Jensen's inequality

Okay kiddo, let me explain Jensen's inequality like you are 5 years old!

Picture a number line with some dots on it. These dots represent numbers. Now, let's say you want to know the average of these numbers. You can add up all the dots and divide by how many dots you have.

But what if you have dots that are really far away from each other? The average might not be a good representation of what the majority of the dots look like.

That's where Jensen's inequality comes in! It tells us that if we have a function (like the average one we just did), and we apply it to some dots, the result will be bigger than if we first group the dots together and then apply the function.

Let me give you an example. Say you have a set of numbers that are all between 0 and 10. You want to find the average of these numbers, but you also know that some of them (let's say the bigger ones) are too far away from the others.

So, you group the numbers that are close together, finding the average within each group, and then find the average of those averages. Jensen's inequality tells us that this average of averages will be smaller than the average of all the numbers together.

See, isn't Jensen's inequality easy to understand when you put it in terms of dots and averages? It's a useful tool in math, and it helps make sure we get accurate results when working with data that isn't evenly distributed.