5  Probability

The usual touchstone of whether what someone asserts is mere persuasion or at least a subjective conviction, i.e., firm belief, is betting. Often someone pronounces his propositions with such confident and inflexible defiance that he seems to have entirely laid aside all concern for error. A bet disconcerts him. Sometimes he reveals that he is persuaded enough for one ducat but not for ten. For he would happily bet one, but at ten he suddenly becomes aware of what he had not previously noticed, namely that it is quite possible that he has erred. -— Immanuel Kant, Critique of Pure Reason

The central tension, and opportunity, in data science is the interplay between the data and the science, between our empirical observations and the models which we use to understand them. Probability is the language we use to explore that interplay; it connects models to data, and data to models.

What does it mean that Donald Trump had a 30% chance of winning election in the fall of 2016? That there is a 90% probability of rain tomorrow? That the dice at the casino are fair?

Probability quantifies uncertainty. Think of probability as a proportion. The probability of an event occurring is a number from 0 to 1, where 0 means that the event is impossible and 1 means that the event is 100% certain.

Begin with the simplest events: coin flips and dice rolls. The set of all outcomes is the sample space. With fair coins and dice, we know that:

If the probability of an outcome is unknown, we will often refer to it as an unknown parameter, something which we might use data to estimate. We usually use Greek letters to refer to parameters. Whenever we are talking about a specific probability (represented by a single value), we will use \(\rho\) (the Greek letter “rho” but spoken aloud as “p” by us) with a subscript which specifies the exact outcome of which it is the probability. For instance, \(\rho_h = 0.5\) denotes the probability of getting heads on a coin toss when the coin is fair. \(\rho_t\) — spoken as “PT” or “P sub T” or “P tails” — denotes the probability of getting tails on a coin toss. This notation can become annoying if the outcome whose probability we seek is less concise. For example, we might write the probability of rolling a 1, 2 or 3 using a fair six-sided dice as:

\[ \rho_{dice\ roll\ is\ 1,\ 2\ or\ 3} = 0.5 \]

We will rarely write out the full definition of an event along with the \(\rho\) symbol. The syntax is just too ugly. Instead, we will define an event a as the case when one rolled dice equals 1, 2 or 3 and, then, write

\[\rho_a = 0.5\]

A random variable is a function which produces a value from a sample set. A random variable can be either discrete — where the sample set has a limited number of members, like H or T for the result of a coin flip, or 2, 3, …, 12 for the sum of two dice — or continuous (any value within a range). Probability is a claim about the value of a random variable, i.e., that you have a 50% probability of getting a 1, 2 or 3 when you roll a fair dice.

We usually use capital letters for random variables. So, \(C\) might be our symbol for the random variable which is a coin toss and \(D\) might be our symbol for the random variable which is the sum of two dice. When discussing random variables in general, or when we grow tired of coming up with new symbols, we will use \(Y\).

Small letters refer to a single outcome or result from a random variable. \(c\) is the outcome from one coin toss. \(d\) is the result from one throw of the dice. The value of the outcome must come from the sample space. So, \(c\) can only take on two possible values: heads or tails. When discussing random variables in general, we use \(y\) to refer to one outcome of the random variable \(Y\). If there are multiple outcomes — if we have, for example, flipped the coin multiple times — then we use subscripts to indicate the separate outcomes: \(y_1\), \(y_2\), and so on. The symbol for an arbitrary outcome is \(y_i\), where \(i\) ranges from 1 through \(N\), the total number of events or experiments for which an outcome \(y\) was produced.

The only package we need in this chapter is tidyverse.

To understand probability more fully, we first need to understand distributions.

5.1 Distributions

A variable in a tibble is a column, a vector of values. We sometimes refer to this vector as a “distribution.” This is somewhat sloppy in that a distribution can be many things, most commonly a mathematical formula. But, strictly speaking, a “frequency distribution” or an “empirical distribution” is a list of values, so this usage is not unreasonable.

5.1.1 Scaling distributions

Consider the vector which is the result of rolling one dice 10 times.

ten_rolls <- c(5, 5, 1, 5, 4, 2, 6, 2, 1, 5)

There are other ways of storing the data in this vector. Instead of reporting every observation, we could record the number of times each value appears or the percentage of the total which this number accounts for.

Distribution of Ten Rolls of a Fair Dice
Counts and percentages reflect the same information
Outcome Count Percentage
1 2 0.2
2 2 0.2
4 1 0.1
5 4 0.4
6 1 0.1

In this case, with only 10 values, it is actually less efficient to store the data like this. But what happens when we have 1,000 rolls?

