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I'm wondering about the meaning of ascribing probabilities to the outcomes of nonrepeating events. As a concrete example, here in the UK we're pollsters are currently predicting the result of the referendum on UK's membership in the European Union. They say things such as "Britain will stay in the EU with 70% probability", but I just don't understand the intuition behind this statement. If we had a referendum every week, and if we could assume that the individual outcomes were independent, then this would mean that, in about 70% of the cases, people would vote to stay. In reality, however, we will actually have just one referendum, so there is no way to talk about 70% of the outcomes. Thus, the number "70%" seems completely arbitrary to me and it doesn't seem to carry much more information than, say, 42!

I assume that my question is related to What does actually probability mean? and the relationship between the frequentist and Bayesian interpretation of probability. I've read the Wikipedia page mentioned in that article, and I think I understand the frequentist interpretation and find it intuitive; however, it seems to me that this interpretation relies on events repeating. Moreover, I just can't get my head around the Bayesian interpretation. So I would really appreciate it if someone could clearly explain the following points:

  • Does it at all make sense to talk of probability of outcomes of one-off events? Maybe people just use the word "probability" without really paying attention to its formal meaning?

  • Am I right in thinking that the frequentist interpretation requires repeating events?

  • Could someone please explain or point me to a nice paper/book explaining the intuition behind the Bayesian probability?

Please note that I'm not asking which interpretation is right; I'd like to understand the merits of either interpretation independently. Many thanks in advance!

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  • $\begingroup$ Minor observation: Take 10 balls, color 3 green and 7 red, place them in a box, shake, and take one out. If Britain based its decision on choosing a red ball, you would agree (perhaps) on the number $70\%$, even though this event is non-repeating. Of course, in principle, we could repeat the experiment over and over. $\endgroup$ – Michael Jun 15 '16 at 21:20
  • $\begingroup$ The actual statement about Britain in the news might be based on random samplings of voter opinions from a small sample of the population and/or intuitions gained based on observations of similar historical events. $\endgroup$ – Michael Jun 15 '16 at 21:22
  • $\begingroup$ I understand the second point. Thank you very much for the first comment: I'm finding it really illustrative! So is the interpretation of probability you're suggesting more frequentist or more Bayesian, and should I even care to understand the difference (in this specific care)? $\endgroup$ – Boris Jun 15 '16 at 21:32
  • $\begingroup$ I do not know enough about the "frequentist versus Bayesian" camps to label myself either way. Do we need to have a 2-party system here? Perhaps we should introduce a "Green party." $\endgroup$ – Michael Jun 15 '16 at 21:51
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There are a number of links you can Google, and most texts in probability discuss this. I remember a nice discussion in this direction in the intro of Bob Gallager's book Discrete Stochastic Processes. The main points I think are:

1) Probability does not need to have repeatable experiments. In the mathematical world, we just need to define an abstract set of possible outcomes and a corresponding probability measure that satisfies certain basic axioms.

2) To relate probability theory to the real world, the probability measures we use might be based on "relative frequency" concepts and real-world observations about repeated experiments. This is not required for the theory, but helps to explain why theoretical results are often useful in the real world.

3) The "Law of Large Numbers" is an interesting bridge between probability theory and relative frequency intuition. Within the theory itself, it is possible to define a notion of "independent and identically distributed experiments." The Law of Large Numbers says the time average over repetitions of independent and identically distributed experiments does indeed converge to the a-priori success probability of one such experiment.

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  • $\begingroup$ Thank you, I'll check the book out you're suggesting. $\endgroup$ – Boris Jun 15 '16 at 21:42
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Unless you want to invoke quantum mechanics, the actual probability of a Yes vote is either $0$ or $1$, we just don't happen to know yet which it is. What is being asserted as being $70\%$ may actually be a "subjective probability". Based on whatever evidence they may have gathered, the pollsters might judge that a bet on the outcome at odds 70 to 30 would be a fair bet. This might involve a Bayesian analysis, or it might just be a number plucked out of the air.

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  • $\begingroup$ Thank you, I always had the impression that such numbers are somewhat arbitrary. $\endgroup$ – Boris Jun 15 '16 at 22:33
  • $\begingroup$ What is meant by "the actual probability of a Yes vote" and why should it be either 0 or 1? $\endgroup$ – Michael Jun 15 '16 at 23:24
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    $\begingroup$ I mean, according to a deterministic view of the universe, either it will happen or it won't. We'll know which when the votes are counted. In principle (according to classical physics) if we could measure precisely enough the positions and velocities of all particles on Earth (or close enough to affect us by voting day) and integrate their equations of motion with a powerful enough computer, we could predict with certainty what the result would be. $\endgroup$ – Robert Israel Jun 16 '16 at 0:23
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    $\begingroup$ Now quantum mechanics actually says that outcomes are not completely determined: you could say the world's wave function on voting day is determined by the wave function now, but that may not be an eigenfunction of the observable "result of the vote", so the probability of a "Yes" outcome would not be $0$ or $1$. But this quantum mechanical probability, in an uncontrolled macroscopic situation such as ours, is practically impossible to measure, and certainly is not accessible to the pollsters. $\endgroup$ – Robert Israel Jun 16 '16 at 0:33
  • $\begingroup$ Thanks. With that perspective, my guess is that a deterministic approximation of all particles in the universe would only be accurate for a fraction of a second, not long enough to predict an election. So the quantum mechanics, and any other as-yet-unknown factors (including free will?) would be non-negligible. $\endgroup$ – Michael Jun 16 '16 at 23:35

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