Right. We looked at the quote unquote meaning of the Bayesian network and 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. So it's conditionally independent of its non-descendants given its

parents so x and zlj is actually conditionally independent given its parents. These parents

here. Yes.

How would this scholastic independency look like?

A what?

A normal independence. You just don't have any parents. So in our example, burglary

and earthquake are independent given their parents which is nothing. So this is a full

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

Okay? Good. So 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

kind of 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. And if that's true we

have to have a way of recovering that. And indeed we can do that via the chain rule thing

and exploit conditional independence and then we get the probability over all variables

is actually the probability distribution over all variables is just the product over these

conditional probability distributions given the parents. Okay? And those things here are

exactly what we have in the conditional probability tables. So take everything in your, all the

tables from your Bayesian network, make a big product of all of them which of course

is a matrix product here and then we're done. We don't want to do that in practice of course

because things are going to explode on us because just this here is a D to the N multi-matrix.

And this here is something where we need, if we want to prove this we need to, we need

the A-siclicity here. Okay? And in practice that's what you do. We have a 5-tuple of events

which gives me a 5-tuple of fact, 5 product which I can just read off my network and get

the result. And of course we have to do that 2 to the 5 times because there are 2 to the

5 probabilities to compute. Now 2 to the 5 is fine but we'll see that normal networks

are going to be at least 30 to 50 nodes if you're modelling anything interesting. And

then things become icky and you don't just want to do this. There were a couple of questions

in and after the last lecture and most of them centred around can we express this and

that? And the answer is maybe. Anything you can express with the full joint probability

distributions you can. But your choice of variables is actually a modelling step. What

you want to realise here that this is a model of the world that can do certain things, that

cannot do. If you want a better model of the world, make a better Bayesian network. Okay?

This can only answer the questions in the language you're addressing here. Any questions

about that is expressible in a propositional logic formula taking these 5 variables? That's

something you can deal with. And otherwise it's outside the model. The model doesn't

know.