# Are you sure?

Let’s say you work in a hospital. One random day, I come to your office and ask, “Is it going to rain today?” You look outside and see that it’s not particularly cloudy but it’s not one of those bright sunny days either. You have seen days like this pass without a single drop of rain and you have seen days that start like this and end in a flood. So you say, “I am not sure.”

Since I like asking questions, I don’t stop here and instead, I say, “What’s the probability that it will rain today?” You think for a while. You try to remember the number of days which started like this in your life and the fraction that ended up in rain. But to your disappointment, you don’t have that sharp a memory. You try to estimate this fraction, but very soon realize that you have absolutely no clue what it is. So you say, “I am not sure.”

Anyway, the point is, when I ask you to assign a probability to the event that it rains today and you say you are not sure, I can ask you to assign a probability distribution to the different values of probability with which it might rain today. This means that I can ask you for the probability that the probability that it rains today lies in a small interval around x, where x is a real number between 0 and 1. And if you claim you are still not sure, you know what I can ask next. Let me still state it, just for the kicks. The question I will ask next is this – “What is the probability that the probability that the probability that it rains today lies in a small interval around x, where x is a real number between 0 and 1?”

Some time ago, I was thinking about these things and was trying to figure out what was wrong when I realized something that made it all clear. What I realized was as follows. When I ask you what’s the probability that a certain event will happen, I am not asking you to give me some magic number so that if I perform a certain experiment in similar conditions a hundred times, then the event in question will happen that many number of times. What I am actually asking you is to give me an estimate of your own uncertainty about the event, based only on the information that you currently have. Different people might assign different values to the probability. But it won’t mean that any one of them is wrong. The probability that one assigns to a certain event’s occurrence shows his own uncertainty about it.

This is why Bayes’ Theorem makes sense. It gives you a precise way to update your own uncertainty about something based on the knowledge you receive and the evidence you assimilate.

So, to sum it up, the problem I tried to explain in the post was that you either know the answer to an infinite number of questions or you don’t know the answer to any one of them. The way it is resolved is that you actually know the answer to all the questions, because the reason the questions sounded unanswerable earlier was that you had not understood them correctly.

This is good understanding for the start. I plan to read E.T. Jaynes’ Probability Theory: The Logic of Science some time soon. I hope I will be wiser after that.