3 Ideas in 2 Minutes on Probabilistic Thinking
Occam’s Razor, Bayesian Thinking & the Linda Problem
I. Occam’s Razor
English philosopher William of Ockham on how to deal with competing hypotheses:
Simpler explanations are more likely to be correct; avoid unnecessary or improbable assumptions.
—William of Occam
II. Bayesian Thinking
Farnam Street on how English statistician Thomas Bayes thought about chance in the 18th century:
The core of Bayesian thinking (or Bayesian updating, as it can be called) is this: given that we have limited but useful information about the world, and are constantly encountering new information, we should probably take into account what we already know when we learn something new. […]
Consider the headline “Violent Stabbings on the Rise.” Without Bayesian thinking, you might become genuinely afraid because your chances of being a victim of assault or murder is higher than it was a few months ago. But a Bayesian approach will have you putting this information into the context of what you already know about violent crime.
You know that violent crime has been declining to its lowest rates in decades. Your city is safer now than it has been since this measurement started. Let’s say your chance of being a victim of a stabbing last year was one in 10,000, or 0.01%. The article states, with accuracy, that violent crime has doubled. It is now two in 10,000, or 0.02%. Is that worth being terribly worried about? The prior information here is key. When we factor it in, we realize that our safety has not really been compromised.
III. The Linda Problem
Consider the following scenario proposed by psychologists Daniel Kahneman and Amos Tversky in the 1980s:
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.
Which is more probable?
Linda is a bank teller.
Linda is a bank teller and is active in the feminist movement.
—Kahnemann & Tversky, Judgments of and by Representativeness
If you chose no. 2 you’ve fallen for the conjunction fallacy. It states that the probability of the conjunctions can never be higher than the probability of its conjuncts.
In other words, both events combined (Linda being a bank teller and active in the feminist movement) cannot be more likely than a single event on its own. No matter how plausible Linda’s ideological leanings may sound. 🐘
Have a great week,