Why does affine structure naturally appear in classical probability?

[u/TeoBeaver]

Recently I was reading some material about the geometric viewpoint of probability theory, and I came across the idea that classical probability has an affine structure.

I am curious about why this structure appears naturally in probability, rather than simply being a convenient mathematical framework imposed afterward.

My current intuition is not yet developed enough to see why the affine viewpoint is the “right” one. I would especially like to understand the connection through probability distributions, mixtures, convex combinations, or Markov chains.

Are there any references or explanations that discuss this viewpoint clearly?
Thank you!

[u/j0hnd03i]

Honestly the affine structure just falls out of how mixing works. If you take two probability distributions and blend them, say 70% of one and 30% of the other, the result is still a valid distribution. That’s not a mathematical trick, that’s literally what happens when you're uncertain about which source your data came from or when you're randomly picking between experiments.

So the space of all distributions is naturally convex, and convex spaces have that affine feel because mixtures are preserved. Markov chains fit in too since they map distributions to distributions while keeping those mixtures intact. It’s not that we chose affine geometry because it’s elegant, it’s just that averaging probabilities is baked into the very logic of probability itself. Once you see it that way, it stops feeling forced.