# Probability interpretations: Examples

## Bet­ting on one-time events

Con­sider eval­u­at­ing, in June of 2016, the ques­tion: “What is the prob­a­bil­ity of Hillary Clin­ton win­ning the 2016 US pres­i­den­tial elec­tion?”

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. 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.

On 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.

On 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.

## A coin with an un­known bias

Sup­pose 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.

The coin is then flipped three times where we can see it. It comes up heads twice, and tails once: HHT.

The coin is then flipped again, where no­body can see it yet. An hon­est and trust­wor­thy ex­per­i­menter lets you spin a wheel-of-gam­bling-odds,noteThe 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.

On 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.

On 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.

To de­cide whether or not to take the bet, a fre­quen­tist might try to ap­ply an 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 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.

In this case, a sim­ple un­bi­ased es­ti­ma­tor is to guess that the coin’s bias $$f$$ is equal to the ob­served pro­por­tion of heads, or 23. 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. $$\1 is fair, meaning that it doesn't yield an expected profit, so we have no reason to take the bet. On a **subjectivist** view, we start out personally unsure of where the bias$$f$$lies within the interval [0, 1]. Unless we have any knowledge or suspicion leading us to think otherwise, the coin is just as likely to have a bias between 33% and 34%, as to have a bias between 66% and 67%; there's no reason to think it's more likely to be in one range or the other. Each coin flip we see is then [22x evidence] about the value of$$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}$$. When we start with a uniform [219 prior], observe multiple flips of a coin with an unknown bias, see M heads and N tails, and then try to estimate the odds of the next flip coming up heads, the result is [21c Laplace's Rule of Succession] which estimates (M + 1) : (N + 1) for a probability of$$\frac{M + 1}{M + N + 2}.$$In this case, after observing HHT, we estimate odds of 2 : 3 for tails vs. heads on the next flip. This makes a gamble that wins \2 on tails and loses \1 on heads a profitable gamble in expectation, so we take the bet. Our choice of a [219 uniform prior] over$$f$$was a little dubious — it's the obvious way to express total ignorance about the bias of the coin, but obviousness isn't everything. (For example, maybe we actually believe that a fair coin is more likely than a coin biased 50.0000023% towards heads.) However, all the reasoning after the choice of prior was rigorous according to the laws of [1bv probability theory], which is the [probability_coherence_theorems only method of manipulating quantified uncertainty] that obeys obvious-seeming rules about how subjective uncertainty should behave. ## Probability that the 98,765th decimal digit of$$\pi$$is$$0$$. What is the probability that the 98,765th digit in the decimal expansion of$$\pi$$is$$0$$? The **propensity** and **frequentist** views regard as nonsense the notion that we could talk about the *probability* of a mathematical fact. Either the 98,765th decimal digit of$$\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$$with 10% frequency in the long run. The **subjectivist** considers probabilities to just refer to their own uncertainty. So if a subjectivist has picked the number 98,765 without yet knowing the corresponding digit of$$\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$$either, and their friend offers a whimsical gamble that costs \1 if the digit is non-zero and pays \20 if the digit is zero, the Bayesian takes the bet. Note that this demonstrates a difference between the subjectivist interpretation of "probability" and Bayesian probability theory. A perfect Bayesian reasoner that knows the rules of logic and the definition of$$\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 prob­a­bil­ity the­ory 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.

Parents:

• Interpretations of "probability"

What does it mean to say that a fair coin has a 50% prob­a­bil­ity of com­ing up heads?

• Probability

The de­gree to which some­one be­lieves some­thing, mea­sured on a scale from 0 to 1, al­low­ing us to do math to it.