7.2: What Does Probability Mean?
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Let’s start with the first of these questions. What is “probability”? It might seem surprising to you, but while statisticians and mathematicians (mostly) agree on what the rules of probability are, there’s much less of a consensus on what the word really means . It seems weird because we’re all very comfortable using words like “chance”, “likely”, “possible” and “probable”, and it doesn’t seem like it should be a very difficult question to answer. If you had to explain “probability” to a five year old, you could do a pretty good job. But if you’ve ever had that experience in real life, you might walk away from the conversation feeling like you didn’t quite get it right, and that (like many everyday concepts) it turns out that you don’t really know what it’s all about.
So I’ll have a go at it. Let’s suppose I want to bet on a soccer game between two teams of robots, Arduino Arsenal and C Milan . After thinking about it, I decide that there is an 80% probability that Arduino Arsenal winning. What do I mean by that? Here are three possibilities…
- They’re robot teams, so I can make them play over and over again, and if I did that, Arduino Arsenal would win 8 out of every 10 games on average.
- For any given game, I would only agree that betting on this game is only “fair” if a $1 bet on C Milan gives a $5 payoff (i.e. I get my $1 back plus a $4 reward for being correct), as would a $4 bet on Arduino Arsenal (i.e., my $4 bet plus a $1 reward).
- My subjective “belief” or “confidence” in an Arduino Arsenal victory is four times as strong as my belief in a C Milan victory.
Each of these seems sensible. However they’re not identical, and not every statistician would endorse all of them. The reason is that there are different statistical ideologies (yes, really!) and depending on which one you subscribe to, you might say that some of those statements are meaningless or irrelevant. In this section, I give a brief introduction the two main approaches that exist in the literature. These are by no means the only approaches, but they’re the two big ones.
frequentist view
The first of the two major approaches to probability, and the more dominant one in statistics, is referred to as the frequentist view , and it defines probability as a long-run frequency . Suppose we were to try flipping a fair coin, over and over again. By definition, this is a coin that has P(H)=0.5. What might we observe? One possibility is that the first 20 flips might look like this:
T,H,H,H,H,T,T,H,H,H,H,T,H,H,T,T,T,T,T,H
In this case 11 of these 20 coin flips (55%) came up heads. Now suppose that I’d been keeping a running tally of the number of heads (which I’ll call N H ) that I’ve seen, across the first N flips, and calculate the proportion of heads N H /N every time. Here’s what I’d get (I did literally flip coins to produce this!):
| number.of.flips | number.of.heads | proportion |
|---|---|---|
| 1 | 0 | 0.00 |
| 2 | 1 | 0.50 |
| 3 | 2 | 0.67 |
| 4 | 3 | 0.75 |
| 5 | 4 | 0.80 |
| 6 | 4 | 0.67 |
| 7 | 4 | 0.57 |
| 8 | 5 | 0.63 |
| 9 | 6 | 0.67 |
| 10 | 7 | 0.70 |
| 11 | 8 | 0.73 |
| 12 | 8 | 0.67 |
| 13 | 9 | 0.69 |
| 14 | 10 | 0.71 |
| 15 | 10 | 0.67 |
| 16 | 10 | 0.63 |
| 17 | 10 | 0.59 |
| 18 | 10 | 0.56 |
| 19 | 10 | 0.53 |
| 20 | 11 | 0.55 |
Notice that at the start of the sequence, the proportion of heads fluctuates wildly, starting at .00 and rising as high as .80. Later on, one gets the impression that it dampens out a bit, with more and more of the values actually being pretty close to the “right” answer of .50. This is the frequentist definition of probability in a nutshell: flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N→∞), the proportion of heads will converge to 50%. There are some subtle technicalities that the mathematicians care about, but qualitatively speaking, that’s how the frequentists define probability. Unfortunately, I don’t have an infinite number of coins, or the infinite patience required to flip a coin an infinite number of times. However, I do have a computer, and computers excel at mindless repetitive tasks. So I asked my computer to simulate flipping a coin 1000 times, and then drew a picture of what happens to the proportion N H /N as N increases. Actually, I did it four times, just to make sure it wasn’t a fluke. The results are shown in Figure 9.1. As you can see, the proportion of observed heads eventually stops fluctuating, and settles down; when it does, the number at which it finally settles is the true probability of heads.
