Bayes’ Theorem and its Implications in Behavioural Economics
Understanding one of the most important theorems in probability
An individual has been described by a friend as follows: “Steve is very shy and withdrawn, invariably helpful but with very little interest in people or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.” Is Steve more likely to be a librarian or a farmer?
In 1974 Daniel Kahneman and Amos Tversky provided incredible insight into how we think. They showed us just how wrong human rationale can be and how flawed our intuition can be. They researched human judgments and, more pointedly, when these judgements irrationally contrasted what the balance of probability suggested.
Based on the fact that Steve is described as meek and tidy, most people would state that Steve is a librarian, drawing from their associative memory. But almost no one would consider the ratio of farmers to librarians: there are 20x more farmers than librarians in the United States. It doesn’t matter if someone knew it or not — but did they even consider the importance of it? As Grant Sanderson puts it, “Rationality is not about knowing facts; it’s about recognizing which facts are relevant”.
Let’s say that we’re looking at a sample of 10 librarians and 200 farmers (abiding by the above ratio) and you estimate that 60% of librarians and only 10% of farmers match Steve’s description. In our sample, that would mean that 6 librarians would match the description. Compare that to 20 farmers. Even though you think that the probability that a librarian matches the description is 6x that of a farmer, the actual amount of librarians matching the description is 70% less than the number of farmers. The true probability that Steve is a librarian given the description is 6/26, or 23.01%. The upshot is that our intuition is often irrational when viewed against the probability; Bayes’ Theorem allows for the calculation of the probability of an event (Steve is a farmer, in this case) while accounting for the known evidence (Steve is a meek and tidy soul).
New evidence does not determine your hypothesis; rather, it should be used to update it. This is the core idea behind Bayes’ Theorem. This theorem is fundamental in conditional probability, allowing for the description of the probability of an event occurring based on known prior conditions. Seeing the evidence restricts the scope of possibilities.
In the Bayesian interpretation of probability, the probability is viewed as a ‘degree of belief’; the theorem links the degree of belief for a hypothesis prior to accounting for evidence and after. For instance, prior to flipping a coin, it is believed that the chance of getting heads is 2x as likely as tails, and this belief is associated with a 50% certainty. After numerous trials, the degree of belief may increase, decrease, or remain unchanged depending on the results.
You can apply Bayes’ Theorem whenever you have a hypothesis or belief and you have observed some evidence and you want to see the probability your hypothesis holds given the evidence is true. Mathematically, this is represented as P(H|E). Bayes’ Theorem states the equation which allows us to determine this probability:
Let’s analyze this formula in terms of our Steve scenario. P(H|E) is the probability that Steve is a librarian given the description. P(H) is the probability that Steve is a farmer (4.76%), which comes from our initial ratio of 20:1. This is known as the prior, as this is the probability our hypothesis holds prior to any evidence. P(E|H) is the probability that Steve matches the description given that he is a librarian i.e. this is the proportion of librarians that fit the description (60% in this case). This value is known as the likelihood. P(E) is the probability that Steve matches the description (12.38%). It is the total probability of seeing the evidence. This is illustrated as:
Solving this equation gives us that P(Farmer|Description) is 0.231, or 23.1% as we previously established. This probability is known as the posterior. Astutely put, this is the probability of a librarian fitting the description out of the total number of people who fit the description. It may not always be clear what the probability of the evidence existing is, so it can be further broken down as
Our interpretation of Bayes’ Theorem is the frequentist interpretation, where the probability represents a proportion of the total outcomes. We can see that at every step we’re limiting our view and looking at a proportion of the entire sample. In fact, probability itself stems from the mathematics of proportions. Bayes’ Theorem can be thought of as the proportion of the number of cases where the hypothesis is true out of the number of cases where the evidence is true.
The upshot is that Bayes’ Theorem allows us to mathematically quantify and calculate the concept of altering the hypothesis (beliefs). Bayes’ Theorem has a myriad of applications, from artificial intelligence to medicine to treasure hunting. One of the most famous use cases is testing for false positive and false negatives. Perhaps most crucially, however, it provides a medium to reframe how you think about thought itself.
Kahneman and Tversky did a similar experiment where they discovered what is known as the conjunction fallacy. The conjunction fallacy is a logical fallacy that arises when it is presumed that specific conditions are more probable than a single general one. Their reasoning for this fallacy was the fact that our judgment is made heuristic representativeness. Here’s the situation:
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
Is it more likely that Linda is a bank teller or a bank teller that is active in the feminist movement? 85% of participants chose the latter, even though that is a subset of the first option. This is due to the fact that option 2 appears more representative of Linda, even though it is less mathematically plausible.
Bayes’ Theorem is a marriage of mathematics and common sense. It allows us to see the probability of our hypothesis being correct, provided some evidence. It also shows us how unreliable our intuition can be. Bayes’ Theorem is one of the most important theorems in probability, allowing for our hypothesis to be updated with the discovery of relevant data.
Bayes’ Theorem is a formula that allows us to quantify a belief, providing rationality to our theories. Evidence should not determine your beliefs — rather, it should update them.
