Understanding Bayes’ Theorem and Its Application to Human Biases

Bayes’ Theorem is a fundamental concept in probability theory and statistics that describes how to update the probability of a hypothesis as more evidence or information becomes available. Named after the Reverend Thomas Bayes, this theorem has profound implications not only in statistics but also in decision-making processes in various fields, including medicine, finance, and even personal beliefs.

What is Bayes Theorem?

Before we get to the math of it (and we will). Let’s discuss how to understand Bayes in an intuitive way. Generally speaking, there are two camps in statistics, frequentists and Bayesian.

Frequentists calculate the probility of an outcome and do not update their belief system as part of the calculation. For example, if I flip a coin, most people would accept that it has a 50% probability landing as heads or tails. But imagine, I look at the coin and ask you is it still 50%? A frequentist will say, the probability doesn’t change, just because you looked at the coin, it’s still as likely to be heads as it is to be tails.

Bayesian perspective will say, well, from your perspective, the probability is 100% certain as heads or tails, but from my perspective, I can’t tell, so I’ll stick to 50%. The point is that you could update your belief system or calculation methods based on your understanding of the situation. If the person flipping a coin has a tell when he sees that it is heads, then I will have near 100% certainty of guessing whether it is heads or tails.

The Basics of Bayes’ Theorem

Bayes’ Theorem is mathematically expressed as:

P(H|E) = (P(E|H) x P(H) ) / P(E)

Where:

• P(H|E) is the probability of the hypothesis (H) given the evidence (E).
• P(E|H) is the probability of observing the evidence (E) given that (H) is true.
• P(H) is the prior probability of the hypothesis (H) being true before observing the evidence.
• P(E) is the probability of the evidence (E) under all possible hypotheses.

Bayesian Calculation: Timid Student in College

To further illustrate the practical application of Bayes’ Theorem, let’s consider a scenario involving a timid student in college. Suppose you meet a student who appears timid. This student is either a Math major or a Business major.

Most people would think that the probability the student is a Math major is significantly higher than being a business major. Given specific probabilities and the distribution of students in each major, we can use Bayes’ Theorem to determine which major the timid student is more likely from.

Problem Setup

Imagine we know the following:

1. Probability of a Math major being timid P(T|M): 90%
2. Probability of a Business major being timid P(T|B): 10%
3. Number of Math majors: 100
4. Number of Business majors: 1000

We want to find the probability that the timid student is a Math major given that they are timid P(M|T).

Step-by-Step Calculation

1. Prior Probabilities:

• Probability of meeting a Math major P(M) : 100 / 1100 ≈ 0.091 (9.1%)
• Probability of meeting a Business major P(B) : 1000/1100 ​≈ 0.909 (90.9%)

2. Likelihoods:

• Probability of a Math major being timid P(T|M) : 0.90
• Probability of a Business major being timid P(T|B): 0.10

3. Total Probability of a student being timid P(T) :

• P(T) = P(T|M) x P(M) + P(T|B) x P(B)
• P(T) = (0.90 x 0.091) + (0.10 x 0.909) )
• P(T) = 0.0819 + 0.0909
• P(T) = 0.1728

4. Posterior Probability P(M|T) :

• Using Bayes’ Theorem: P(M|T) = (P(T|M) x P(M)) / P(T)
• P(M|T) = (0.90 x 0.091) / 0.1728
• P(M|T) = 0.0819/0.1728
• P(M|T) ≈ 0.474

Interpretation of the Results

Even though the probability of being timid is much higher for Math majors than for Business majors, the fact that there are far more Business majors significantly impacts the overall probability. The posterior probability P(M|T) ≈ 0.474 suggests that given a timid student, there is approximately a 47.4% chance that the student is a Math major. This is less than 50%, meaning it is still more likely that the timid student is a Business major despite the higher likelihood of timidity among Math majors.

Applying Bayes’ Theorem to Human Beliefs

One of the classic applications of Bayes’ Theorem is in updating beliefs in light of new evidence. This is particularly relevant when considering biases in human understanding. People often rely on majority opinion or frequentist interpretations, which can lead to misconceptions and biases.

