A Better Bayes’ Rule
(Let me say upfront that this post has a little bit of math in it. For those of you who are not mathematically inclined, stay with this: it’s is a very useful trick that you can do in 10 seconds and will help you make better decisions. I’ll explain everything and keep it simple.)
Let’s say I’m considering investing in an early-stage startup (Company X) and I want to assess the probability that it will succeed. One the one hand I know that most early startups fail, so investing in them is always risky. On the other hand, this particular company company seems to have a lot going for it, so the evidence is compelling. How should I weigh these two things?
Answer: Bayes Rule. As a refresher, Bayes’ Rule allows you to answer two related questions: (a) what is the probability of ‘x’ being true given some evidence; and (b) if I had a prior belief about the probability of ‘x’ being true, how should I update that believe given new evidence. It should be pretty obvious why/how this could be helpful.
When I first learned about Bayes’ Rule in college, it intuitively struck me as both extremely important and useful. Over the years I’ve revisited it occasionally in an attempt to really drill it into my brain and hopefully get to a point where I would just naturally use it. But it never quite happened as the mental math involved was just a bit complex for me (I am not good at mental math).
Then one day a couple of years ago I came across the odds form of Bayes’ Rule. And it simplified everything. A lot. I won’t go through a detailed explanation on how it works or how it’s derived (if you want that, see here) but let me just show how I practically can use it now and how easy it is.
Play the Odds
First, a quick refresher on odds for those who need it. Let’s say I think there’s a 10% change that my favorite team is going to win the game (and therefore a 90% chance that they wont). The odds are 10:90 or 1:9. In other words, odds are just p(x will happen)/p(x won’t happen). To convert from odds of a:b back to a percentage, just calculate a/(a+b). In this example, it’s 1/(1+9) = 1/10 = 10%.
OK, with that done, let’s move on to Bayes’ Rule. Saying with the startup example, the odds form of Bayes’ Rule says:
Let’s go through the example of trying to determine the probability that the startup will be a success:
Start with the ‘prior odds’ or the ‘base rate’: p(success)/p(failure) (i.e. the rightmost term in the above equation). From previous reading say I know that the probability for a seed stage startup being successful is 10%, so the odds are 10% : 90% = 1:9.
Next, assess the likelihood ratio, p(E|S) : p(E|F) (i.e. the middle term above). Let’s say this startup has a strong team, a compelling idea in a large and growing market, seems to have a unique take on the space, and is moving quickly.
To evaluate the numerator, p(E|S), I ask myself “assume a random startup company ends up being a success. What is the probability that this company had all of the things in place that Company X has (at the same phase in their lifecycles)?” The answer is probably ” almost all”. So let’s say 95%.
For the denominator of the likelihood ratio, it’s the almost the same question: “assume a random startup company ends up being a failure. What is the probability that such company had all of the things in place that Company X has? (again, at the same place in their lifecycles)” Actually, the answer is still probably “most” – even great teams fail regularly etc. – so let’s say it’s 70%.
So now we have the likelihood ratio p(E|S) / p(E|F) = 95% / 70%, or ~ 1.35 : 1
So now we just multiply:
So the odds that the company will be a success are 1.35:9. To convert that to a percentage, we calculate numerator / (numerator + denominator), so we have 1.35/(10.35) = ~13%.
This means that – given my rough assumptions – I should expect that this company has a 13% chance of succeeding.
Why so low? To get a better understanding you can read the full articles, but I think of it this way: the base rate of success is very low. You have some evidence and you have to evaluate how much more likely that evidence is to show up for successful companies than for unsuccessful ones. In this example we’ve estimated that while almost all successful companies will demonstrate that evidence….so will most unsuccessful ones. For that reason, the evidence doesn’t sway us much away from our base rate.
The good thing about this odds form calculation is that I can do it very quick on a napkin, excel, or (sometimes) even in my head. After I tried it once I found it easy to use. And now I use it all the time.