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--- bayes_theorem [2018/12/31 19:50]
paul
+++ bayes_theorem [2019/03/31 14:49] (current)
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-Probability
+====== Bayes Theorem ======
+Probability discovered by Thomas Bayes in the 18th century.
+[[Conditional Probability]]
-[[Conditional Probability]]
+Probability of a proposition is the chance or likelihood that a proposition is true.
+If 1 student in 20 has the flue, the probability is 1 in 20.
+Prior:
+$P(Sally\ has\ the\ flu) = 0.05$
+Suppose there are 5 girls and 15 boys.
+==== Conditional Probabilities ====
+Suppose we learn some more information about the situation.
+The probability that Sally has the flu conditional on the patient being a girl is 0.2.
+The probability that Sally has the flu conditional on the patient being a boy is 0.
+We write this as:
+$P(Sally\ has\ the\ flu\ |\ the\ flu\ patient\ is\ a\ girl) = 0.20$\\
+$P(Sally\ has\ the\ flu\ |\ the\ flu\ patient\ is\ a\ boy) = 0$
+**Sometimes you know that you might encounter some new evidence in the future, but you don't know how that evidence should effect the probability**
+===== Bayes Theorem =====
+Bayes Theorem gives you a way to figure out what your conditional probabilities should be.
+$P(H) =$ Prior probability, the probability that the hypothesis is correct.
+$P(H|E) =$ Probability of hypothesis H, given condition E.
+$P(E|H) =$ Probability of the condition E, given hypothesis E.
+$P(E) =$ Probability of the condition E.
+The probability of a hypothesis, $H$ conditional on a new piece of evidence, $E$
+$P(H|E) = \frac{P(E|H)P(H)}{P(E)}$
+To summarize, Bayes Theorem tells us how to calculate the probability of a hypothesis given a condition. This depends on three things:
+      - Conditional probability given H
+      - The prior probability of the hypothesis
+      - The prior probability of the evidence
-Probability of A given B is the probab