Now we start doing inference and the only real problem here is that if we had all probabilities,

if we had the full joint probability distribution, we could in theory do everything we want.

In practice, these objects are just much too big.

So we want to use the fact that empirically these high dimensional tables of probabilities

have identical values in many places.

And they have these identical values in these many, sometimes funny places, for reasons

of how the world works.

One of the most important is independence.

When you have cavities, it's independent of weather.

If you have two dice, their outcomes are independent.

What you do today is probably independent of what Susie Parker does in San Francisco.

You don't even know her, I hope.

It might not be independent of what Trump does.

You hear something about him, you need lots of beer in the evening because you're so

frustrated.

But very often, if we have independent events, that leads to many identical values in the

joint probability distribution and we want to take advantage of these.

And the full joint probability distributions are not the right tools to do this, not the

right tools to express that.

And that's what we're developing now.

And we want to have ways of dealing with these that are shorter than just writing down this

huge table.

And the way we'll do it will be based on networks.

Before we can do that, we have to do a little bit more work.

First of all, we have to find independence, which just means conjunction of events, events

being the case at the same time, really translates into product of probabilities.

This is not the case everywhere.

Not any two just have this nice product probability, which this tells us is really we don't have

to know the joint probability of A and B. We can compute it.

Which actually, if we think from a joint probability, a full distribution point of view, the cell

for this event and the cell can be computed from something else.

So we don't really actually have to write it down.

And of course, if we have independent events, then by the definition of conditional probability,

we can do better for those as well.

So B doesn't, if A and B are independent, B doesn't play a role.

That's really where this idea of independence comes from.