The frequentist definition of probability has some desirable characteristics. Firstly, it is objective: the probability of an event is necessarily grounded in the world. The only way that probability statements can make sense is if they refer to (a sequence of) events that occur in the physical universe. 142 Secondly, it is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer. However, it also has undesirable characteristics. Firstly, infinite sequences don’t exist in the physical world. Suppose you picked up a coin from your pocket and started to flip it. Every time it lands, it impacts on the ground. Each impact wears the coin down a bit; eventually, the coin will be destroyed. So, one might ask whether it really makes sense to pretend that an “infinite” sequence of coin flips is even a meaningful concept, or an objective one. We can’t say that an “infinite sequence” of events is a real thing in the physical universe, because the physical universe doesn’t allow infinite anything. More seriously, the frequentist definition has a narrow scope. There are lots of things out there that human beings are happy to assign probability to in everyday language, but cannot (even in theory) be mapped onto a hypothetical sequence of events. For instance, if a meteorologist comes on TV and says, “the probability of rain in Adelaide on 2 November 2048 is 60%” we humans are happy to accept this. But it’s not clear how to define this in frequentist terms. There’s only one city of Adelaide, and only 2 November 2048. There’s no infinite sequence of events here, just a once-off thing. Frequentist probability genuinely forbids us from making probability statements about a single event. From the frequentist perspective, it will either rain tomorrow or it will not; there is no “probability” that attaches to a single non-repeatable event. Now, it should be said that there are some very clever tricks that frequentists can use to get around this. One possibility is that what the meteorologist means is something like this: “There is a category of days for which I predict a 60% chance of rain; if we look only across those days for which I make this prediction, then on 60% of those days it will actually rain”. It’s very weird and counterintuitive to think of it this way, but you do see frequentists do this sometimes. And it will come up later in this book (see Section 10.5).
Bayesian view
The Bayesian view of probability is often called the subjectivist view, and it is a minority view among statisticians, but one that has been steadily gaining traction for the last several decades. There are many flavours of Bayesianism, making hard to say exactly what “the” Bayesian view is. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. From that perspective, probabilities don’t exist in the world, but rather in the thoughts and assumptions of people and other intelligent beings. However, in order for this approach to work, we need some way of operationalising “degree of belief”. One way that you can do this is to formalise it in terms of “rational gambling”, though there are many other ways. Suppose that I believe that there’s a 60% probability of rain tomorrow. If someone offers me a bet: if it rains tomorrow, then I win $5, but if it doesn’t rain then I lose $5. Clearly, from my perspective, this is a pretty good bet. On the other hand, if I think that the probability of rain is only 40%, then it’s a bad bet to take. Thus, we can operationalise the notion of a “subjective probability” in terms of what bets I’m willing to accept.
What are the advantages and disadvantages to the Bayesian approach? The main advantage is that it allows you to assign probabilities to any event you want to. You don’t need to be limited to those events that are repeatable. The main disadvantage (to many people) is that we can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an intelligent agent out there that believes in things. To many people this is uncomfortable: it seems to make probability arbitrary. While the Bayesian approach does require that the agent in question be rational (i.e., obey the rules of probability), it does allow everyone to have their own beliefs; I can believe the coin is fair and you don’t have to, even though we’re both rational. The frequentist view doesn’t allow any two observers to attribute different probabilities to the same event: when that happens, then at least one of them must be wrong. The Bayesian view does not prevent this from occurring. Two observers with different background knowledge can legitimately hold different beliefs about the same event. In short, where the frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to), the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).
What’s the difference? And who is right?
Now that you’ve seen each of these two views independently, it’s useful to make sure you can compare the two. Go back to the hypothetical robot soccer game at the start of the section. What do you think a frequentist and a Bayesian would say about these three statements? Which statement would a frequentist say is the correct definition of probability? Which one would a Bayesian do? Would some of these statements be meaningless to a frequentist or a Bayesian? If you’ve understood the two perspectives, you should have some sense of how to answer those questions.
Okay, assuming you understand the different, you might be wondering which of them is right ? Honestly, I don’t know that there is a right answer. As far as I can tell there’s nothing mathematically incorrect about the way frequentists think about sequences of events, and there’s nothing mathematically incorrect about the way that Bayesians define the beliefs of a rational agent. In fact, when you dig down into the details, Bayesians and frequentists actually agree about a lot of things. Many frequentist methods lead to decisions that Bayesians agree a rational agent would make. Many Bayesian methods have very good frequentist properties.
For the most part, I’m a pragmatist so I’ll use any statistical method that I trust. As it turns out, that makes me prefer Bayesian methods, for reasons I’ll explain towards the end of the book, but I’m not fundamentally opposed to frequentist methods. Not everyone is quite so relaxed. For instance, consider Sir Ronald Fisher, one of the towering figures of 20th century statistics and a vehement opponent to all things Bayesian, whose paper on the mathematical foundations of statistics referred to Bayesian probability as “an impenetrable jungle [that] arrests progress towards precision of statistical concepts” Fisher (1922b). Or the psychologist Paul Meehl, who suggests that relying on frequentist methods could turn you into “a potent but sterile intellectual rake who leaves in his merry path a long train of ravished maidens but no viable scientific offspring” Meehl (1967). The history of statistics, as you might gather, is not devoid of entertainment.
In any case, while I personally prefer the Bayesian view, the majority of statistical analyses are based on the frequentist approach. My reasoning is pragmatic: the goal of this book is to cover roughly the same territory as a typical undergraduate stats class in psychology, and if you want to understand the statistical tools used by most psychologists, you’ll need a good grasp of frequentist methods. I promise you that this isn’t wasted effort. Even if you end up wanting to switch to the Bayesian perspective, you really should read through at least one book on the “orthodox” frequentist view. And since R is the most widely used statistical language for Bayesians, you might as well read a book that uses R. Besides, I won’t completely ignore the Bayesian perspective. Every now and then I’ll add some commentary from a Bayesian point of view, and I’ll revisit the topic in more depth in Chapter 17.