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  text: '[summary: Consider evaluating, in June of 2016, the question:  "What is the probability of Hillary Clinton winning the 2016 US presidential election?"\n\n- On the **propensity** view, Hillary has some fundamental chance of winning the election.  To ask about the probability is to ask about this objective chance.\n- On the **subjective** view, saying that Hillary has an 80% chance of winning the election summarizes our *knowledge about* the election, or, equivalently, our *state of uncertainty* given what we currently know.\n- On the **frequentist** view, we cannot formally or rigorously say anything about the 2016 presidential election, because it only happens once.]\n\n## Betting on one-time events\n\nConsider evaluating, in June of 2016, the question:  "What is the probability of Hillary Clinton winning the 2016 US presidential election?"\n\nOn the **propensity** view, Hillary has some fundamental chance of winning the election.  To ask about the probability is to ask about this objective chance.  If we see a prediction market in which prices move after each new poll — so that it says 60% one day, and 80% a week later — then clearly the prediction market isn't giving us very strong information about this objective chance, since it doesn't seem very likely that Clinton's *real* chance of winning is swinging so rapidly.\n\nOn the **frequentist** view, we cannot formally or rigorously say anything about the 2016 presidential election, because it only happens once.  We can't *observe* a frequency with which Clinton wins presidential elections.  A frequentist might concede that they would cheerfully buy for \\$1 a ticket that pays \\$20 if Clinton wins, considering this a favorable bet in an *informal* sense, while insisting that this sort of reasoning isn't sufficiently rigorous, and therefore isn't suitable for being included in science journals.\n\nOn the **subjective** view, saying that Hillary has an 80% chance of winning the election summarizes our *knowledge about* the election or our *state of uncertainty* given what we currently know.  It makes sense for the prediction market prices to change in response to new polls, because our current state of knowledge is changing.\n\n## A coin with an unknown bias\n\nSuppose we have a coin, weighted so that it lands heads somewhere between 0% and 100% of the time, but we don't know the coin's actual bias.\n\nThe coin is then flipped three times where we can see it.  It comes up heads twice, and tails once:  HHT.\n\nThe coin is then flipped again, where nobody can see it yet.  An honest and trustworthy experimenter lets you spin a wheel-of-gambling-odds,%note:The reason for spinning the wheel-of-gambling-odds is to reduce the worry that the experimenter might know more about the coin than you, and be offering you a deliberately rigged bet.% and the wheel lands on (2 : 1).  The experimenter asks if you'd enter into a gamble where you win \\$2 if the unseen coin flip is tails, and pay \\$1 if the unseen coin flip is heads.\n\nOn a **propensity** view, the coin has some objective probability between 0 and 1 of being heads, but we just don't know what this probability is.  Seeing HHT tells us that the coin isn't all-heads or all-tails, but we're still just guessing — we don't really know the answer, and can't say whether the bet is a fair bet.\n\nOn a **frequentist** view, the coin would (if flipped repeatedly) produce some long-run frequency $f$ of heads that is between 0 and 1.  If we kept flipping the coin long enough, the actual proportion $p$ of observed heads is guaranteed to approach $f$ arbitrarily closely, eventually.  We can't say that the *next* coin flip is guaranteed to be H or T, but we can make an objectively true statement that $p$ will approach $f$ to within epsilon if we continue to flip the coin long enough.\n\nTo decide whether or not to take the bet, a frequentist might try to apply an [unbiased_estimator unbiased estimator] to the data we have so far.  An "unbiased estimator" is a rule for taking an observation and producing an estimate $e$ of $f$, such that the [4b5 expected value] of $e$ is $f$.  In other words, a frequentist wants a rule such that, if the hidden bias of the coin was in fact to yield 75% heads, and we repeat many times the operation of flipping the coin a few times and then asking a new frequentist to estimate the coin's bias using this rule, the *average* value of the estimated bias will be 0.75.  This is a property of the _estimation rule_ which is objective.  We can't hope for a rule that will always, in any particular case, yield the true $f$ from just a few coin flips; but we can have a rule which will provably have an *average* estimate of $f$, if the experiment is repeated many times.