Case Study: Quran Verse 9:128-129

To illustrate this, let’s consider the example of Quran verses 9:128-129. A minuscule fraction of people believe these verses are not part of the original Quran, whereas the overwhelming majority believe they are authentic. Let’s denote:

• ( H ) as the hypothesis that “Quran verses 9:128-129 are not part of the original Quran.”
• ( E ) as the evidence or new information supporting this hypothesis.

Initially, the prior probability P(H) is very low because most people believe in the authenticity of these verses.

Bayesian Updating

Suppose new evidence ( E ) emerges that supports the hypothesis ( H ). According to Bayes’ Theorem, we need to update our belief in ( H ) based on this evidence.

Step-by-Step Calculation

1. Prior Probability P(H): The initial probability of the hypothesis (H). Given the minuscule fraction of people, let’s assume P(H) = 0.01 (1%).
2. Likelihood P(E|H) : The probability of observing the evidence ( E ) if (H) is true. For the sake of argument, let’s assume P(E|H) = 0.8 (80%).
3. Evidence Probability P(E) : The total probability of observing the evidence ( E ). This includes both the scenarios where ( H ) is true and where ( H ) is false. Let’s assume this is 0.05 (5%).
4. Calculate the Posterior Probability P(H|E) :

P(H|E) = (P(E|H) x P(H)) / P(E) = (0.8 x 0.01) / 0.05 = 0.008/0.05 = 0.16

After incorporating the new evidence, the probability of the hypothesis (H) being true increases to 16%. This is a significant update from the initial 1%.

What if there are 5 evidence observed that the verses were falsely injected, 10, 20. Each evidence unearthed will increase the probability that the hypothesis is true, that is, the verses are indeed false.

For 5 pieces of evidence:

P(H|E₅) = (0.8⁵ × 0.01) / 0.05⁵ = (0.32768 × 0.01) / 0.0000003125 = 10.49

To express it as a probability:

P(H|E₅) = 10.49 / (10.49 + 1) = 0.913 or 91.3%

For 10 pieces of evidence:

P(H|E₁₀) = (0.8¹⁰ × 0.01) / 0.05¹⁰ = 110.00

To express it as a probability:

P(H|E₁₀) = 110.00 / (110.00 + 1) = 0.99 or 99%

For 20 pieces of evidence:

P(H|E₂₀) = (0.8²⁰ × 0.01) / 0.05²⁰ = 1.209 × 10¹⁸

As a probability:

P(H|E₂₀) = 1.209 × 10¹⁸ / (1.209 × 10¹⁸ + 1) ≈ 1 or 100%

The video and articles below present this as evidence, increasing the count and probability. I’ll leave it to the reader to assess their significance.

The History Behind 9:128 – 9:129

https://qurantalk.gitbook.io/false-verses

Contrast with Frequentist Approach

A frequentist approach might simply look at the number of people who believe in the authenticity of the verses and conclude that the hypothesis (H) is highly unlikely without considering new evidence. This method does not adjust beliefs based on new data but rather relies on the majority’s belief.

Bayesian reasoning, on the other hand, provides a structured way to update our beliefs. It emphasizes the importance of new evidence and its impact on our understanding, irrespective of the initial or majority belief.

Overcoming Bias with Bayesian Thinking

Human cognition is prone to biases such as confirmation bias, where individuals favor information that confirms their pre-existing beliefs. Bayesian updating helps counteract this by quantitatively integrating new evidence into our belief system. This method encourages a more flexible and evidence-based approach to understanding, leading to more rational decision-making.

Bayes’ Theorem is a powerful tool that underscores the importance of updating our beliefs in light of new evidence. It provides a mathematical framework to navigate through biases and misconceptions, promoting a more accurate and nuanced understanding of the world. Whether it’s re-evaluating historical texts, making medical diagnoses, or investing in stocks, Bayesian thinking offers a robust way to refine our beliefs and decisions based on the continuous influx of information.