Okay, next thing is we can actually measure the size of a Bayesian network.

I've been hand waving about this.

I've been appealing to your intuition about that this is a bad network and our little

star-shaped network was a good one.

So how would we make this quantitative?

Yes?

We try to find the network with the least degree because the degree is the amount of

arrows arriving at one node and we just want the degree where the number of network were

in the node.

What we're really interested in is that is kind of a derived measure in a way.

What we're really interested in is the work we have to do now, namely assessing the conditional

probability tables.

That's where the real work is.

You can imagine writing those things down.

Here's an ugly one.

Really what we're interested in is how many numbers do we have to put here?

That's essentially how we're going to define the size.

We have n random variables with their domains, then we can define the size of the corresponding

of a Bayesian network on these variables as being essentially the size of the CPTs, which

is I have a big sum.

Every single node has a CPT.

That actually has size of DI many rows.

I have to see what, so that's this size, and I have to see what's incoming, and that's

actually the product over the parent's sizes, of the ranges of the parent's sizes.

The size of the ranges of the parent.

That way it makes sense.

I guess you'll have to stare at it a little bit.

If you have smaller Bayesian networks via this number here, then we have to assess less

probabilities.

That's very good for the modeling phase, and we'll have more efficient inference.

How much better is that than using the full joint probability distribution?

The full joint probability distribution is this big.

You have to look at all the possible values in all the nodes in all the random variables.

That's a big product.

If we have a fully Boolean network like we do in our example, then you basically have

2 to the n here.

If you have other outcomes, then of course that comes in as a product.

We know that we have at most k parents in the network.

Here in this network, k would be 4.

In this network, k would be 2.

In our nice star-shaped network, we would have a maximum of k equals 2 as well.

Then of course, then we know that the size of the network is the sum here.

We have the sum is n times the maximum of all the d i's, where I can estimate to the

big here.

This product is at most k long and this size is also less or equal to d max.

We have something like n times d max to the k plus 1.

If we have a fully Boolean network, n being 20, k being 4, then we have 2 to the 20 possibilities.

The Bayesian network size in this example is 20 times 2 to the 5, which is 640.

There is a slight gain here.

That's a network with just 2, with just a Boolean network.