Introduction to Bayes’ Theorem

Introduction

• Last week we began with covering probability theory.

• Today we will continue talking about conditional probability, Bayes’ Theorem, and its application.

• After formal lecture, we will have an “introduction to R and Quarto” informal lecture.

Conditional Probability

• Conditional probability of an event A given that an event B has occurred is as follows

P[A|B] = \frac{P[A \cap B]}{P[B]},

• so long as P[B] > 0.

• Algebraically equivalent,

P[A \cap B] = P[A|B] \times P[B] \ \ \ \ \ \& \ \ \ \ \ P[B] = \frac{P[A \cap B]}{P[A|B]}

Conditional Probability

• Initech employees sometimes forget TPS cover sheets. In fact, those in charge claim that 30% of TPS reports are missing their cover sheet.

• Milton is famous for forgetting his cover sheets – when a cover sheet is missing, Milton is the responsible employee 60% of the time.

• What is the proability that a cover sheet is missing and Milton was the author?

Conditional Probability

• Initech employees are also late – 40% of the employees were late during the last quarter.

• Looking at individual employee records, we can see that Peter is late to work 80% of the time.

• If an employee shows up late, what is the probability that the employee in question is Peter?

Bayes’ Theorem

• The general idea of Bayes’ Theorem (and Bayesian analysis in general) is that we will update our beliefs based on new evidence.

\text{current belief} \to \text{new evidence} \to \text{new belief}

• Bayes’ Theorem:

P[A|B] = \frac{P[B|A]\times P[A]}{P[B]}

Bayes’ Theorem

• Bayes’ Theorem:

P[A|B] = \frac{P[B|A]\times P[A]}{P[B]}

• where
  • P[A|B] = posterior probability = probability that hypothesis A is true given the evidence B.
  • P[B|A] = likelihood = the probability of observing evidence B given that hypothesis A is true.
  • P[A] = prior probability = our initial belief about the probability of hypothesis A being true without considering new evidence.
  • P[B] = evidence probability = the overall probability of observing evidence B.

Bayes’ Theorem

• Initech leadership wants to revisit the missing cover sheet problem. They’ve discovered:
  • Milton is assigned 10% of TPS reports and forgets cover sheets 80% of the time.
  • Peter is assigned 70% of TPS reports and forgets cover sheets 10% of the time.
  • Samir is assigned 20% of TPS reports and forgets cover sheets 30% of the time.
• What is the probability that Milton forgot a cover sheet, given that one was missing?

Bayes’ Theorem

• Initech has two departments: Accounting and Engineering.

• 60% of TPS reports come from Accounting.

  • In the Accounting Department, Milton writes 25% of Accounting TPS reports but forgets the cover sheet 90% of the time, while other Accounting employees forget the TPS cover sheet 15% of the time.

• 40% of TPS reports come from Engineering.

  • In the Engineering Department, Peter writes 50% of Engineering TPS reports but forgets the cover sheet 20% of the time, while other Engineering employees forget the cover sheet 5% of the time.

• If a TPS report is found without a cover sheet, what is the probability that Milton was the author?

Bayes’ Theorem

• If a TPS report is found without a cover sheet, what is the probability that Milton was the author?

Bayes’ Theorem

• Your turn!

• If a TPS report is found without a cover sheet, what is the probability that Peter was the author?

Wrap Up

• No additional homework than what was in the slides from last week.

• This is “only” the first portion of class today.

• We will take a 15 minute break, then return for an Introduction to R, RStudio, and Quarto.

  • See .qmd file on Canvas for today’s activity.