Bayes' Theorem & Conditional Probability PDF
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This document provides explanations and visual examples of Bayes' theorem and conditional probability. These concepts are explored using diagrams. The content is suitable for undergraduate-level students studying probability and statistics.
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### Venn Diagram of my Life: This is a Venn Diagram with 3 overlapping circles. - The first circle is labeled *Things I like to do*. - The second circle is labeled *Things I'm good at*. - The third circle is labeled *Things that make money*. The overlapping sections of the circles represent thi...
### Venn Diagram of my Life: This is a Venn Diagram with 3 overlapping circles. - The first circle is labeled *Things I like to do*. - The second circle is labeled *Things I'm good at*. - The third circle is labeled *Things that make money*. The overlapping sections of the circles represent things that the individual both likes to do and is good at, things that are good at and make money, and things that the individual likes to do and make money. ### A Visual Depiction of Conditional Probability: A visual depiction of conditional probability is shown. The image depicts two overlapping circles, one labeled *A* and the other labeled *B*. The circle *B* is filled yellow while the circle *A* has only the overlapping section filled yellow. The caption above the image states: > The Entire Yellow Space is *P(B)* The text below the image explains the concept of conditional probability: > Given *B*, what's the probability of *A*? > > *P(A \ B)* = *P(A ∩ B)* / *P(B)* > > Intuitively we are asking -- *What Share of B contains the overlap with A?* > > In a conditional probability problem, the sample space is "reduced" to the "space" of the given outcome (e.g. if given *B*, we now just care about the probability of *A* occurring "inside" of *B*). ### Bayes’ Law/ Theorem: This image shows the formula for Bayes' Theorem, which is as follows: *P(H|E)* = *P(E|H)* x *P(H)* / *P(E)* Where: - *P(H|E)* represents the posterior, or the probability of event *H* occurring, given that event *E* has already occured. - *P(E|H)* represents the likelihood, or the probability of event *E* occurring, given event *H*. - *P(H)* represents the prior, or the probability of event *H* occurring. - *P(E)* represents the evidence, or the probability of event *E* occurring. ### Bayes’ Law/ Theorem Continued: This image is a break down of Bayes' Theorem: **Pr(H<sub>p</sub> | Evidence) / Pr(H<sub>d</sub> | Evidence) = Pr(Evidence | H<sub>p</sub>) x Pr(H<sub>p</sub>) / Pr(Evidence | H<sub>d</sub>) x Pr(H<sub>d</sub>)** Where: - The left hand side of the equation is referred to as the *Posterior Odds*. - The right hand side of the equation is divided into two components: the *Likelihood Ratio* and the *Prior Odds*. ### Bayes’ Law/ Theorem Example: The image depicts a simple example of Bayes’ Theorem. It states: > ROBBERS: HONEST > **Prior Odds Ratio** 1:100 <= **Wrong!!!** > **Likelihood Ratio** 80:10 > **Posterior Odds Ratio** 8:100 This example shows that Bayes’ theorem can be used to update beliefs about an event. The prior odds ratio represents the odds of an event happening before any new information is considered. The likelihood ratio represents the amount of evidence supporting the occurrence of the event. The posterior odds ratio represents the updated odds of the event after considering new evidence. The image highlights the difference between prior odds and posterior odds. The example states that the prior odds ratio was assumed to be 1:100, which is incorrect as the actual ratio should be 8:100. This highlights how important it is to carefully consider all available prior information when using Bayes’ theorem.