CH5 - Bayes Theorem

This note discusses the Bayes Theorem and how it can be used to calculate conditional probabilities.

This should be noted as the theorem used in statistics as a tool to solve probability equations. For the field of statistics where Bayes’ Theorem is treated as the underlying principle of probability, see Bayesian Inference.

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Consider the following:

$$P(B\mid A)=\frac{P(A\cap B)}{P(A)}{\to }^{P(A)}P(A\cap B)=P(A)\times P(B\mid A)$$

Also consider:

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}{\to }^{P(A)}P(A\cap B)=P(B)\times P(A\mid B)$$

Now we apply this to isolate for events:

$$P(A\mid B)=\frac{P(A)\times P(B\mid A)}{P(B)\leftarrow canbepartitionedoverB/B^C}$$

Same Can be done for other:

$$P(B\mid A)=\frac{P(A)\times P(A\mid B)}{P(A)}$$

Bayes Theorem

Finally, this leads to Bayes Theorem:

$$P(A\mid B)=\frac{P(A)\times P(B\mid A)}{P(A)\times (B\mid A)+P(A^C)+P(B\mid A^C)}$$

Where the bottom part of the fraction is called the Law of Total Probability