Matt Lane← Writing
Full article · Patterns of Perception · August 25, 2024

How to Update Your Cognitive and Cultural Operating System

Using Probabilistic Logic Every piece of evidence that affects our confidence in an outcome changes its probability and should influence our actions. Probabilities reflect our ignorance of the world, not the world itself. As new information reduces our ignorance, it alters the probability.

The key idea is that you must constantly update your beliefs as you get new information, rather than relying solely on your initial assumptions or the latest piece of data. This is core to doing critical thinking right and how to reduce bias.

Probability is a fundamental concept in mathematics that helps us understand and predict the likelihood of events occurring. In this article, we'll explore three important types of probability: conjunctive, disjunctive, and conditional. We'll break down each type, provide real-world examples to illustrate how these concepts apply to everyday life, and discuss common fallacies associated with each.

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Conjunctive probability refers to the likelihood of two or more events occurring together. For two events A and B, the probability of their conjunction, P(A∩B), is calculated as follows:

P(A∩B)=P(A)×P(B)

P(A∩B)=P(A)×P(B∣A), where P(B∣A) is the conditional probability of B given A.

The conjunction fallacy occurs when people incorrectly judge the probability of a conjunction P(A∩B) to be higher than the probability of a single event P(A).

Example: Linda is described as politically active and socially conscious. People might mistakenly judge P(Bank Teller∩Feminist) (Linda is both a bank teller and a feminist) as more likely than P(Bank Teller) (Linda is a bank teller).

However, mathematically: P(Bank Teller∩Feminist)≤P(Bank Teller) The probability of the conjunction cannot exceed the probability of its individual components.

Disjunctive probability measures the likelihood of at least one of several events occurring. For two events A and B: P(A∪B)=P(A)+P(B)−P(A∩B)

P(A∩B)=P(A)×P(B), which simplifies the calculation.

Suppose there’s a 40% chance of visiting a museum (P(Museum)=0.4) and a 50% chance of going to the beach (P(Beach)=0.5). If these events are independent: P(Museum∪Beach)=0.4+0.5−(0.4×0.5)=0.7 There’s a 70% chance of visiting either the museum or the beach.

Conditional probability quantifies the likelihood of an event occurring given that another event has already occurred. For events A and B:

The probability of rain tomorrow is 30% (P(Rain)=0.3). If clouds are present (P(Clouds)=0.6), the probability of rain rises to P(Rain∣Clouds)=0.5.

Bayes’ theorem allows for further refinement of probabilities when new evidence arises:

Bayes’ theorem is a foundational tool for updating beliefs based on new evidence:

Consider a spam filter analyzing emails containing the word "free":

Using Bayes’ theorem:

where: P(Free)=P(Free∣Spam)⋅P(Spam)+P(Free∣Not Spam)⋅P(Not Spam)

Plugging in the values: P(Spam∣Free)≈0.087 Despite the association between "free" and spam, the low prior probability of spam keeps the posterior probability relatively low.

  1. Neglecting Prior Probabilities (Base Rate Fallacy): People often focus on the likelihood (P(B|A)) and ignore the prior probability (P(A)). In our spam example, focusing only on the high percentage of spam emails containing "free" (95%) while ignoring the low overall prevalence of spam (1%) leads to overestimating the probability of an email being spam.
  2. Confusion of the Inverse: This fallacy involves mistaking P(A|B) for P(B|A). In our example, it would be like assuming the probability of an email being spam given it contains "free" (8.76%) is the same as the probability of an email containing "free" given it's spam (95%).
  3. Overconfidence in New Information: People tend to give too much weight to new evidence and undervalue prior probabilities. In the spam filter context, this might lead to overreacting to the presence of a single word like "free" without considering other factors.
  4. Failure to Consider Alternative Hypotheses: When applying Bayes' theorem, it's crucial to consider all possible hypotheses. In our example, we considered both spam and legitimate emails. Failing to do so can lead to incorrect probability assessments.

The following fallacies and concepts all relate to our human tendency to see patterns and meaning where none exist, or to misinterpret the true nature of random events and probability. They highlight the importance of critical thinking, understanding basic probability, and being aware of our cognitive biases when interpreting events and data.

Cluster Illusion:

The cluster illusion is the tendency to see patterns or clusters in random events where none exist. This is a type of cognitive bias where people perceive meaningful patterns or trends in data that are, in fact, completely random.

Example: Imagine flipping a coin 100 times. You might see sequences like "HHHHHH" (six heads in a row) and think it's a significant pattern. However, in truly random sequences, such "clusters" are not only possible but are expected to occur occasionally.

Opportunities for Coincidences:

The concept of "opportunities for coincidences" relates to the fact that in our daily lives, we encounter numerous events and possibilities, making seemingly unlikely coincidences more probable than we intuit.

Example: Consider the "birthday problem." In a room of just 23 people, there's a 50% chance that at least two people share a birthday. This seems counterintuitive, but it's due to the large number of possible pairs in the group (253 pairs in 23 people).

Sharpshooter Fallacy:

The sharpshooter fallacy, also known as the Texas sharpshooter fallacy, occurs when differences in data are ignored, but similarities are emphasized. The name comes from the image of a shooter firing at a barn wall and then painting a target around the largest cluster of hits, making it appear as if they were aiming for that spot all along.

Example: A company might launch ten products and find that one is selling particularly well in a specific demographic. They might then claim they "targeted" that demographic, ignoring the fact that they had no such intention initially and that the other nine products failed.

Confirmation Bias:

While not strictly a probability fallacy, confirmation bias often plays a role in these misinterpretations. It's the tendency to search for, interpret, favor, and recall information in a way that confirms or supports one's prior beliefs or values. In other words, once we expect a pattern, we seek out examples and ignore the counterexamples.

Example: If you believe in astrology, you might remember all the times your horoscope seemed accurate while forgetting or dismissing the times it wasn't.

Gambler's Fallacy:

This fallacy involves believing past events influence the likelihood of independent events. For example, expecting a coin toss to land heads because of a previous streak of tails. Independent events do not affect each other.

Hot Hand Fallacy:

This is the belief that a person who has experienced success with a random event has a greater probability of further success in additional attempts.

Example: In basketball, players and fans often believe a player's chance of hitting a shot is greater following a hit than following a miss. Studies have shown this to be a fallacy in truly random sequences.

As powerful as our cognitive systems are in systematically isolating and reducing problems, in the complex world we live in today, and especially when we are making decisions in high-risk situations, we must know when to discount them and hand our reasoning and intuition over to instruments, e.g., the tools of probabilistic logic and critical thinking, which stretch our ability to reason beyond what nature gave us.

A note on hypothesis testing