Whether you’ve formally studied probability or not, we all use probabilistic reasoning to make decisions. For example, when you ride in a car you (hopefully) wear a seat belt, in case you get into an accident. This is a decision based on a probability: if you knew was a 0% chance of getting into an accident you wouldn’t bother wearing a seat belt, and if this chance was a 100% chance then you wouldn’t even get into the car. This is just one of the countless number of decisions you have to make to just to get through a regular day, and humans evolved this ability because it was necessary for our survival.
For many low-stakes decisions, going by intuition is good enough. But when you need to make a difficult and high-stakes decision, you need to use more rigorous probabilistic reasoning to make the smartest decision possible. Furthermore, we all like to think that we’re rational people with rational beliefs, who integrate new evidence into our beliefs as it comes along. But most people don’t understand what evidence really is, on a true probabilistic level.
Now, becoming comfortable working with probability equations is not something that everyone can or is willing to do. But I hope to at least give you an understanding of the basics of Bayesian reasoning on an intuitive level, so that you can:
It is now unavoidable that we must go into some math. I imagine this will be some of the driest content for many readers, but it’s really important so I encourage you to power through! Think of it like brushing your teeth, learning this stuff is necessary for mental hygiene.
Here I’m going to explain the basic probability definitions and ideas that are necessary to understand beliefs and decision making. Don’t worry, I won’t be asking you to solve things like difficult deck-of-card puzzles you might see in a probability class. The notation may look complicated at first but don’t be scared away, it’s just a way to simplify statements that would be unwieldy to describe in words each time.
Firstly, we write $P(A)$ to denote the probability that event $A$ occurs. For example, if $A$ is a random coin flip coming up heads, then $P(A) = \frac{1}{2}$. In words, this means that the probability that a random coin flip comes up heads is $\frac{1}{2}$.
We write $P(A\wedge B)$ to denote the probability that $A$ and $B$ occur. Some examples:
Independence
Notice that in the first example above, $P(A\wedge B) = P(A)\times P(B) = \frac{1}{2}\times \frac{1}{2}$, but in the second example this is not the case. This is because in the first example $A$ and $B$ are independent: the outcome of $A$ has no effect on the outcome of $B$, and vice-versa. So we can just multiple the probabilities of each of the two outcomes happening to get the probability they both happen. But, in the second example they are completely dependent: the outcome of $B$ is always the opposite of the outcome of $A$.
The idea of two outcomes being independent is something we all intuitively understand, even if we don’t all have the mathematical language to describe it. If you want to know the weather outside you don’t check the stock market, because you know these two things are almost entirely unrelated. But you might look out a window and see if the ground is wet, because the ground being wet is definitely related to the weather.
The mathematical definition of two outcomes $A$ and $B$ being independent is simple, it just means that
$$ P(A\wedge B) = P(A)\times P(B). $$