Distribution of One Thousand Rolls of a Fair Dice
Counts and percentages reflect the same information
Outcome Count Percentage
1 190 0.190
2 138 0.138
3 160 0.160
4 173 0.173
5 169 0.169
6 170 0.170

Instead of keeping around a vector of length 1,000, we can just keep 12 values — the 6 possible outcomes and their frequency — without losing any information.

Two distributions can be identical even if they are of very different lengths. Let’s compare our original distribution of 10 rolls of the dice with another distribution which just features 100 copies of those 10 rolls.

more_rolls <- rep(ten_rolls, 100)

The two graphs have the exact same shape because, even though the vectors are of different lengths, the relative proportions of the outcomes are identical. In some sense, both vectors are from the same distribution. Relative proportions, not the total counts, are what matter.

5.1.2 Normalizing distributions

If two distributions have the same shape, then they only differ by the labels on the y-axis. There are various ways of “normalizing” distributions so as to place them all the same scale. The most common scale is one in which the area under the distribution adds to 1, e.g., 100%. For example, we can transform the above plots:

We sometimes refer to a distribution as “unnormalized” if the area under the curve does not add up to 1.

5.1.3 Simulating distributions

There are two distinct concepts: a distribution and a set values drawn from that distribution. But, in typical usage, we employ “distribution” for both. When given a distribution (meaning a vector of numbers), we often use geom_histogram() or geom_density() to display it. But, sometimes, we don’t want to look at the whole thing. We just want some summary measures which report the key aspects of the distribution. The two most important attributes of a distribution are its center and its variation around that center.

We use summarize() to calculate statistics for a variable, a column, a vector of values, or a distribution. Note the language sloppiness. For the purposes of this book, “variable,” “column,” “vector,” and “distribution” all mean the same thing. Other popular statistical functions include: mean(), median(), min(), max(), n() and sum(). Functions which may be new to you include three measures of the “spread” of a distribution: sd() (the standard deviation), mad() (the scaled median absolute deviation) and quantile(), which is used to calculate an interval which includes a specified proportion of the values.

Think of the distribution of a variable as an urn from which we can pull out, at random, values for that variable. Drawing a thousand or so values from that urn, and then looking at a histogram, can show where the values are centered and how they vary. Because people are sloppy, they will use the word distribution to refer to at least three related entities:

  1. the (imaginary!) urn from which we are drawing values.
  2. all the values in the urn
  3. all the values which we have drawn from the urn, whether that be 10 or 1,000

Sloppiness in the usage of the word distribution is universal. However, keep three distinct ideas separate:

  • The unknown true distribution which, in reality, generates the data which we see. Outside of stylized examples in which we assume that a distribution follows a simple mathematical formula, we will never have access to the unknown true distribution. We can only estimate it. This unknown true distribution is often referred to as the data generating mechanism, or DGM. It is a function or black box or urn which produces data. We can see the data. We can’t see the urn.

  • The estimated distribution which, we think, generates the data which we see. Again, we can never know the unknown true distribution. But, by making some assumptions and using the data we have, we can estimate a distribution. Our estimate may be very close to the true distribution. Or it may be far away. The main task of data science to to create and use these estimated distributions. Almost always, these distributions are instantiated in computer code. Just as there is a true data generating mechanism associated with the (unknown) true distribution, there is an estimated data generating mechanism associated with the estimated ditribution.

  • A vector of numbers drawn from the estimated distribution. Both true and estimated distributions can be complex animals, difficult to describe accurately and in detail. But a vector of numbers drawn from a distribution is easy to understand and use. So, in general, we work with vectors of numbers. When someone — either a colleague or a piece of R code — creates a distribution which we want to use to answer a question, we don’t really want the distribution itself. Rather, we want a vectors of “draws” from that distribution. Vectors are easy to work with! Complex computer code is not.

Again, people (including us!) will often be sloppy and use the same word, “distribution,” without making it clear whether they are talking about the true distribution, the estimated distribution, or a vector of draws from the estimated distribution. The same sloppiness applies to the use of the term data generating mechanism. Try not to be sloppy.

Much of the rest of the Primer involves learning how to work with distributions, which generally means working with the draws from those distributions. Fortunately, the usual rules of arithmetic apply. You can add/subtract/multiply/divide distributions by working with draws from those distributions, just as you can add/subtract/multiply/divide regular numbers.

5.2 Probability distributions

Bruno de Finetti, an Italian statistician who wrote a famous treatise on the theory of probability that began with the statement “PROBABILITY DOES NOT EXIST.”

For the purposes of this Primer, a probability distribution is a mathematical object which maps a set of outcomes to probabilities, where each distinct outcome has a chance of occurring between 0 and 1 inclusive. The probabilities must sum to 1. The set of possible outcomes, i.e., the sample space — heads or tails for the coin, 1 through 6 for a single dice, 2 through 12 for the sum of a pair of dice — can be either discrete or continuous. Remember, discrete data can only take on certain values. Continuous data, like height and weight, can take any value within a range. The set of outcomes is the domain of the probability distribution. The range is the associated probabilities.

