Rain in Rome?

From a summer day in Rome to Bayes Theorem

Let’s continue with the rain example.

It’s a beautiful summer day in Rome, 30 degrees, blue sky. Question: What is P(H), the probability that it will rain tomorrow? Answer: Very low – let’s say 1%. Rain is highly unlikely.

But then suppose that in the evening we start seeing some grey clouds on the horizon. The TV weather reporter says there is a slow-moving precipitation band 50 km upwind, aligned with prevailing winds, and warns that, as a result, it is probable that we are going to see some rain tomorrow. That’s confirmed by the cloud icons we see on our phone weather app.

Question: What is P(H|E), the probability that it will rain tomorrow in the light of this new evidence E? Clearly, it is going to be higher that 1%. But how much higher? Let’s see.

Before we do that, I warn you: you will see some equations. Famously, in the preface to A Brief History of Time, Steven Hawking wrote: Someone told me that each equation I included in the book would halve the sales. I therefore resolved not to have any equations at all. But I will not heed his advice. Instead, I will ask you to make an investment. None of what follows is complicated – you just need to sit down with pencil and paper and understand the main concepts. To the mathophobes among you: please don’t be afraid. Think of maths as nothing more than a shorthand language. Instead of saying: the probability of the hypothesis is one in a hundred, we write P(H)=1%. This has three key advantages:

1. It is compact

2. It is universal – the same in all languages

3. It makes it easier to draw logical consequences that would mostly be lost in a natural language discourse.

So here we go. Write to me if you need help.

H: It will rain tomorrow

E: The weather report predicts probable rain