Project One

PROJECT ONE

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Bayes Rule

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Dealing with uncertainty requires people to think of the world in probabilities. And the foundation of probability theory is Bayes rule. But what exactly is Bayes rule?

Thomas Bayes was a Presbyterian minister who in the 18th century gave a very elegant solution to a problem of inverse probability. Without geeking out too much it gave rise to a much more general rule known as Bayes rule.

The formula can be written as follows.

f(θ|y)=f(y|θ)f(θ) / f(y)

To understand most things that use probability we need to understand Bayes rule. Not just applying the formula but really understanding the intuition behind it.

Let’s unpack what that means with a relevant (but fictitious) example.

Let’s say we are back in February of 2020, and we are trying to figure out what is the likelihood someone will die if they contracted COVID. Morbid I know, but bear with me. We believe at the time that the most similar disease we know has a death rate of 20%. That is our prior. We think, with no extra evidence that this will be the same.

So if a person comes into our ICU with COVID we can treat him like he has a 20% chance of dying.
Now let’s say he lives. What is the next person’s likelihood to die if he contracted COVID? Surely it’s less than 20%. But maybe you say that’s just one person. They could have been part of that 80% who survive. OK.

Now let’s say that 20 more people come in and none of them die. Will we still think that the likelihood that the next person coming in will die is 20%? What if 1000 come in next and 5 of them die? With each person coming in, we are updating our beliefs.

What we are doing is incorporating new evidence into our beliefs about the disease. We are essentially using Bayes rule. Not to do so would be ignoring evidence.

Two things happen here. One, with enough evidence the system is less and less dependent on our initial beliefs. That means that we use them to get started but then can acclimate to the evidence without relying on our seed or any one piece of evidence. Two, if the world changes (as we have seen in the past few months with consumer spending) this method incorporates and learns about that change.

Now why are we interested in this at Project One? Well we use some version of Bayes rule in every step to finally get at an expected return (and estimate of risk). This is important because model selection, risk optimization, portfolio balancing, even searching over thousands of leading indicators of consumer demand are all governed at some point by an update of our beliefs using Bayes rule.

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