Assume that a probability distribution is created by a probability function, a set function which maps outcomes to probabilities. The concept of a “probability function” is often split into two categories: probability mass functions (for discrete random variables) and probability density functions (for continuous random variables). As usual, we will be a bit sloppy, using the term probability distribution for both the mapping itself and for the function which creates the mapping.

We discuss three types of probability distributions: empirical, mathematical, and posterior.

The key difference between a distribution, as we have explored them in Section 5.1, and a probability distribution is the requirement that the sum of the probabilities of the individual outcomes must be exactly 1. There is no such requirement for a distribution in general. But any distribution can be turned into a probability distribution by “normalizing” it. In this context, we will often refer to a distribution which is not (yet) a probability distribution as an “unnormalized” distribution.

Pay attention to notation. Recall that when we are talking about a specific probability (represented by a single value), we will use \(\rho\) (the Greek letter “rho”) with a subscript which specifies the exact outcome of which it is the probability. For instance, \(\rho_h = 0.5\) denotes the probability of getting heads on a coin toss when the coin is fair. \(\rho_t\) — spoken as “PT” or “P sub T” or “P tails” — denotes the probability of getting tails on a coin toss. However, when we are referring to the entire probability distribution over a set of outcomes, we will use \(P()\). For example, the probability distribution of a coin toss is \(P(\text{coin})\). That is, \(P(\text{coin})\) is composed of the two specific probabilities (50% and 50%) mapped from the two values in the domain (Heads and Tails). Similarly, \(P(\text{sum of two dice})\) is the probability distribution over the set of 11 outcomes (2 through 12) which are possible when you take the sum of two dice. \(P(\text{sum of two dice})\) is made up of 11 numbers — \(\rho_2\), \(\rho_3\), …, \(\rho_{12}\) — each representing the unknown probability that the sum will equal their value. That is, \(\rho_2\) is the probability of rolling a 2.

5.2.1 Flipping a coin

Data science problems start with a question. Example:

What are the chances of getting three heads in a row when flipping a fair coin?

Questions are answered with the help of probability distributions.

An empirical distribution is based on data. Think of this as the probability distribution created by collecting data in the real world or by running a simulation on your computer. In theory, if we increase the number of coins we flip (either in reality or via simulation), the empirical distribution will look more and more similar to the mathematical distribution. The mathematical distribution is the Platonic form. The empirical distribution will often look like the mathematical probability distribution, but it will rarely be exactly the same.

In this simulation, there are 44 heads and 56 tails. The outcome will vary every time we run the simulation, but the proportion of heads to tails should not be too different if the coin is fair.

A mathematical distribution is based on a mathematical formula. Assuming that the coin is perfectly fair, we should, on average, get heads as often as we get tails.

The distribution of a single observation is described by this formula.

\[ P(Y = y) = \begin{cases} 1/2 &\text{for }y= \text{Heads}\\ 1/2 &\text{for }y= \text{Tails} \end{cases}\] We sometimes do not know that the probability of heads and the probability of tails both equal 50%. In that case, we might write:

\[ P(Y = y) = \begin{cases} \rho_H &\text{for }y= \text{Heads}\\ \rho_T &\text{for }y= \text{Tails} \end{cases}\]

Yet, we know that, by definition, \(\rho_H + \rho_T = 1\), so we can rewrite the above as:

\[ P(Y = y) = \begin{cases} \rho_H &\text{for }y= \text{Heads}\\ 1- \rho_H &\text{for }y= \text{Tails} \end{cases}\]

Coin flipping (and related scenarios with only two possible outcomes) are such common problems, that the notation is often simplified further, with \(\rho\) understood, by convention, to be the probability of heads. In that case, we can write the mathematical distribution is two canonical forms:

\[P(Y) = Bernoulli(\rho)\] and

\[y_i \sim Bernoulli(\rho)\] All five of these versions mean the same thing! The first four describe the mathematical probability distribution for a fair coin. The capital \(Y\) within the \(P()\) indicates a random variable. The fifth highlights one “draw” from that random variable, hence the lower case \(y\) and the subscript \(i\).

Most probability distributions do not have special names, which is why we will use the generic symbol \(P\) to refer to them. But some common probability distributions do have names, like “Bernoulli” in this case.

If the mathematical assumptions are correct, then, as your sample size increases, the empirical probability distribution will look more and more like the mathematical distribution.

