Bayes’ Theorem 🕵

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<a href="https://psychmethods.github.io/coursenotes/" target="_blank">Methods in Psychological Research</a>
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# Bayes' Theorem (Revisted)

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#  Introduction to Bayes' Rule

- Bayes' Rule is a foundational concept in statistics that provides a framework for updating our beliefs based on new evidence.

- It is widely used in fields such as medicine, finance, and social sciences to help make decisions under uncertainty.

- The general formula for Bayes' Rule helps to calculate posterior probabilities by combining prior knowledge and new data.

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## Bayes' Theorem Formula

- Bayes' Theorem allows us to "flip" conditional probabilities:

$$
P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)}
$$
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It relates:
- What we want to know: P(A|B)  
- What we know: P(B|A), P(A), and P(B)
]

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<img src="data:image/png;base64,#Bayes_files/figure-html/unnamed-chunk-2-1.png" width="50%" style="display: block; margin: auto;" />
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## Bayes' Theorem: Intuition

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- Starts with our initial belief about A: P(A)
- Updates that belief based on new evidence B
- Gives us a new, updated belief: P(A|B)

This mirrors the scientific process of updating hypotheses based on evidence!
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<img src="data:image/png;base64,#../img/bayes_science.png" width="65%" style="display: block; margin: auto;" />
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.footnote[van de Schoot, R., Depaoli, S., King, R. et al. Bayesian statistics and modelling. Nat Rev Methods Primers 1, 1 (2021). https://doi.org/10.1038/s43586-020-00001-2]

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# Gardner's Boy or Girl Paradox

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## The Paradox

Martin Gardner presented two seemingly similar problems in Scientific American (1959):

1. "Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?"

2. "Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?"

- Gardner gave the answers as 1/2 and 1/3 respectively, leading to much debate.

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## Problem 1: The Older Child is a Girl

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"Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?"

- Only the outcomes where the older child is a girl are considered:
  - Girl, Girl (GG)
  - Girl, Boy (GB)

P(both girls | older is girl) = 1/2
]

Sample space outcomes:

| Option 1 | Option 2 |
|----------|----------|
| Girl     | Girl     |
| Girl     | Boy      |

| Option 3 | Option 4 |
|----------|----------|
| Boy      | Girl     |
| Boy      | Boy      |

]

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## Problem 2: At Least One Boy

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"Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?"

- Possible outcomes:
  - Boy, Boy (BB)
  - Boy, Girl (BG)
  - Girl, Boy (GB)

P(both boys | at least one boy) = 1/3
]

.pull-right[
<table class="table" style="color: black; width: auto !important; margin-left: auto; margin-right: auto;">
<caption>All Outcomes</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> Child 1 </th>
   <th style="text-align:left;"> Child 2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Boy </td>
   <td style="text-align:left;"> Boy </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Boy </td>
   <td style="text-align:left;"> Girl </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Girl </td>
   <td style="text-align:left;"> Boy </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Boy </td>
   <td style="text-align:left;"> Boy </td>
  </tr>
</tbody>
</table>
]

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## The Paradox Explained

The key difference lies in the information given:

1. In the first problem, we know the order (older child is a girl), which eliminates two possibilities (BB and BG).

2. In the second problem, we don't know which child is a boy, so we only eliminate one possibility (GG).

This subtle difference in information leads to different sample spaces and thus different probabilities.

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## Bayesian Perspective

We can also approach these problems using Bayes' Theorem:

For Problem 1:

`$$P(GG \mid \text{older is G}) = \frac{P(\text{older is G} \mid GG) \cdot P(GG)}{P(\text{older is G})} = \frac{1 \times \frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}$$`

For Problem 2:

`$$P(BB \mid \text{at least one B}) = \frac{P(\text{at least one B} \mid BB) \cdot P(BB)}{P(\text{at least one B})} = \frac{1 \times \frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}$$`

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# Applying Bayes' Theorem

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## Example: Testing for a Rare Disease

Imagine a test for a disease that affects 1% of the population:
- 99% accurate for people who have the disease (sensitivity)
- 99% accurate for people who don't have the disease (specificity)

- We can use two approaches: a **tree diagram** to break down all possible outcomes and **Bayes' Theorem** mathematically.

