It is not just customers who bought X also bought Y, even though that is how many online retailers phrase their recommendations (see the Amazon.com screenshot given earlier); a real system cannot work like this. Why not? Because such a system would get fooled by very frequently bought items and would simply recommend that which is popular without any personalization.

For example, at a supermarket, many customers buy bread every time they shop, or almost every time (for the sake of argument, let us say that 50 percent of visits end with the purchase of bread). So, if you focus on any particular item, say, dishwasher soap, and look at what is frequently bought with dishwasher soap, you might find that bread is frequently bought with dishwasher soap. In fact, just by random chance, let's say that 50 percent of the times when someone buys dishwasher soap, they buy bread. However, bread is frequently bought with anything else, just because people frequently buy bread.

What we are really looking for is customers who bought X are statistically more likely to buy Y than the average customer who has not bought X. If you buy dishwasher soap, you are likely to buy bread, but not more so than the baseline. Similarly, a bookstore that simply recommended bestsellers no matter which books you had already bought would not be doing a good job of personalizing recommendations.