A posterior distribution is based on beliefs and expectations. It displays your beliefs about things you can’t see right now. You may have posterior distributions for outcomes in the past, present, or future.

In the case of the coin toss, the posterior distribution changes depending on your beliefs. For instance, let’s say your friend brought a coin to school and asked to bet you. If the result is heads, you have to pay them $5. In that case, your posterior probability distribution might look like this:

The fact that your friend wants to bet on heads suggests to you that the coin is not fair. Does it prove that the coin is unfair? No! Much depends on the sort of person you think your friend is. Your posterior probability distribution is your opinion, based on your experiences and beliefs. My posterior probability distribution will often be (very) different from yours.

The full terminology is mathematical (or empirical or posterior) probability distribution. But we will often shorten this to just mathematical (or empirical or posterior) distribution. The word “probability” is understood, even if it is not present.

Recall the question with which we started this section: What are the chances of getting three heads in a row when flipping a fair coin? To answer this question, we need to use a probability distribution as our data generating mechanism. Fortunately, the rbinom() function allows us to generate the results for coin flips. For example:

rbinom(n = 10, size = 1, prob = 0.5)
 [1] 1 1 0 1 1 0 0 0 0 0

generates the results of 10 coin flips, where a result of heads is presented as 1 and tails as 0. With this tool, we can generate 1,000 draws from our experiment:

tibble(toss_1 = rbinom(n = 1000, size = 1, prob = 0.5),
       toss_2 = rbinom(n = 1000, size = 1, prob = 0.5),
       toss_3 = rbinom(n = 1000, size = 1, prob = 0.5))
# A tibble: 1,000 × 3
   toss_1 toss_2 toss_3
    <int>  <int>  <int>
 1      0      1      1
 2      0      1      1
 3      0      1      0
 4      0      0      1
 5      1      1      0
 6      1      0      1
 7      1      0      0
 8      1      0      1
 9      0      0      1
10      0      1      1
# … with 990 more rows

Because the flips are independent, we can consider each row to be a draw from the experiment. Then, we simply count up the proportion of experiments in which resulted in three heads.

tibble(toss_1 = rbinom(n = 1000, size = 1, prob = 0.5),
       toss_2 = rbinom(n = 1000, size = 1, prob = 0.5),
       toss_3 = rbinom(n = 1000, size = 1, prob = 0.5)) |> 
  mutate(three_heads = toss_1 + toss_2 + toss_3 == 3) |> 
  summarize(chance = mean(three_heads))
# A tibble: 1 × 1
  chance
   <dbl>
1  0.104

This is close to the “correct” answer of \(1/8\)th. If we increase the number of draws, we will get closer to the “truth.” The reason for the quotation marks around “correct” and “truth” is that we are uncertain. We don’t know the true probability distribution for this coin. If this coin is a trick coin — like the one we expect our friend to have brought to school — then the odds of three heads in a row would be much higher:

tibble(toss_1 = rbinom(n = 1000, size = 1, prob = 0.95),
       toss_2 = rbinom(n = 1000, size = 1, prob = 0.95),
       toss_3 = rbinom(n = 1000, size = 1, prob = 0.95)) |> 
  mutate(three_heads = toss_1 + toss_2 + toss_3 == 3) |> 
  summarize(chance = mean(three_heads))
# A tibble: 1 × 1
  chance
   <dbl>
1   0.87

This is our first example of using a data generating mechanism — meaning rbinom() — to answer a question. We will see many more in the chapters to come.

5.2.2 Rolling two dice

Data science begins with a question:

What is the probability of rolling a 7 or an 11 with a pair of dice?

We get an empirical distribution by rolling two dice a hundred times, either by hand or with a computer simulation. The result is not identical to the mathematical distribution because of the inherent randomness of the real world and/or of simulation.

We might consider labeling the y-axis in plots of empirical distributions as “Proportion” rather than “Probability” since it is an actual proportion, calculated from real (or simulated) data. We will keep it as “Probability” since we want to emphasize the parallels between mathematical, empirical and posterior probability distributions.

Our mathematical distribution tells us that, with a fair dice, the probability of getting 1, 2, 3, 4, 5, and 6 are equal: there is a 1/6 chance of each. When we roll two dice at the same time and sum the numbers, the values closest to the middle are more common than values at the edge because there are more combinations of numbers that add up to the middle values.

\[ P(Y = y) = \begin{cases} \dfrac{y-1}{36} &\text{for }y=1,2,3,4,5,6 \\ \dfrac{13-y}{36} &\text{for }y=7,8,9,10,11,12 \\ 0 &\text{otherwise} \end{cases} \]

The posterior distribution for rolling two dice a hundred times depends on your beliefs. If you take the dice from your Monopoly set, you have reason to believe that the assumptions underlying the mathematical distribution are true. However, if you walk into a crooked casino and a host asks you to play craps, you might be suspicious, just as in the coin flip example above. The word “suspicious” means that you suspect that the data generating mechanism for these dice is not like that for honest dice. For example, in craps, a “come-out” roll of 7 and 11 is a “natural,” resulting in a win for the “shooter” and a loss for the casino. You might expect those numbers to occur less often than they would with fair dice. Meanwhile, a come-out roll of 2, 3 or 12 is a loss for the shooter. You might also expect values like 2, 3 and 12 to occur more frequently. Your posterior distribution might look like this:

Someone less suspicious of the casino would have a posterior distribution which looks more like the mathematical distribution.

