Bayesian Thinking: A Primer

In the 17th century, mathematician and philosopher Thomas Bayes developed a way of thinking that has been both misunderstood and misused for centuries.

In this article, we will explore what Bayesian thinking is, why it’s so powerful, how it can be used to make better decisions and understand the world around us better.

1. What is Bayesian thinking

Bayes theorem provides an equation for updating a hypothesis based on new evidence. Bayesian thinking has been successfully applied to fields outside of statistics including education, philosophy, economics and law. To put it simply, it's a mental model which allows you which allows you to adapt your thinking reactively to new evidence.

“Critical thinking is an active and ongoing process. It requires that we all think like Bayesians, updating our knowledge as new information comes in.” Daniel J. Levitin,

While Alan Turing is most known for using the Bayes theorem to crack the German Enigma code during World War II, the theory has a wide range of applications in a wide range of disciplines in today's modern world. However, the understanding that underpins this seemingly complex idea (like with all arithmetic and statistics) is actually rather straightforward. In truth, we are continuously implementing this theorem in our daily lives without even knowing it.

In the 17th century, mathematician and philosopher Thomas Bayes developed a way of thinking that has been both misunderstood and misused for centuries.

In this article, we will explore what Bayesian thinking is, why it’s so powerful, how it can be used to make better decisions and understand the world around us better.

This mental model works in contrast to another statistical method called frequentism which determines how frequently some event occurs within a certain time period and then predicts future events using this information.

2. How does Bayesian thinking work?

Many professions, such as physicians and judges, are expected to make critical decisions based on data. When evaluating a positive mammography screening, for example, Bayesian conclusions are frequently required. Several empirical studies have revealed poor judgments and even cognitive illusions among medical professionals who need to apply Bayesian thinking (Hoffrage et al., 2000; Operskalski and Barbey, 2016).

When analyzing evidence based on a fragmented DNA sample, jury mis-convictions or acquittals may result from a lack of statistical comprehension in general and erroneous Bayesian reasoning in particular. As in the instance of Sally Clark, these faults might put innocent lives in jeopardy (Schneps and Colmez, 2013; Barker, 2017).

Statistical data in probability format, i.e. fractions or percentages representing the possibility of a certain occurrence, such as the prevalence of breast cancer in the population, is frequently presented to these professions.

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3. Examples of Bayesian thinking

Bayesian thinking is a form of statistical reasoning. It involves calculating and updating probabilities as new information becomes available to make the best possible predictions.

Bayes’ Theorem states that: “The probability of an event happening A, given that it has happened B, is equal to the probability of the event happening B, given that it has happened A times the probability of both events occurring together divided by the sum total for all possibilities.”

For example, if you have observed 100 people with red hair and blue eyes then you would be justified in assuming that there are more people with red hair than without red hair because your sample size is large enough to justify this assumption. If on occasion you observe a person with red hair and blue eyes, you would be justified in believing that there are also people without red hair who have blue eyes because there isn't enough data to make an accurate prediction.

Three statistics are supplied when Bayesian conclusions are required: the base rate (or a priori probability), sensitivity, and false alarm rate.

Take, for example, the probability of heroin addiction:

A random person in a population has a 0.01 percent probability of being addicted to heroin (base rate). If someone randomly selected from this group is addicted to heroin, they will have new needle pricks (sensitivity). If a non-heroin user is picked at random from this group, there is a 0.19 percent chance of having fresh needle pricks (false alarm rate). What is the likelihood that a person with fresh needle pricks is addicted to heroin (posterior probability)?

In reality, just 4% of participants in a comprehensive meta-analysis (McDowell and Jacobs, 2017) are capable of making the necessary conclusions to arrive at the correct conclusion. The majority of people struggle, which can lead to catastrophic errors. The most common problem is a misapprehension of the base rate.

The outcome, determined using the Bayes method, is just 5% given the probabilistic information (low base rate, high sensitivity, and low false alarm rate). To most people, this appears to be a shockingly low figure.

However, there are several limitations to Bayes theorem.

It doesn't tell you anything new.

It may not be applicable in every situation (i.e., there needs to be some existing evidence).

It also doesn't tell you how likely it is that a hypothesis will be true - Bayes theorem only tells us the probability of some event given our existing evidence.

Conclusion

Bayesian thinking is a type of cognitive reasoning that has been around for centuries. The idea behind Bayesian decision-making is to update your beliefs about the world based on new information you’ve encountered. If you are trying to decide between two options, it means updating your belief in one option if there's evidence against it and increasing your belief in the other because of what you've learned.

“Under Bayes' theorem, no theory is perfect. Rather, it is a work in progress, always subject to further refinement and testing.” Nate Silver

It also means changing how much uncertainty or risk you think each choice entails (maybe one seems riskier than before). You can apply this type of reasoning in many aspects of life: from solving math problems to deciding whether or not to buy something on Amazon.