Bayes' Theorem

#Math

$\displaystyle P(B|A)=\frac{P(A|B)P(B)}{P(A)}=\frac{P(A\cap B)}{P(B)}$

• $\displaystyle P(A)$ may be substituted using Law of Total Probability

$\displaystyle P(D+|T+)=\frac{P(T+|D+)P(D+)}{P(T+|D+)P(D+)+P(T+|D-)P(D-)}$

• Positive predictive value of a diagnostic test given test sensitivity, test specificity, and proportion of sick/healthy people in a population

$\displaystyle P(\theta|x)= \frac{P(x|\theta)P(\theta)}{P(x)}$

• $\displaystyle P(\theta|x)$ is the posterior
• $\displaystyle P(x|\theta)$ is the likelihood
• $\displaystyle P(\theta)$ is the prior
• $\displaystyle P(x)$ is the marginal