Last week we learned about conditional independence as a way of modeling probabilistic facts about

the world and move towards the kind of main reasoning tool, namely Bayesian networks that

allow us to actually go the step from math to modeling. So the basic idea was that we build

up these networks, networks where the nodes are random variables and the arrows are dependencies.

More importantly, non-arrows mean conditional independence. This is kind of the prototypical

kind of topology that we're interested in. We have two conditionally independent things given

their parents. So that's the idea and if that's the case then we can start the mill grinding

doing normalization and marginalization. We're actually interested in situations. We have a

certain evidence the agent has seen or perceived something and we want to compute certain probability

distributions given that evidence. So we do the normalization and marginalization trick

to get at these kind of probabilities and then these we compute with a chain rule and here is where

the Bayesian networks kicks in. Basically in these conditional probabilities we can drop

lots of stuff, namely everything except the parents of a node. That's going to be where the

good stuff happens, namely things get less complex. That's where we exploit conditional independence.

Conditional independence. Here we've talked about applications and kind of entered the math of Bayesian

networks and so you should think of Bayesian networks being a couple of things at the same time.

It's an inference procedure, something that allows us to program with probabilities and computing

these probability distributions given some evidence. It's a very well engineered substitute for working

with the full joint probability distribution which is unwieldy and gets too big and it's also a

graphical tool where you can just basically look at the model of the world and see whether it makes sense.

So we extensively looked at this example which is up here. We have a Bayesian networks. You remember

we have some causes. We have an alarm that can go off if there's a burglary that's what it's for

and there's an earthquake that's what it's not for and we kind of have this remote

supervision system of the neighbours which call if they hear the alarm and now the question is

John or Mary call you at work and now the question is do you call the police or not?

We have this very simple Bayesian networks that have the intuition that burglary and earthquake

cause the alarm and the alarm causes the calls and we have certain probabilities involved with this.

In the simplest case on the roots we have the prior probabilities on the single

inflow nodes we have a very simple probability transformation. If the alarm is true then

John calls with 90% and if it's false he still calls with 0.5% and if it's false he still calls

with 0.5% and here we have this kind of conditional probability table that tells me the probabilities

of whatever the four possibilities of what burglary and earthquake is.

It's a very succinct and simple representation of the world and

it tells me we have this conditional independence situation here that we want to exploit.

We looked at the quote unquote meaning of the Bayesian network and the idea

of the Bayesian network is that it's a very simple representation of the world.

The idea is that one of the possible meanings is that it's a representation of conditional

independencies and the kind of catchphrase here is that any node or random variable x

is conditionally independent of its non-descendants. Descendants are this,

the downward cones and the non-descendants are everything else.

It's conditionally independent of its non-descendants given its parents so x and zlj

is actually conditionally independent given its parents.

How would this scholastic independency look like?

A normal independence. You just don't have any parents.

In our example burglary and earthquake are independent given their parents which is nothing.

This is a full independence where we only have a conditional independent because we have parents there.

Another way of expressing the meaning and if this was more an inferential meaning,

what can I do with it? What does that help me? Another way of thinking of the meaning,

a more mathematical way of the meaning is that the Bayesian network is a way of representing

the full joint probability distribution of all these variables.