Let (\Omega, \mathcal{F}, P) be a probability space, and let X: \Omega -> S be a random variable taking values in some measurable space (S, \mathcal{S}).
Then the expectation is \int X(\omegs)dP
In computer science terms, do an experiment with every possible random seed and average the outcome (set \Omega to be the set of all seeds, and set P to be the uniform measure on them).
Probability is far from clear. Very briefly, there are two main camps:
1. Bayesian probability is about degrees of belief. But that's always subjective and belief about what, if not probability? It's circular.
2. Frequentist probability is about, after X >> 1 runs of an experiment, an outcome with odds of Y occurs Y/X times. But it's only exact with an infinite number of runs, which never happens. And what's the odds of exactly Y x 1000 outcomes after 1000 runs? Again, that's circular.
My favourite way to think about probability is the multiverse kind:
3. Assuming there are an infinite number of fungible identical worlds, if a coin flip has 50% of heads, it means observers in exactly half the worlds see heads. However, this isn't actually probability at all - from a god's eye view it's objectively certain what happens.
Your "3" is a Bayesian view. Specifically, from the Jaynesian school, which views probability as ignorance.
When we can't calculate which of those world's we're in, we express our remaining uncertainty with probability. The connection to subjective "beliefs" is recognizing that these probabilities are all in our own heads. Believing otherwise is the "mind projection fallacy"; in reality -- as you noted -- these things are certain from the god's eye view, and we fall somewhere in between that and total ignorance/entropy.
(I'm not a physicist, but I know some use the Many Worlds interpretation to apply this determinism even to quantum physics.)
E.T. Jaynes fleshes out his worldview in "Probability Theory: The Logic of Science", which was published posthumously in 2003.
Dear downvoters: Even if we assume an infinite number of (real or imaginary) universes, what does “in exactly half of the worlds" mean? This definition doesn’t seem at all less problematic that the usual ones.
I think "infinite" is just sloppy language. If every possible universe exists in some sense, that is a large number, but not infinite - because nothing about a universe has infinite precision. Thus, "half" would still mean something.
Lets not forget the non Kolmogorovian notion of probability - or quantum mechanics. I personally believe that we would want to accommodate a more generalized notion of probability to significantly improve our statistical models of the world. You certainly hint at it in #3
I don't even understand how the frequentist view is a valid alternative. It always seemed to me like either you are honest about your priors, and use Bayesian logic to take them into account, or you sweep it under the rug. Lying to yourself always produces bad results, is my overriding heuristic. But I'm not good at math.
As far as we know, there is no underlying theory of probability for them to be techniques of. So maybe they are equivalent in some sense, but on the face of it, they are separate ideas.
Probability was understood long before computers, true, but it waited for Kolmogorov’s axiomatic formulation to actually become the coherent field of math that it is today, rather than a hodge lodge of definitions, tricks and theorems.
And that only happened in 1933, which is around the time that computers became a thing. Not general purpose ones yet - I agree it was before computers were widespread, but definitely not far before they were a thing.
Finally! I thought I was alone (and stupid) for thinking like this.
Is there any literature or any meta-work that discusses the notion of probability itself? What is expectation? What is dependence?