# Bayes Theorem Rationality is not about knowing facts, it's about recognizing which facts are relevant. ~ Grant Sanderson Evidence should not determine beliefs, but update them. ~ Grant Sanderson $ p(y|x) = \frac{p(x|y)p(y)}{p(x)} $ $p(y)$ is the prior probability of $y$ - observed before observing $x$ $p(y|x)$ is the posterior probability of $y$ - after observing $x$ $p(x|y)$ is the likelihood of $X=x$ given $Y=y$ $p(x)$ is the evidence for $X=x$ ### Frequentist vs Bayesian interpretation Frequentist - Probaility is the fraction of times an event occurs in an experiment Bayesian - Probability is the quantification of plausibility or stength of the belief on an event. ## Derrivation of Bayes theorem From product rule, $p(x,y) = p(x|y)p(y)$ From symmetry property, $p(x,y) = p(y,x) = p(y|x)p(x)$ Thus, $p(y|x) = \frac{p(x|y)p(y)}{p(x)}$ Using the sum rule, the denominator in Bayes theorem can be expressed in terms of quantities from the numerator: $p(x) = \sum_Y p(Y|X)p(X)$ ## References: 1. 3Blue1Brown video: https://www.youtube.com/watch?v=HZGCoVF3YvM ---