> Is the posterior probability that you'll buy a mattress again shortly after buying one, a small chance? Yes, undoubtedly. But is it smaller than the prior probability of a random person who hasn't bought any mattress at all recently, suddenly being on the market for a mattress? Apparently not.

This makes no sense. If I just bought a mattress, the prior probability of a random person buying it is irrelevant. What's relevant is the posterior distribution of what people buy after buying a mattress. Yes, you might buy another one, but what about other bedroom accessories for example?

A mattress might not be the best example though, because what really matters here is the expected value from the posterior probability, which includes the price. So a more expensive product with a slightly lower probability of purchase might yield a higher expected value than cheaper products with higher probability.

Anyway, that's orthogonal to the discussion: my point is that it's the posterior distribution conditioned on the mattress purchase that should be modeled, not the comparison with random people that may or may not just have bought a mattress.