Good. Next thing. Very simple observation. We have this identity. The probability of

A given B can be computed from the probability of B given A. If we know the priors for A and B.

Just something you can get from the product rule. What do you do? You use the definition

probability of A given B is just the probability of A and B over B and by the product rule the

probability of A and B is just A of B given A times P of A. Very simple. Just a little rewriting.

Turns out that this is extremely powerful. It's extremely powerful. Why? Because we can turn

around the conditional probabilities and very often one of them is either easy to get

remember cavity given toothache and the other one is really interesting. How is my probability

of feeling pain when I have a cavity? That's interesting. Or one of those is really

stable under time and the other is not. This is really what we're going to use a lot as well.

If you look at that, we know that toothache given cavity is 0.6, P of cavity is 0.2,

P of toothache is 0.2, then we can actually compute the other way around which also happens to be 0.6.

What you want to realize here is that this here is really a causal

probability. We know that you have a toothache because there's something wrong in your teeth.

The cavity causes the pain. Whereas for diagnosis, the thing the toothache does,

you actually have, you look at what's the probability of having a cavity given a toothache.

It's kind of the diagnostic direction of reading this. Remember when we started out, we tried out logic.

We tried out to write down the diagnostic rule and the causal rule and they both didn't really work.

Well, this is really what we have here, what the relations between causal and diagnostic is.

The nice thing is that causal probabilities, causal conditional probabilities are really stable.

It doesn't really matter whether you have a huge population or you have a tiny population.

It's always, causal is how things work. How thick your teeth are, how long it takes to wear out the tooth,

how nerves work and so on. The frequency of causes really don't matter much.

If there's a cavity epidemic, Coca-Cola changes their formula, adds a little bit of sulfuric acid or something like this.

Everybody has a toothache. Everybody has cavities. The diagnostic direction really changes a lot.

Dentists are seeing people with sore teeth all the time. This is difficult.

Of course, how your teeth work don't change at all. It just happened not to work well anymore because of the sulfuric acid.

The nice thing is that with Bayes' rule, we can still do diagnosis even if causes change a lot.

Let's make another example out of this. Are there any questions?

Let's make, instead of cavities, another example. You all know about meningitis, which is if you're bitten by a tick,

then that's what you can catch. One of the symptoms that is used for diagnosis is whether you have a stick neck.

Go home and tell your mother, I've been bitten by a tick. The first thing she does is this.

She wants to know whether you have to go to the doctor or not.

We have to know the prior probabilities of meningitis and stiff neck.

They just happen to be meningitis, which is fortunately quite rare.

Having a stiff neck is less rare, which is not really a problem.

You can get it from all kinds of things, not just being bitten by a tick.

What does it mean when your mother does this to you?

The important thing here is that meningitis causes a stick neck, which is why it's a good thing to do diagnostically.

But it's not a sure thing. You don't always have a stick neck if you have meningitis.

So we have this kind of a 70% chance that meningitis causes a stiff neck.

Those are things that doctors know, those are the things that your mother knows.

Actually, typically a doctor knows these as well and your mother basically knows.

It's in the majority of cases. At least that's what my mother knows.

So I'm assuming something here.

But the doctor actually uses Bayes' rule. That's something they actually learn.

They are only given many, many more examples than I'm giving you.

So what you want to know is, given the evidence that you have a stiff neck,

what is the chance that you actually have meningitis?

Given Bayes' rule, you just turn it around into the causal direction, which you know is 0.7,