We have a couple of rules that become extremely useful for computing.

We have the product rule, right?

P of A and B is P of A given B times P of B.

So if you want to know whether you have a cavity and a toothache, you have to see what

the probability is of having a toothache given that you have a cavity times the probability

of having a cavity.

So this is interesting in the case where you don't know, where you don't have independence.

In the case of independence, this actually, we know if cavity or toothache were independent,

then P of toothache given cavity would just be P of toothache and then you would just

have toothache times cavity.

So that's a good plausibility check.

And conditional probabilities are very often very much easier to assess.

For P and B, you have to look at a huge set of things.

For P of A given B, you only have to look at the set of all Bs.

I shouldn't have erased this, so I'm going to draw it again.

If I have A and B, for the probability of A and B, you basically have to look at all

of this.

Whereas, compare this and that, whereas if you have P of A given B, you only have to

compare this set with that set.

You have to look at less stuff.

So conditional probabilities are easier to assess.

If you want to have the probability of somebody having a cavity given that they're male, that's

much easier because you only have to look at the male population.

Whereas, for the probability of having a toothache and being male at the same time, you have

to look at males and females together.

Which is why this product rule is so useful.

Of course, there's a lot of symmetry here.

You have this product rule as well.

Again, you can assume this as a default, whenever we can do it for one, we can do it for whole

vectors.

Here we have systems of equations which we can write down as matrices.

They behave in the right way.

Yes?

So we know that they are independent, but how would AI know that?

Depends on whether you have a hard coded agent in a forever unchanging world, then you can

actually determine, then you can compile in the probabilities.

But most world aren't unchanging.

So you would probably have an agent that has to assess the probabilities.

If you think about humans, intelligence agents in an interesting world.

At some point, somebody had to do the statistics and say, well, how many cavities are there?

Then you have to do it again when companies like Coke and McDonald's appear on the scene

and people go from eating beets and roots and meat to Big Macs and drinking Diet Coke

or whatever.

So the probabilities will change.

You have to assess them.

You have to learn them.

You have to find out about your world.

That might actually be a good idea because you know more about the world and can predict

the world much better.

And so the answer is unfortunately, no, you can't hard code typically in a more interesting