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Bayes Theorem

Tags: probability

Bayes Theorem (Thomas Bayes 1761) allows calculation of true probability based on a sample of prior knowledge and a base rate1.

\begin{equation} P(A|B)=\frac{P(B|A) * P(A)}{P(B|A)P(A)+P(\neg{A})P(B|\neg{A})} \end{equation}

\begin{equation} P(A|B)=P(B|A)P(A)+P(\neg{A})P(B|\neg{A}) \end{equation}

  • Prob of hypothesis * prob of seeing evidence w/ Hypothesis + Probability hypothesis false * prob of seeing evidence with hypothesis false

\begin{equation} P(A|B)=\frac{P(B|A) * P(A)}{P(B)} \end{equation}

  • P(A|B) Posterior (Given Hypothesis, and some New Evidence, what is the probability that the hypothesis is true)
  • P(B) = Total probability of seeing the evidence
  • P(A) = Prior (base rate, probability that the hypothesis is true)
  • P(B|A) = Likelihood: (Probability that we would see this evidence give that the hypothesis is true)

Seeing evidence should update prior beliefs, i.e. it should restrict the space of possibilities and the ratio you need to consider after that 2.

References

1.
Lewis, M. The Undoing Project: A Friendship That Changed Our Minds. (W. W. Norton & Company, 2016).
2.
3Blue1Brown. Bayes Theorem. (2019).