A more intuitive formulation of Bayes theorem

There’s a lot to say about Bayes theorem. And most of it’s been said. In my little echo chamber of the internet, just about every YouTuber that I watch or blogger that I read has talked about Bayes theorem. So the critical question is: what do I have to add?

Just this for now. Almost every introduction to Bayes theorem introduces the formula like so:

$$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$

I think the reason that this formulation is so common is that when you want to use Bayes theorem, you’re probably trying to compute $P(A|B)$. The fact that this formula has isolated $P(A|B)$ on one side makes it practical to use.

However, I’d argue that an introduction to a topic should prioritize giving the reader an intuition for why the formula is true. If possible, the reader should come away thinking that it’s so obvious that it’s almost uninteresting. I think the following formulation does a better job at that.

$$P(A|B)P(B)=P(A\mathrm{\&}B)=P(B\mathrm{\&}A)=P(B|A)P(A)$$

Derivation

One way to compute the probability of two events $A$ and $B$ happening is as follows:

$$P(A\mathrm{\&}B)=P(A|B)P(B)$$

To make this concrete, let’s say $A$ is “I go to the park tomorrow” and $B$ is “it rains tomorrow”. The probability that it rains tomorrow and I go to the park is [the probability that it rains] times [the probability that I go to the park given it rains]. Stated that way, it seems kind of obvious.

From there it’s a small leap to notice that $P(A\mathrm{\&}B)=P(B\mathrm{\&}A)$, so we could also write the probability like so:

$$P(A\mathrm{\&}B)=P(B|A)P(A)$$

And the very last step is to combine the two equations:

$$\begin{array}{rl}P(A|B)P(B)& =P(A\mathrm{\&}B)=P(B|A)P(A)\\ P(A|B)P(B)& =P(B|A)P(A)\end{array}$$

Divide each side by $P(B)$ and you will get the standard formulation of Bayes theorem.

I think this route to deriving Bayes theorem makes the connection to $P(A\mathrm{\&}B)$ more clear and, at least in my opinion, the fact that you can rewrite $P(A\mathrm{\&}B)$ in two ways is a nice intuition for why Bayes theorem works.