We began this section with a question about the probability (or odds) of rolling a 7 or 11 — i.e., a “natural” — with a pair of dice. The answer to the question depends on whether or not we think the dice are fair. In other words, we need to know which distribution to use to answer the question.

Assume that the dice are fair. In that case, we can create a data generating mechanism by hand. (Alas, there is not a built-in R function for dice like there is for coin flips with rbinom().)

set.seed(7)

# Creating a variable like rolls makes our code easier to read and modify. Of
# course, we could just hard code the 4 into the size argument for each of the
# two calls to sample, but that is much less convenient.

rolls <- 4

# The details of the code matter. If we don't have replace = TRUE, sample will
# only use each of the 6 possible values once. That might be OK if we are just
# rolling the dice 4 times, but it won't work for thousands of rolls.

tibble(dice_1 = sample(x = 1:6, size = rolls, replace = TRUE),
       dice_2 = sample(x = 1:6, size = rolls, replace = TRUE)) |> 
  mutate(result = dice_1 + dice_2) |> 
  mutate(natural = ifelse(result %in% c(7, 11), TRUE, FALSE))
# A tibble: 4 × 4
  dice_1 dice_2 result natural
   <int>  <int>  <int> <lgl>  
1      2      2      4 FALSE  
2      3      6      9 FALSE  
3      4      3      7 TRUE   
4      2      6      8 FALSE  

This code is another data generating mechanism or dgm. It allows us to simulate the distribution of the results from rolling a pair of fair dice. To answer our question, we simply increase the number of rolls and calculate the proportion of rolls which result in a 7 or 11.

rolls <- 100000

# We probably don't need 100,000 rolls, but this code is so fast that it does
# not matter. Generally 1,000 (or even 100) draws from the data generating
# mechanism is enough for most practical purposes.

tibble(dice_1 = sample(x = 1:6, size = rolls, replace = TRUE),
       dice_2 = sample(x = 1:6, size = rolls, replace = TRUE)) |> 
  mutate(result = dice_1 + dice_2) |> 
  summarize(natural_perc = mean(result %in% c(7, 11)))
# A tibble: 1 × 1
  natural_perc
         <dbl>
1        0.221

The probability of rolling either a 7 or an 11 with a pair of fair dice is about 22%.

5.2.3 Presidential elections

Data science begins with a question:

What is the probability that the Democratic candidate will win the Presidential election?

Consider the probability distribution for a political event, like a presidential election. We want to know the probability that Democratic candidate wins X electoral votes, where X comes from the range of possible outcomes: 0 to 538. (The total number of electoral votes in US elections since 1964 is 538.)

The empirical distribution in this case would involve counting the number of electoral votes that the Democratic candidate won in each of the Presidential elections in the last 50 years or so. For the empirical distribution, we create a tibble with electoral vote results from past elections. Looking at elections since 1964, we can observe that the number of electoral votes that the Democratic candidate received in each election is different.

Given that we only have 15 observations, it is difficult to draw conclusions or make predictions based off of this empirical distribution. But “difficult” does not mean “impossible.” For example, if someone, more than a year before the election, offered to bet us 50/50 that the Democratic candidate was going to win more than 475 electoral votes, we would take the bet. After all, this outcome has only happened once in the last 15 elections, so a 50/50 bet seems like a great deal.

We can build a mathematical distribution for X which assumes that the chances of the Democratic candidate winning any given state’s electoral votes is 0.5 and that the results from each state are independent.

If our assumptions about this mathematical distribution are correct — they are not! — then, as the sample size increase, the empirical distribution should look more and more similar to our mathematical distribution.

However, the data from past elections is more than enough to demonstrate that the assumptions of our mathematical probability distribution do not work for electoral votes. The model assumes that the Democrats have a 50% chance of receiving each of the 538 votes. Just looking at the mathematical probability distribution, we can observe that receiving 13 or 17 or 486 votes out of 538 would be extreme and almost impossible if the mathematical model were accurate. However, our empirical distribution shows that such extreme outcomes are quite common. Presidential elections have resulted in much bigger victories or defeats than this mathematical distribution seems to allow for, thereby demonstrating that our assumptions are false.

