Wait, IlyaShipitser—I think you overestimate my knowledge of the field of statistics. From what it sounds like, there's an actual, quantitative difference between Bayesian and Frequentist methods. That is, in a given situation, the two will come to totally different results. Is this true?

I should have made it more clear that I don't care about some abstract philosophical difference if said difference doesn't mean there are different results (because those differences usually come down to a nonsensical distinction [à la free will]). I was under the impression that there is a claim that some interpretation of the philosophy will fruit different results—but I was missing it, because everything I've been introduced to seems to give the same answer.

Is it true that they're different methods that actually give different answers?

Sorry, I didn't mean to imply that probabilities only apply to the future. Probabilities apply only to uncertainty.

That is, given the same set of data, there should be no difference between event A happening, and you having to guess whether or not it happened, and event A not having happened yet—and you having to guess whether or not it will happen.

When you say "apply a probability to something," I think:

"If one were to have to make a decision based on whether or not event A will happen, how would one consider the available data in making this decision?"

The only time event A happening matters is if it happening generated new data. In the Bob-Alice situation, Alice rolling a die in separate room gives zero information to Bob—so whether or not she already rolled it doesn't matter. Here are a couple of different situations to illustrate:

A) Bob and Alice are in different rooms. Alice rolls the die and Bob has to guess the number she rolled. B) Bob has to guess the number that Alice's die will roll. Alice then rolls the die. C) Bob watches alice roll the die, but did not see the outcome. Bob must guess the number rolled. D) Bob is a supercomputer which can factor in every infinitesimal fact about how Alice rolls the die, and the die itself upon seeing the roll. Bob-the-supercomputer watches Alice roll the die, but did not see the outcome.

In situations A, B, and C—whether or not Alice rolls the die before or after Bob's guess is irrelevant. It doesn't change anything about Bob's decison. For all intents and purposes, the questions "What did Alice roll?" and "What will Alice roll?" are exactly the same question. That is: We assume the system is simple enough that rolling a fair die is always the same. In situation D, the questions are different because there's different information available depending on whether or not Alice rolled already. That is, the assumption of a simple-system isn't there because Bob is able to see the complexity of the situation and make the exact same kind of decision. Alice having actually rolled the dice does matter.

I don't quite understand your "likely or not likely" question. To try to answer: If an event is likely to happen, then your uncertainty that it will happen is low. If it is not likely, then your uncertainty that it will happen is high.

(Sorry, I totally did not expect this reply to be so long.)

I'm having a hard time answering this question with "yes" or "no":

The event in question is "Alice rolling a particular number on a 6-sided die." Bob, not knowing what Alice rolled, can talk about the probabilities associated with rolling a fair die many times, and base whatever decision he has to make from this probability (assuming that she is, in fact, using a fair die). Depending on the assumed complexity of the system (does he know that this is a loaded die?), he could convolute a bunch of other probabilities together to increase the chances that his decision is accurate.

Yes... I guess?

(Or, are you referring to something like: If Alice rolled a 5, then there is a 100% chance she rolled a 5?)

A quantitative thing that indicates how likely it is for an event to happen.

This ends up being somewhat circular then, doesn't it?

Olbers' paradox is only a paradox in an infinite, static universe. A fininte, expanding universe explains the night sky very well. One can't use Olbers' paradox to discredit the idea of an expanding universe when Olbers' paradox depends on the universe being static.

Furthermore, upon re-reading MazeHatter's "The way I see it is..." comment, Theory B does not put us at some objective center of reality. An intuitive way to think about it is: Imagine "space" being the surface of a balloon. Place dots on the surface of the balloon, and blow the balloon up. The distance between dots in all directions expands. One can arbitrarily consider one dot as the "center," but that doesn't change anything.

I'm beginning to think that MazeHatter's comments do not warrant as much discussion as has taken place in this thread. =\

Trial-and-error.

There are, of course, inconsistencies that I'm unaware of: These are known unknowns. The idea, though, is that when I'm presented with a situation, any such relevant inconsistencies come up and are eliminated (either by a change of the foundation or a change of the judgement).

That is, inconsistencies that exist but don't come up aren't relevant.

An example—extreme but illustrative: Say an element of this foundational set is "I want to 'treat everyone equally'". I interview a Blue man for a job and, upon reflecting, think very negatively of him, even though he's more qualified than others. When I review the interview as if I were a 3rd party [ignorant of any differences between Blue people and regular people], I come to the conclusion that the interview was actually pretty solid.

I now have a choice to make. Do I actually want to treat people equally? If so, then I must think differently of this Blue man, his Blue people, give him this job, and make a very conscious effort to incorperate Blue people into my "everybody" perception. This is a change in judgement. Or, maybe I don't want to treat everyone equally—maybe I want to treat everyone who's not Blue equally. This is a change in foundation (but this change in foundation would have to coincide with the other elements in the foundation-set; or those, too, would change).

But, until now, my perception of Blue people was irrelevant.

Perhaps it would have been best to say: The process by which I make moral decisions is built to maximize for consistency. A lot goes into this. everything from honing the ability to look at a situation as a 3rd party, to comparing a decision with decisions I've made in the past. As a result, there's a very practiced part of me that immediately responds to nigh all situations with "Is this inconsistent?"

(An unrelated note: Are there things in this post I could have eliminated to get the same point across, but be more succint? I often feel as if my responses [in general] are too long.)

Haha, that's what I do.

If my cost is $14.32, I know $1.43 is 10%, and half of that is about $0.71, so the tip's $2.14 (though I tip 20%, which is even easier).

Yes and no. It's a different experience—like taking a bath and going swimming.