A friend of mine was recently traveling overseas and encountered a rather stressful 24 hours during which he thought the keys to his apartment had been stolen and he had potentially been robbed. To help calm his nerves, we stepped through a Bayesian probability analysis which showed that – despite his keys being missing – he was unlikely to have been robbed.
First the situation: while my friend was abroad, he had another person staying in his apartment. He employed the use of a lockbox out the front of his apartment to share his keys between his houseguest and his cleaner that came weekly. Yet when the cleaner last visited there were no keys in the lockbox. Furthermore, his houseguest assured him that the keys were indeed left in the lockbox.
We can use a framework called Bayes’ Theorem to understand how the new information of the keys disappearing impacts the likelihood of a robbery. (For a great explanation of Bayes’ Theorem, see Nate Silver’s excellent book: The Signal and the Noise).
To use Bayes’ Theorem, we need to make three input estimations:
- The probability of a robbery prior to the new information that the keys are missing (known as the “base rate”): statistically, 1% of apartments in the area are robbed annually. We conservatively assumed a 5% probability given the keys were left in a lockbox out the front of the apartment.
- The probability that the keys would go missing in a world in which a robbery took place: we assumed 80% here given that this would be the easiest way for a robbery to occur.
- The probability that the keys would go missing in a world in which the robbery did not take place: we assumed 30% here given it would have to mean that either the cleaner or the houseguest were mistaken.
With these inputs, Bayes’ Theorem tells us that the starting probability of a robbery of 5% increases to 12% – or a one in eight chance – given the new information that the keys had gone missing. This was still a very low chance that my friend had been robbed and helped serve to calm his nerves. Naturally, his mind was imagining the worst which was quite distressing. But analysing the situation rationally suggested the emotional stress was likely unwarranted.
On the next day, it turned out by chance, the real-estate agent of my friend’s apartment needed to make a visit to the premises to conduct an inspection. Following the inspection, the agent said the place looked immaculate. Given this new information, we again updated our probability of a robbery.
For the inputs, we used the following:
- For the base rate we used 12%, based on the outcome of the prior analysis.
- For the probability that the apartment was immaculate in a world in which a robbery took place, we assumed 15%. After all, if someone went to all the trouble of robbing the apartment, why would they take so much care in not messing it up?
- For the probability that the apartment was immaculate in a world in which a robbery did not take place, we assumed 100%. No robbery means the apartment is as it was left.
As shown above, with these inputs, we can use Bayes’ Theorem to calculate that the probability that the apartment had been robbed, given the new information that it was immaculate, had fallen from 12% to just 2%. At a 2% probability, it was very unlikely there was any robbery and this certainly calmed the nerves of my friend.
(As it turned out, the apartment was not robbed. The houseguest had mistakenly thought he had returned the keys to the lockbox, but had inadvertently taken them with him).
The point of this true story is to demonstrate the correct way to update probabilities when new information presents itself. The key point to note is that the base rates typically dominate the equation. So if the base rate is low, one would need a lot of new compelling evidence to lift it to any high-lever probability.
Investors – whether they realize it or not – are constantly estimating the probabilities of future events happening. And when new information presents itself, these probabilities are updated in our minds. The only problem is: an untrained human mind is typically not great at intuitively updating these probabilities. To train your mind, we recommend learning about Bayes’ Theorem and putting it into practice. This is a requirement for all members of the Montaka research team.