Bayes’ Theorem Runs the World
If you have any interest in reasoning correctly but have never taken the time to learn Bayes’ theorem, you should delay no longer as Bayes’ theorem is the correct way to update your beliefs based on the evidence gathered. I use this reasoning very frequently and only haven’t attempted to explain it here because it is math and jargon heavy. To get around this issue let’s first try a visual example and later see what the generalities are which make up the most important equation you’ll ever learn.
Say I have two jars each containing 100 marbles. One has a 50/50 split of red and green marbles and the other has 90 red marbles and 10 green marbles. Now suppose we were going to blindly draw marbles out of one of the jars without knowing which jar we are drawing from. If we draw a green marble on the first try my intuition tells me that it is much more likely to be that we are drawing from the even jar than the majority red jar because there are five times as many green marbles in the even jar. In my experience this makes sense to most people but why is this the case? Well this is where the math comes in.
To answer this the question is we must know what is the probability of my hypothesis (H) given my observations (O) which is P(H|O). According to Bayes’ theorem P(H|O) is equal to: P(O|H) P(H) / P(O).
For those weary of math this can look intimidating but P(O|H) is just the probability of seeing the observations (O) given the hypothesis (H), in other words the degree that H predicts O. And P(H) is just the probability of the hypothesis (H) before the observations (O) are taken into account. All of that is then divided by P(O) which is the probability of seeing those observations (O) irrespective of any particular hypothesis. This giant equation really breaks down to how much a hypothesis is inferred by an observation which is, intuitively, proportional to the amount that hypothesis predicts the observation while accounting for how likely you are to see the observation under any circumstance and how likely you already consider the hypothesis to be. This is the technical explanation for why if you draw one green marble it’s likely you’ve drawn from the even jar and if on the second try you draw another green marble you become even more confident that you are not pulling from the majority red jar.
Confused? I hope not, but admittedly this is one of the few areas where I find myself unable to recall what all that jargon looked like to me before I grasped the concepts. For those new to Bayes’ theorem I think seeing the concepts with Venn diagrams and Eliezer Yudkowsky’s (lengthy) intuitive explanation are the most useful but I find the best way to learn this idea is to apply it with visual or familiar examples. In upcoming posts, I’ll be doing both.