\n\nIn this case, a simple unbiased estimator is to guess that the coin's bias $f$ is equal to the observed proportion of heads, or 2/3. In other words, if we repeat this experiment many many times, and whenever we see $p$ heads in 3 tosses we guess that the coin's bias is $\\frac{p}{3}$, then this rule definitely is an unbiased estimator. This estimator says that a bet of \\$2 vs. $\\1 is fair, meaning that it doesn't yield an expected profit, so we have no reason to take the bet.\n\nOn a **subjectivist** view, we start out personally unsure of where the bias $f$ lies within the interval [0, 1].  Unless we have any knowledge or suspicion leading us to think otherwise, the coin is just as likely to have a bias between 33% and 34%, as to have a bias between 66% and 67%; there's no reason to think it's more likely to be in one range or the other.\n\nEach coin flip we see is then [22x evidence] about the value of $f,$ since a flip H happens with different probabilities depending on the different values of $f,$ and we update our beliefs about $f$ using [1zj Bayes' rule]. For example, H is twice as likely if $f=\\frac{2}{3}$ than if $f=\\frac{1}{3}$ so by [1zm Bayes's Rule] we should now think $f$ is twice as likely to lie near $\\frac{2}{3}$ as it is to lie near $\\frac{1}{3}$.\n\nWhen we start with a uniform [219 prior], observe multiple flips of a coin with an unknown bias, see M heads and N tails, and then try to estimate the odds of the next flip coming up heads, the result is [21c Laplace's Rule of Succession] which estimates (M + 1) : (N + 1) for a probability of $\\frac{M + 1}{M + N + 2}.$\n\nIn this case, after observing HHT, we estimate odds of 2 : 3 for tails vs. heads on the next flip.  This makes a gamble that wins \\$2 on tails and loses \\$1 on heads a profitable gamble in expectation, so we take the bet.\n\nOur choice of a [219 uniform prior] over $f$ was a little dubious — it's the obvious way to express total ignorance about the bias of the coin, but obviousness isn't everything. (For example, maybe we actually believe that a fair coin is more likely than a coin biased 50.0000023% towards heads.) However, all the reasoning after the choice of prior was rigorous according to the laws of [1bv probability theory], which is the [probability_coherence_theorems only method of manipulating quantified uncertainty] that obeys obvious-seeming rules about how subjective uncertainty should behave.\n\n## Probability that the 98,765th decimal digit of $\\pi$ is $0$.\n\nWhat is the probability that the 98,765th digit in the decimal expansion of $\\pi$ is $0$?\n\nThe **propensity** and **frequentist** views regard as nonsense the notion that we could talk about the *probability* of a mathematical fact.  Either the 98,765th decimal digit of $\\pi$ is $0$ or it's not.  If we're running *repeated* experiments with a random number generator, and looking at different digits of $\\pi,$ then it might make sense to say that the random number generator has a 10% probability of picking numbers whose corresponding decimal digit of $\\pi$ is $0$.  But if we're just picking a non-random number like 98,765, there's no sense in which we could say that the 98,765th digit of $\\pi$ has a 10% propensity to be $0$, or that this digit is $0$ with 10% frequency in the long run.\n\nThe **subjectivist** considers probabilities to just refer to their own uncertainty.  So if a subjectivist has picked the number 98,765 without yet knowing the corresponding digit of $\\pi,$ and hasn't made any observation that is known to them to be entangled with the 98,765th digit of $\\pi,$ and they're pretty sure their friend hasn't yet looked up the 98,765th digit of $\\pi$ either, and their friend offers a whimsical gamble that costs \\$1 if the digit is non-zero and pays \\$20 if the digit is zero, the Bayesian takes the bet.\n\nNote that this demonstrates a difference between the subjectivist interpretation of "probability" and Bayesian probability theory. A perfect Bayesian reasoner that knows the rules of logic and the definition of $\\pi$ must, by the axioms of probability theory, assign probability either 0 or 1 to the claim "the 98,765th digit of $\\pi$ is a $0$" (depending on whether or not it is). This is one of the reasons why [bayes_intractable perfect Bayesian reasoning is intractable]. A subjectivist that is not a perfect Bayesian nevertheless claims that they are personally uncertain about the value of the 98,765th digit of $\\pi.$ Formalizing the rules of subjective probabilities about mathematical facts (in the way that [-1bv] formalized the rules for manipulating subjective probabilities about empirical facts, such as which way a coin came up) is an open problem; this in known as the problem of [-logical_uncertainty].\n',
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