Bayes’ Formula Made Easy

The Bayesian way of thinking about probability has been mentioned in several previous articles on Rational Numbers. Scientists sometimes use Bayes’ formula to interpret the results of their experiments, but the formula doesn’t just apply in the realm of science. Bayes’ formula can also be useful in thinking about many situations encountered in daily life[1]. Even if you can only make educated guesses about the probabilities involved, Bayes’ formula can improve the accuracy and consistency of those guesses.

The formula[2] is a simple equation that allows you to take your current degree of belief in the truth of an idea and combine it with new information to update that degree of belief. People who suffer from math anxiety will also be pleased to learn that in its simplest form, applying Bayes’ formula requires only the math you learned in primary school. It’s about as difficult as the math you do when figuring out what tip to leave at a restaurant.

Bayes’ formula can be written in many different ways that are applicable to different situations, but here we’ll look at one of the simplest ways to use Bayes’ formula by using odds. Odds are the ratio of how probable something is to happen to how probable it is to not happen. If you roll a fair six-sided die the probability of rolling a four is 1/6, and the probability of not rolling a four is 5/6. We then say the odds of rolling a four are 1:5 (read as “one to five” or “five to one against”). Since odds is a ratio it doesn’t matter if you write it as 1/6:5/6, 1:5, 2:10, or 10:50; these are all the same ratio.

Similarly, if we flip a fair coin the probability of getting heads is 1/2 and the probability of not getting heads is also 1/2, so the odds of getting heads is 1:1. You can easily convert back and forth between odds and probability using the following formulas:

odds=\frac{probability}{1-probability}                   probability=\frac{odds}{1+odds}

Imagine you have a friend who’s phone often runs out of battery power. You call him and find there is no answer. This makes you wonder how this new information should affect your degree of belief that his phone’s battery is dead.

First you have to know what you believed before you tried calling him, your prior belief. You guesstimate that on average his phone is charged 70% of the time and is dead the other 30% of the time. You had no other information before you tried calling your friend, so your prior odds were 3:7 that his phone was dead:

  1. Hypothesis: His battery is dead. (3:7 odds)
  2. Alternative: His battery is not dead (7:3 odds)

Now we update our beliefs based on the new observation. To do this we imagine each of these cases were true, and then ask what the probability of our observation would be in that case. In this case our observation is that there was no answer when we tried calling.

First we imagine the hypothesis is true – his battery is dead. If this were the case there’s no way he could answer his phone, so the probability of what you observed would effectively be 100%. Next we imagine the alternative is true – his battery is not dead. You guesstimate that when his phone is charged he will pick up 80% of the time and miss the call 20% of the time. So if the alternative were true, the the probability of getting no answer would be 20%.

To update our belief we just multiply our prior odds by the first probability and divide it by the second probability. This gives our new odds, the posterior odds.

\dpi{100} 3:7\times \frac{100\%}{20\%}=15:7

So after trying to call him and get no answer, the odds that his phone’s battery is dead went from 3:7 (30% probability) to 15:7, (~68% probability).

To summarize, here’s what happened. What we observed was five times more probable if his phone was dead than if it was charged, and so the odds that his phone is dead became five times greater. When you do it this way the math is just that easy.

The new odds, 15:7, is your new degree of belief. It also becomes your prior odds for the next time you encounter new information. For example, imagine now somebody else tells you that your friend’s phone is actually not dead. You guesstimate that this person is reliable about 90% of the time, so when we imagine the case that the phone is dead, hearing this news is 10% probable, and when we imagine the case that the phone is not dead, hearing this news is 90% probable. Now we can update our belief again:

\dpi{100} 15:7\times \frac{10\%}{90\%}=5:21

Now our degree of belief that his phone is dead is 5:21 (~19% probability).

One important thing to keep in mind is that you must consider all the previous information each time you imagine the alternative scenarios and estimate the probability of your observation. For example, if you try calling your friend again, and again get no answer, the odds don’t go up by five times again. If you already tried calling him and he didn’t pick up then, it’s now more likely that he won’t pick up again this time. The probabilities may sometimes change with your previous information.

Using odds, Bayesian reasoning becomes incredibly simple. All we need is an idea about the world – a hypothesis – and a relevant observation. Then we compare how likely that observation would be if the hypothesis were true to how likely the observation would be if the hypothesis were false. The odds that the hypothesis is true gets multiplied by this ratio.