But don’t worry, we won’t be using any big words or equations just some simple examples and jokes to help you understand what it all means!

So let’s say you’re trying to predict something based on a bunch of data points (like stock prices, for example). You might use a fancy algorithm called “smoothing” to make your predictions more accurate. But did you know that some smoothing schemes are better than others when it comes to handling permutations?

Permutation what now? Well, let’s say you have two sets of data points: one with the prices in order (let’s call this Set A), and another set where the same prices are scrambled up (Set B). If your smoothing scheme is “permutation equivariant,” it means that when you apply it to both Sets A and B, you should get the same result.

Here’s an example: let’s say Set A looks like this: 10, 20, 30, 40, 50. If we use a simple smoothing scheme that just takes the average of each set of three numbers (called “moving averages”), our predicted price for the fourth data point would be (20 + 30 + 40) / 3 = 30.

But what if we scrambled up Set A to get Set B: 50, 10, 40, 30, 20? If our smoothing scheme is permutation equivariant, the predicted price for the fourth data point (which would be the third in Set B) should still be 30. And that’s exactly what happens!

Now let’s say we use a different smoothing scheme called “expected value” instead of moving averages. This time, our predicted price is based on the average of all possible output values (which might involve some fancy math). If we apply this to Set A and Set B, you might think that the results would be different but they’re not! That’s because expected value smoothing is also permutation equivariant.

So why does it matter if our smoothing scheme is permutation equivariant? Well, in some cases (like when we have missing data or errors in our measurements), the order of our data points might not be important but the values themselves are still meaningful. By using a permutation equivariant smoothing scheme, we can make more accurate predictions without worrying about whether the data is scrambled up or not!

And that’s all there is to it! Permutation equivariance might sound fancy and complicated, but in reality it just means that our smoothing schemes are fair and consistent no matter how we shuffle around our data points.