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## Setting Up the Problem

Let's define our events:
- D: Having the disease (Case of the Mondays)
- T: Testing positive

We want to know: P(D|T)

We know:
- P(D) = 0.01 (1% of population has the disease)
- P(T|D) = 0.99 (99% sensitivity)
- P(T|not D) = 0.01 (1 - 99% specificity)

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## Applying Bayes' Theorem mathematically

`$$P(T) = P(T \mid D) \cdot P(D) + P(T \mid \neg D) \cdot P(\neg D) = 0.99 \times 0.01 + 0.01 \times 0.99 = 0.0198$$`

`$$P(D \mid T) = \frac{0.99 \times 0.01}{0.0198} \approx 0.50 \text{ or 50\%}$$`
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## Applying Tree Diagrams for Rare Disease Testing

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- A **tree diagram** is a graphical way to represent all possible outcomes of an event, making it easier to understand conditional probabilities.
- Let's define the population for simplicity:
  - **D**: Having the disease (1%)
  - **No D**: Not having the disease (99%)
- We will branch out from each state (having the disease or not) to show the possible test outcomes (Positive or Negative), which depend on sensitivity and specificity.
]

<img src="data:image/png;base64,#tree_diagram.png" width="90%" style="display: block; margin: auto;" />
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## Interpretation Using the Tree Diagram

- **Path 1 (Disease -> Positive Test):** Probability is **0.01 * 0.99 = 0.0099**
- **Path 2 (Disease -> Negative Test):** Probability is **0.01 * 0.01 = 0.0001**
- **Path 3 (No Disease -> Positive Test):** Probability is **0.99 * 0.01 = 0.0099**
- **Path 4 (No Disease -> Negative Test):** Probability is **0.99 * 0.99 = 0.9801**
]

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Using Bayes' Theorem to find P(Disease | Positive Test):
`$$P(D \mid T) = \frac{P(T \mid D) \cdot P(D)}{P(T)}$$`

`$$P(T) = P(T \mid D) \cdot P(D) + P(T \mid Not D) \cdot P(Not D) = 0.0099 + 0.0099 = 0.0198$$`

`$$P(D \mid T) = \frac{0.0099}{0.0198} \approx 0.50 \text{ or 50\%}$$`
]

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## Wrapping Up

- Both approaches—the tree diagram and Bayes' Theorem—help arrive at the same conclusion: there's a 50% chance that someone who tests positive actually has the disease.
- The tree diagram provides a visual breakdown, while Bayes' Theorem offers a mathematical perspective.
- This example demonstrates how rare diseases can lead to high rates of false positives, even when the test accuracy is high.

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# Bayesian Thinking in Psychology

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## Fun Example: Updating a Belief

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- A pet owner initially thinks their cat is limping due to an injury (80% sure).
- After observing the cat immediately stop limping when the door opens, how should they update their belief?

$$P(\text{injury} \mid \text{cat stops limping when door opens}) = $$
`$$\frac{P(\text{cat stops limping when door opens} \mid \text{injury}) \cdot P(\text{injury})}{P(\text{cat stops limping when door opens})}$$`
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<iframe src="https://www.youtube.com/embed/OHwQedAx8jM" width="533" height="300" frameborder="0" allowfullscreen="" data-external="1"></iframe>
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## Research Example: P-hacking

- P-hacking: Trying multiple analyses until you get a "significant" p-value
- Bayesian perspective: This is like ignoring your prior probabilities and only focusing on P(data | hypothesis)
- Better approach: Consider P(hypothesis) and calculate P(hypothesis | data)

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## Pros and Cons of Bayesian Approaches

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Pros:
- Allows incorporation of prior knowledge
- Provides direct probabilities of hypotheses
- Handles small samples and rare events well
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Cons:
- Selecting priors can be subjective
- Computationally intensive for complex problems
- Not as widely used (yet) in psychology
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Wrapping Up...

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## Simulations

- Simulations are used to understand probabilities in complex situations.

- **Blackjack**: Understanding dealer's odds.
  - [Dealer’s Odds in Blackjack](http://demonstrations.wolfram.com/DealersOddsInBlackjack/)

- **Rock Paper Scissors**: Using AI Player.
  - [Rock Paper Scissors with AI Player](http://demonstrations.wolfram.com/RockPaperScissorsWithAIPlayer/)

- **Texas Hold'em**: Exploring probabilities of winning.
  - [Probabilities of Winning in Texas Hold'em](http://demonstrations.wolfram.com/ProbabilitiesOfWinningInTexasHoldEm/)

- **Lottery (Adam Lamers)**: Running simulations to determine lottery odds.
  - [Lottery Simulator by Adam Lamers](http://adamlamers.com/lottery_simulator)

- **Lottery (Ian Neath)**: Another lottery simulation.
  - [Lottery Simulator by Ian Neath](https://memory.psych.mun.ca/models/lottery/index.shtml)

# Wrapping Up...