The posterior distribution of electoral votes is a popular topic, and an area of strong disagreement, among data scientists. Consider this posterior from FiveThirtyEight.

Below is a posterior probability distribution from the FiveThirtyEight website for August 13, 2020. This was created using the same data as the above distribution, but is displayed differently. For each electoral result, the height of the bar represents the probability that a given event will occur. However, there are no labels on the y-axis telling us what the specific probability of each outcome is. And that is OK! The specific values are not that useful. If we removed the labels on our own y-axes, would it matter? Probably not. Anytime there are many possible outcomes — 539, in this case — we stop looking at specific outcomes and, instead, look at where most of the “mass” of the distribution lies.

Below is the posterior probability distribution from The Economist, also from August 13, 2020. This looks confusing at first because they chose to combine the axes for Republican and Democratic electoral votes. The Economist was less optimistic, relative to FiveThirtyEight, about Trump’s chances in the election.

These two models, built by smart people using similar data sources, have reached fairly different conclusions. Data science is difficult! There is not one “right” answer. Real life is not a problem set.

Watch the makers of these two models throw shade at each other on Twitter! Eliot Morris is one of the primary authors of the Economist model. Nate Silver is in charge of 538. They don’t seem to be too impressed with each other’s work! More smack talk here and here.

There are many questions you could explore with posterior distributions. They can relate to the past, present, or future.

  • Past: How many electoral votes would Hilary Clinton have won if she had picked a different VP?
  • Present: What are the total campaign donations from Harvard faculty?
  • Future: How many electoral votes will the Democratic candidate for president win in 2024?

5.2.4 Height

Question: What is the height of the next adult male we will meet?

The three examples above are all discrete probability distributions, meaning that the outcome variable can only take on a limited set of values. A coin flip has two outcomes. The sum of a pair of dice has 11 outcomes. The total electoral votes for the Democratic candidate has 539 possible outcomes. In the limit, we can also create continuous probability distributions which have an infinite number of possible outcomes. For example, the average height for an American male could be any real number between 0 inches and 100 inches. (Of course, an average value anywhere near 0 or 100 is absurd. The point is that the average could be 68.564, 68.5643, 68.56432 68.564327, or any real number.)

All the characteristics for discrete probability distributions which we reviewed above apply just as much to continuous probability distributions. For example, we can create mathematical, empirical and posterior probability distributions for continuous outcomes just as we did for discrete outcomes.

The empirical distribution involves using data from the National Health and Nutrition Examination Survey (NHANES).

Mathematical distribution is completely based on mathematical formula and assumptions, as in the coin flip example. In the coin-flip example, we assumed that the coin was perfectly fair, meaning that the probability of landing on heads or tails was equal. In this case, we make three assumptions. First, a male height follows a Normal distribution. Second, the average height of men is 175 cm. Third, the standard deviation for male height is 9 cm. We can create a Normal distribution using the rnorm() function with these two parameter values.

Again, the Normal distribution which is a probability distribution that is symmetric about the mean described by this formula.

\[y_i \sim N(\mu, \sigma^2)\]

Each value \(y_i\) is drawn from a Normal distribution with parameters \(\mu\) for the mean and \(\sigma\) for the standard deviation. If the assumptions are correct, then, as our sample size increases, the empirical probability distribution will look more and more like the mathematical distribution.

The posterior distribution for heights depends on the context. Are we considering all the adult men in America? In that case, our posterior would probably look a lot like the empirical distribution using NHANES data. If we are being asked about the distribution of heights among players in the NBA, then our posterior might look like:

Caveats:

  • Continuous variables are a myth. Nothing that can be represented on a computer is truly continuous. Even something which appears continuous, like height, actually can only take on a (very large) set of discrete variables.

  • The math of continuous probability distributions can be tricky. Read a book on mathematical probability for all the messy details. Little of that matters in applied work.

  • The most important difference is that, with discrete distributions, it makes sense to estimate the probability of a specific outcome. What is the probability of rolling a 9? With continuous distributions, this makes no sense because there are an infinite number of possible outcomes. With continuous variables, we only estimate intervals.

Don’t worry about the distinctions between discrete and continuous outcomes, or between the discrete and continuous probability distributions which we will use to summarize our beliefs about those outcomes. The basic intuition is the same in both cases.

5.2.5 Joint distributions

Recall that \(P(\text{coin})\) is the probability distribution for the result of a coin toss. It includes two parts, the probability of heads (\(\rho_h\)) and the probability of tails (\(\rho_t\)). This is a univariate distribution because there is only one outcome, which can be heads or tails. If there is more than one outcome, then we have a joint distribution.

