[Stats4EA] Expectations are not Outcomes
This is the first in what might become a bunch of posts picking out issues from statistics and probability of relevance to EA. The format will be informal and fairly bite-size. None of this will be original, hopefully.
Expectations are not outcomes
Here we attempt to trim back the intuition that an expected value can be safely thought of as a representative value of the random variable.
A Rademacher random variable X takes the value 1 with probability 1⁄2 and otherwise −1. Its expectation is zero. We will almost surely never see any value other than −1 or 1.
This means that the expected value might not even be a number the distribution could produce. We might not even be able to get arbitrarily close to it.
Imagine walking up to a table in a casino and betting that the next roll of a die will be 7⁄2.
Researchers create a natural language simulation model. Upon receiving a piece of text as stimulus it outputs a random short story. What is the expectation of the story?
Let’s think about the first word. There will be some implied probability distribution over a dictionary. Its expectation is some fractional combination of every word in the dictionary. Whatever that means, and whatever it is useful for, it is not the start of a legible story—and should not be used as such.
What is the expected length of the story? What would a solution to that problem mean? Could one, for example, print the expected story?
Distributions with very fat tails. For instance, the Cauchy distribution has an undefined expectation.
It is tempting to freely substitute an expectation in as a representative of a random variable. Suppose we used the following procedure in a blanket fashion:
We are faced with a decision depending on an uncertain outcome.
We take the expected value of the outcome.
We use the expectation as a scenario to plan around.
Step three is unsafe in principle—even if sometimes not in practice.
If there is a next time (the length of this series is currently fractional) I hope to touch on some scenarios less easily dismissed as the concerns of a pedant.