Last week, we discussed Bayes’s Theorem briefly. We used a simple example of walking into a classroom and observe three people wearing different shirts.
This week, we are going to utilize this same example, but we will examine the mechanics of what’s happening through the lens of Bayes’s Theorem.
To do this, we will take a quick look at Bayes’s Theorem and then use its predictive insights to shed some light on our previous example and provide additional understanding of our own intuition.
Author Nate Silver, a champion of Bayesian statistics and founder of FiveThirtyEight, explain that Bayes’s Theorem utilizes conditional probability to “tell us the probability that a theory or hypothesis is true if some event has happened.” We can calculate probabilities with the understanding that this event occurs, either through evidence or through assumption.
We can calculate probabilities with the understanding that this event occurs, either through evidence or through assumption.
To utilize Bayes’s Theorem, we need to have a condition and a hypothesis. The condition is the event that occurs; the new evidence that is presented to us. The hypothesis is an outcome, which is tested using the theorem.
Bayes’s Theorem has three known variables and one unknown variable. Our unknown variable, that we’re solving for, is the Posterior Possibility; the probability of our hypothesis occurring.
The Prior Probability (x), is the probability of a hypothesis being true prior to examination of the evidence. To find our prior, we must utilize a probability of the hypothesis being true before we even knew of the evidence.
We need to have a (y), or the probability of evidence arising given that the hypothesis is true. To find this, we must assume the outcome is true and estimate the probability of the evidence occurring as a condition of the hypothesis being true.
We also need to have a (z), the probability of evidence arising given that the hypothesis is false. To find this, we must assume the outcome is false and estimate the probability of the evidence occurring as a condition of the hypothesis being false.
Now we will return to our example from last week…
Three different shirts
Let us revisit the basic framework of our example from last week. We will assume that you are an avid Iron Maiden fan. It’s the first day of classes and you must decide who you would like to sit next to in order to have the greatest chance of connecting with someone.
Consider the following scenario: You walk into a classroom and you observe three people. Person A is wearing a plain white tee shirt. Person B is wearing an Iron Maiden tee shirt. Person C is wearing a Justin Bieber tee shirt. Who do you sit next to?
Last time, we figured that you would sit next to Person B because you are an avid Iron Maiden fan and they’re wearing an Iron Maiden tee shirt. It makes intuitive sense; it seems as though you have the best chance of connecting with Person B because their choice to wear an Iron Maiden tee shirt leads you to assume you share some commonalities with each other.
We can use Bayes’s Theorem to show why it makes intuitive sense.
We’re going to be using estimates for probabilities here because it is difficult to quantify social probabilities precisely. The model will still illustrate the point regardless, and this is often how it actually occurs in our minds.
In this example, the condition is that you see each person wearing their respective tee shirt; the hypothesis you are evaluating is the probability that you have a connection with this person.
Our prior probability (x) is going to be the same for each individual, and we’re going to estimate it to be around 70%. This means that there’s a 70% chance that we will make a connection with the person we talk to. This number could be different for each individual person and each individual scenario, but we’re going to use this baseline of 70% to show that connection is likely.
Now we need the probability of evidence arising given that the hypothesis is true (y). Given that we do end up making a connection, what is the likelihood that this person is wearing that tee shirt?
For example, given that you make a connection, what is the probability that person is wearing an Iron Maiden tee shirt? A lot of your friends and people you connect with probably enjoy Iron Maiden, but Iron Maiden shirts are not exactly popular. So given that we do end up making a connection, we estimate that there’s a 25% chance that this person is wearing an Iron Maiden tee shirt.
We do the same for Person A, whom we assign a 25% chance, and Person C, whom we assign a 5% chance.
Finally, we need the probability of evidence arising given that the hypothesis is true (y). Given that we don’t end up making a connection, what is the likelihood that this person is wearing that tee shirt?
For example, given that you don’t make a connection, what is the probability that person is wearing a Justin Bieber tee shirt? You don’t particularly care for Justin Bieber’s music and you tend to lack commonalities with his followers, but his shirts are quite popular. So given that we don’t end up making a connection, we estimate that there’s a 20% chance that this person is wearing a Justin Bieber shirt.
We do the same for Person A, whom we assign a 25% chance, and Person B, whom we assign a 10% chance.
Plugging these values into our equation, we get the following results:
The outcome of Bayes’s Theorem tells us that we have the greatest likelihood of connection with Person B. This is the same conclusion we arrived at last week when we just made a gut assumption.
So why should we even care about Bayes’s Theorem in this context?
While we may have arrived at similar conclusions via Bayes’s Theorem and our own intuition, the point is not that we should favor one over the other. The point is that Bayes’s Theorem is a model that we are already embracing intuitively, and understanding Bayes’s Theorem call help us better understand our own intuition.
Our intuition is often mystical and elusive; we cannot always explain why our intuition leads us down a particular path.
As we learn the mechanics of Bayes’s Theorem, we can also learn the mechanics of how our assumptions drive the decisions we make.
All of the probabilities that we illustrated in this example of Bayes’s Theorem are probabilities that we estimate, either consciously or subconsciously, throughout our daily interactions.
Last week, we illustrated the importance of working to improve our assumptions so as to create more accurate models of the world around us.
This week, we used Bayes’s Theorem to examine the importance of understanding how probabilities intermingled with the assumptions we make affect our social interactions. Playing around with the numbers in the model can be an interesting way to learn more about how each assumption affects the outcome. I have provided an Excel spreadsheet with a barebones model to play around with right here.