Joint distributions are also mathematical objects that cover a set of outcomes, where each distinct outcome has a chance of occurring between 0 and 1 and the sum of all chances must equal 1. The key to a joint distribution is that it measures the chance that both outcome \(a\) from the set of events A and outcome \(b\) from the set of events B will occur. The notation is \(P(A, B)\).

Let’s say that you are rolling two six-sided dice simultaneously. Dice 1 is weighted so that there is a 50% chance of rolling a 6 and a 10% chance of each of the other values. Dice 2 is weighted so there is a 50% chance of rolling a 5 and a 10% chance of rolling each of the other values. Let’s roll both dice 1,000 times. In previous examples involving two dice, we cared about the sum of results and not the outcomes of the first versus the second dice of each simulation. With a joint distributions, the outcomes for individual dice matter; so instead of 11 possible outcomes on the x-axis of our distribution plot (ranging from 2 to 12), we have 36 outcomes. Furthermore, a 2D probability distribution is not sufficient to represent all of the variables involved, so the joint distribution for this example is displayed using a 3D plot.

5.2.6 Conditional distrubutions

Imagine that 60% of people in a community have a disease. A doctor develops a test to determine if a random person has the disease. However, this test isn’t 100% accurate. There is an 80% probability of correctly returning positive if the person has the disease and 90% probability of correctly returning negative if the person does not have the disease.

The probability of a random person having the disease is 0.6. Since each person either has the disease or doesn’t (those are the only two possibilities), the probability that a person does not have the disease is \(1 - 0.6 = 0.4\).

  • If a person has the disease, then we go up the top branch. The probability of an infected person testing positive is 0.8 because the test is 80% sure of correctly returning positive when the person has the disease.

  • By the same logic, if a person does not have the disease, we go down the bottom branch. The probability of the person incorrectly testing positive is 0.1.

We decide to go down the top branch if our random person has the disease. We go down the bottom branch if they do not. This is conditional probability. The probability of testing positive is dependent on whether the person has the disease.

How would you express this in statistical notation? \(P(A|B)\) is the same thing as the probability of A given B. \(P(A|B)\) means the probability of A if we know for sure the value of B. Note that \(P(A|B)\) is not the same thing as \(P(B|A)\).

There are three main categories of probability distributions: univariate, joint and condictional. \(p(A)\) is the probability distribution for event A. This is a univariate probability distribution because there is only one random variable. \(p(A, B)\) is the joint probability distribution of A and B. \(p(A | B)\) is the conditional probability distribution of A given that B has taken on a specific value. This is often written as \(p(A | B = b)\).

5.3 List-columns and map functions

We need to expand our collection of R tricks by learning about list-columns and map_* functions. Recall that a list is different from an atomic vector. In atomic vectors, each element of the vector has one value. Lists, however, can contain vectors, and even more complex objects, as elements.

x <- list(c(4, 16, 9), c("A", "Z"))
x
[[1]]
[1]  4 16  9

[[2]]
[1] "A" "Z"

x is a list with two elements. The first element is a numeric vector of length 3. The second element is a character vector of length 2. We use [[]] to extract specific elements.

x[[1]]
[1]  4 16  9

There are a number of built-in R functions that output lists. For example, ggplot objects store all of the plot information in a list. Any function that returns multiple values can be used to create a list output by wrapping that returned object with list().

x <- rnorm(10)

# range() returns the min and max of the argument 

range(x)
[1] -1.841155  1.098223
# We can create a tibble which includes the results of range(x)

tibble(col_1 = list(range(x))) 
# A tibble: 1 × 1
  col_1    
  <list>   
1 <dbl [2]>

Notice this is a 1-by-1 tibble with one observation, which is a list of one element. Voila! You have just created a list-column.

If a function returns multiple values as a vector, like range() does, you must use list() as a wrapper if you want to create a list-column.

A list column is a column of your data which is a list rather than an atomic vector. As with stand-alone list objects, you can pipe to str() to examine the column.

# tibble() is what we use to generate a tibble, it acts sort of like the
# mutate(), but mutate() needs a data frame to add new column, tibble can
# survive on itself.

tibble(col_1 = list(range(x))) |>
  str()
tibble [1 × 1] (S3: tbl_df/tbl/data.frame)
 $ col_1:List of 1
  ..$ : num [1:2] -1.84 1.1

We can use map_* functions to both create a list-column and then, much more importantly, work with that list-column afterwards.

