The modern philosopher Nassim Nicholas Taleb says that one should never cross a river that is on average 4 feet deep. The intuition is clear: Averages are a borderline useless metric when the payoff structure (i.e. being in a certain depth of water) is non-linear. To put it another way, being in 8 feet of water is not twice as bad as being in 4 feet. Jensen’s Inequality captures this idea in a single elegant statement:
Let be a convex function and be an arbitrary random variable. Then
In the example above we could think of to be water depth and to be our relative discomfort. When pain is convex (disproportionately uncomfortable) with respect to some input, our expected pain is much worse than the pain of the average input.
This can be flipped into a more positive outcome. Think of how much more enjoyable it is to eat a single large meal in the evening than trying to micromanage your calorie intake with 10 smaller meals evenly spread out throughout the day. I guarantee your total enjoyment from all those little snacks won’t add up to be anywhere close to enjoyable as that dinner at the end of the day. Or to paraphrase Socrates, hunger is the best spice.
Try to think of more applications of Jensen’s Inequality. When else is the average utility greater than the utility of the average?