So, independence is nice but rare.

Conditional independence is what we're essentially going to go for.

Eventually, but before we do that,

we need a couple of reasoning methods.

First of all, product rule.

We've just seen for independent events, A and B.

We've seen we can just multiply P of A and P of B.

This doesn't hold in the general case,

where in the general case,

we can multiply P of A if B and P of B.

If we have independence,

then P of A if B given B is the same as P of A.

So, we get independence rule back.

So, we have a product rule,

very useful in the general case.

That almost directly can be

chained into what we call the chain rule,

which is what we're really going to use most of the time.

Because in the situations we're interested in,

we have more than two outcomes we want to get together.

But then we can compute in this chain product

of conditional probabilities.

Actually here, I've written it down

as the level of distributions,

which actually means it's again a big system of the query.

You want to implement these things,

you better have a good matrix library somewhere.

The thing to note here,

I've written it down,

is that it sometimes

pays to think about the ordering of the variables,

because that makes a difference.

Because if you order them differently,

you have different conditional probabilities,

some of them which might be easy.

So, chain rule was the first thing that allows us to do stuff.

The second one is what we call marginalization,

which is you can extract the sub-distribution.

It's essentially take your n-dimensional big cuboid table,

and then you can basically take

a high-dimensional or low-dimensional subspace of this.

What you do is you can,

if you have a subset of the random variables that you have,

then you can just sum over all the topples of a subset Y,

of the outcomes of the subset Y of random variable.

You have to just sum over all combinations.

Very simple. We've already looked at that.

I can get the probability of having

a toothache by basically summing up over all the possible outcomes.

The probability of cavity by summing up over