# .x is col_1 from tibble and ~ sum(.) is the formula

tibble(col_1 = list(range(x))) |>
  mutate(col_2 = map_dbl(col_1, ~ sum(.))) |> 
  str()
tibble [1 × 2] (S3: tbl_df/tbl/data.frame)
 $ col_1:List of 1
  ..$ : num [1:2] -1.84 1.1
 $ col_2: num -0.743

map_* functions, like map_dbl() in this example, take two key arguments, .x (the data which will be acted on) and .f (the function which will act on this data). Here, .x is the data in col_1, which is a list-column. .f is the function sum(). However, we can not simply write map_dbl(col_1, sum). Instead, each use of map_* functions requires the use of a tilde — a ~ — to indicate the start of the function and the use of a dot — a . — to specify where the data goes in the function.

map_* functions are a family of functions, with the suffix specifying the type of the object to be returned. map() itself returns a list. map_dbl() returns a double. map_int() returns an integer. map_chr() returns a character, and so on. Example:

tibble(ID = 1) |> 
  mutate(col_1 = map(ID, ~range(rnorm(10)))) |>
  mutate(col_2 = map_dbl(col_1, ~ sum(.))) |> 
  mutate(col_3 = map_int(col_1, ~ length(.))) |> 
  mutate(col_4 = map_chr(col_1, ~ sum(.))) |> 
  str()
Warning: There was 1 warning in `mutate()`.
ℹ In argument: `col_4 = map_chr(col_1, ~sum(.))`.
Caused by warning:
! Automatic coercion from double to character was deprecated in purrr 1.0.0.
ℹ Please use an explicit call to `as.character()` within `map_chr()` instead.
tibble [1 × 5] (S3: tbl_df/tbl/data.frame)
 $ ID   : num 1
 $ col_1:List of 1
  ..$ : num [1:2] -0.741 0.916
 $ col_2: num 0.176
 $ col_3: int 2
 $ col_4: chr "0.175759"

Consider a more detailed example:

# This simple example demonstrates the workflow which we will often follow.
# Start by creating a tibble which will be used to store the results. (Or start
# with a tibble which already exists and to which you will be adding more
# columns.) It is often convenient to get all the code working with just a few
# rows. Once it is working, we increase the number of rows to a thousand or
# million or whatever we need.

tibble(ID = 1:3) |> 
  
  # The big convenience is being able to store a list in each row of the tibble.
  # Note that we are not using the value of ID in the call to rnorm(). (That is
  # why we don't have a "." anywhere.) But we are still using ID as a way of
  # iterating through each row; ID is keeping count for us, in a sense.
  
  mutate(draws = map(ID, ~ rnorm(10))) |> 
  
  # Each succeeding step of the pipe works with columns already in the tibble
  # while, in general, adding more columns. The next step calculates the max
  # value in each of the draw vectors. We use map_dbl() because we know that
  # max() will returns a single number.
  
  mutate(max = map_dbl(draws, ~ max(.))) |> 
  
  # We will often need to calculate more than one item from a given column like
  # draws. For example, in addition to knowing the max value, we would like to
  # know the range. Because the range is a vector, we need to store the result
  # in a list column. map() does that for us automatically.
  
  mutate(min_max = map(draws, ~ range(.)))
# A tibble: 3 × 4
     ID draws        max min_max  
  <int> <list>     <dbl> <list>   
1     1 <dbl [10]> 1.68  <dbl [2]>
2     2 <dbl [10]> 0.364 <dbl [2]>
3     3 <dbl [10]> 1.44  <dbl [2]>

This flexibility is only possible via the use of list-columns and map_* functions. This workflow is extremely common. We start with an empty tibble, using ID to specify the number of rows. With that skeleton, each step of the pipe adds a new column, working off a column which already exists.

5.4 Two models

The simplest possible setting for inference involves two models — meaning two possible states of the world — and two outcomes from an experiment. Imagine that there is a disease — Probophobia, an irrational fear of probability — which you either have or don’t have. We don’t know if you have the diseases, but we do assume that there are only two possibilities.

We also have a test which is 99% accurate when given to a person who has Probophobia. Unfortunately, the test is only 50% accurate for people who do not have Probophobia. In this experiment, there only two possible outcomes: a positive and a negative result on the test.

Question: If you test positive, what is the probability that you have Probophobia?

More generally, we are estimating a conditional probability. Conditional on the outcome of a postive test, what is the probability that you have Probophobia? Mathematically, we want:

\[ P(\text{Probophobia | Test = Postive} ) \]

To answer this question, we need to use the tools of joint and conditional probability from earlier in the Chapter. We begin by building, by hand, the joint distribution of the possible models (you have the Probophobia or you do not) and of the possible outcomes (you test positive or negative). Building the joint distribution involves assuming that each model is true and then creating the distribution of outcomes which might occur if that assumption is true.

For example, assume you have Probophobia. There is then a 50% ch