Bayesian Fundamentals

Bayes Rule

$$ \begin{aligned} P[\theta|x] &= \frac{P[x|\theta] \cdot P[\theta]}{P[x]} \end{aligned} $$

• Prior Knowledge: $P[\theta]$
• Poster Knowledge: $P[\theta | x]$
• Likelihood: $P[x | \theta]​$

Estimators

Maximum Likelihood Estimator(MLE): $$ \begin{aligned} \arg\max_{\theta} P[x | \theta] \end{aligned} $$

Maximum A Posteriori Estimator(MAP):

$$ \begin{aligned} \arg\max_{\theta} P[x | \theta] \cdot P[\theta] \end{aligned} $$

by Jon