\begin{aligned} P (y | x) & = \frac{P (x, y)}{P (x)} \\ P (x, y) & = P (y | x) P (x) . . . . (i i) \end{aligned}

From (i) and (ii),

\begin{aligned} P (x | y) p (y) & = p (y | x) p (x) \\ p (x | y) & = \frac{p (y | x) p (x)}{p (y)} \end{aligned}

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

Bayes Theorem

$p (x | y) = \frac{p (y | x) p (x)}{p (y)}$

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