70 lines
1.9 KiB
ReStructuredText
70 lines
1.9 KiB
ReStructuredText
Bayes' Theorem
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---------------
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This page is inspired from:
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* `MathIsFun <https://www.mathsisfun.com/data/bayes-theorem.html>`__
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* `Wikipedia <https://en.wikipedia.org/wiki/Bayes%27_theorem>`__
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Theorem
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=========
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The Bayes's theorem describes the probability of an event, based on prior knowledge of conditions
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that might be related to the event.
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To compute :math:`P(A|B)`, in other words, the probability of the event :math:`A` to happend, knowing that :math:`B`
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already happend, the following drawing can be used:
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.. image:: figures/bayes_theorem.svg
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:align: center
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:width: 400px
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The red area represents :math:`P(B)=a3+a4`. To compute :math:`P(A|B)` we do:
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.. math::
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P(A|B)=\frac{a3}{a3+a4}=\frac{P(A)P(B|A)}{P(A)P(B|A)+P(\neg A)P(B|\neg A)}=\frac{P(A)P(B|A)}{P(B)}
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This final formula is the Baye's Theorem.
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See the `3Blue1Brown video <https://www.youtube.com/watch?v=HZGCoVF3YvM>`__ for a better understanding.
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.. note::
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The following notation is also used:
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.. math::
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P(A|B)=\frac{a3}{a3+a4}=\frac{P(A \cap B)}{P(B)}
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Example
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=========
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Problem
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^^^^^^^^
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.. image:: figures/cat.jpg
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:align: center
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:width: 150px
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Anna says she is itchy. There is a test for Allergy to Cats, but this test is not always right:
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* For people that do have the allergy, the test says *Yes* 80% of the time
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* For people that do not have the allergy, the test says *Yes* 10% of the time *(false positive)*
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If 1% of the population have the allergy, and Anna's test says *Yes*, what are the chances that she really has the allergy?
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**--- Solution ---**
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Let's draw the little diagram:
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.. image:: figures/bayes_theorem_example.svg
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:width: 400px
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:align: center
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.. math::
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P(Allergy|Yes)&=\frac{a3}{a3+a4}
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&=\frac{P(Allergy)P(Yes|Allergy)}{P(Allergy)P(Yes|Allergy)+P(\neg Allergy)*P(Yes|\neg Allergy)}
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&=\frac{0.01 \times 0.80}{0.01 \times 0.80 + 0.99 \times 0.10} = 0.0747
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The chances that Anna really has allergy is about 7%.
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